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Balinskis theorem and Regularity of Line Arrangements Bruno - - PowerPoint PPT Presentation

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Balinskis theorem and Regularity of Line Arrangements Bruno Benedetti (University of Miami) CombinaTeXas, May 7, 2016 Bruno Benedetti (University of Miami) Balinskis theorem and Regularity of Line Arrangements Joint work with Bruno

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Balinski’s theorem and Regularity of Line Arrangements

Bruno Benedetti (University of Miami) CombinaTeXas, May 7, 2016

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 2

Joint work with

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 3

Joint work with

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 4

Joint work with

Michela di Marca, Matteo Varbaro (U Genova), 2016

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 5

Joint work with

Michela di Marca, Matteo Varbaro (U Genova), 2016

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 6

Joint work with

Michela di Marca, Matteo Varbaro (U Genova), 2016

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 7

Joint work with

Michela di Marca, Matteo Varbaro (U Genova), 2016 (curve arrangements) Barbara Bolognese (Northeastern), 2015

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 8

Joint work with

Michela di Marca, Matteo Varbaro (U Genova), 2016 (curve arrangements) Barbara Bolognese (Northeastern), 2015

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 9

Joint work with

Michela di Marca, Matteo Varbaro (U Genova), 2016 (curve arrangements) Barbara Bolognese (Northeastern), 2015

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 10

Warming up: Linear Optimization in five minutes

Given a linear function f : Rd → R, and a region P ⊂ Rd, suppose we want to find max{f (x) : x ∈ P}.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 11

Warming up: Linear Optimization in five minutes

Given a linear function f : Rd → R, and a region P ⊂ Rd, suppose we want to find max{f (x) : x ∈ P}. If P is a polytope, i.e. the convex hull of finitely many points in Rd, two dreams come true:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 12

Warming up: Linear Optimization in five minutes

Given a linear function f : Rd → R, and a region P ⊂ Rd, suppose we want to find max{f (x) : x ∈ P}. If P is a polytope, i.e. the convex hull of finitely many points in Rd, two dreams come true:

1 max{ f (x) : x ∈ P } = max{ f (v) : v vertex of P }; Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 13

Warming up: Linear Optimization in five minutes

Given a linear function f : Rd → R, and a region P ⊂ Rd, suppose we want to find max{f (x) : x ∈ P}. If P is a polytope, i.e. the convex hull of finitely many points in Rd, two dreams come true:

1 max{ f (x) : x ∈ P } = max{ f (v) : v vertex of P }; 2 because of convexity, every local maximum is also a global

maximum.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 14

Warming up: Linear Optimization in five minutes

Given a linear function f : Rd → R, and a region P ⊂ Rd, suppose we want to find max{f (x) : x ∈ P}. If P is a polytope, i.e. the convex hull of finitely many points in Rd, two dreams come true:

1 max{ f (x) : x ∈ P } = max{ f (v) : v vertex of P }; 2 because of convexity, every local maximum is also a global

maximum.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 15

Warming up: Linear Optimization in five minutes

Given a linear function f : Rd → R, and a region P ⊂ Rd, suppose we want to find max{f (x) : x ∈ P}. If P is a polytope, i.e. the convex hull of finitely many points in Rd, two dreams come true:

1 max{ f (x) : x ∈ P } = max{ f (v) : v vertex of P }; 2 because of convexity, every local maximum is also a global

maximum. (Naif) SIMPLEX METHOD: Start at a (random) vertex;

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 16

Warming up: Linear Optimization in five minutes

Given a linear function f : Rd → R, and a region P ⊂ Rd, suppose we want to find max{f (x) : x ∈ P}. If P is a polytope, i.e. the convex hull of finitely many points in Rd, two dreams come true:

1 max{ f (x) : x ∈ P } = max{ f (v) : v vertex of P }; 2 because of convexity, every local maximum is also a global

maximum. (Naif) SIMPLEX METHOD: Start at a (random) vertex; move to an adjacent vertex that is higher (under f );

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 17

Warming up: Linear Optimization in five minutes

Given a linear function f : Rd → R, and a region P ⊂ Rd, suppose we want to find max{f (x) : x ∈ P}. If P is a polytope, i.e. the convex hull of finitely many points in Rd, two dreams come true:

1 max{ f (x) : x ∈ P } = max{ f (v) : v vertex of P }; 2 because of convexity, every local maximum is also a global

maximum. (Naif) SIMPLEX METHOD: Start at a (random) vertex; move to an adjacent vertex that is higher (under f ); keep climbing and you’ll reach the top!

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 18

Warming up: Linear Optimization in five minutes

Given a linear function f : Rd → R, and a region P ⊂ Rd, suppose we want to find max{f (x) : x ∈ P}. If P is a polytope, i.e. the convex hull of finitely many points in Rd, two dreams come true:

1 max{ f (x) : x ∈ P } = max{ f (v) : v vertex of P }; 2 because of convexity, every local maximum is also a global

maximum. (Naif) SIMPLEX METHOD: Start at a (random) vertex; move to an adjacent vertex that is higher (under f ); keep climbing and you’ll reach the top!

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Balinski’s theorem

From Neil’s talk this morning: A graph is d-connected if it has at least d + 1 vertices, and the deletion of d − 1 or less vertices, however chosen, leaves it connected.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Balinski’s theorem

From Neil’s talk this morning: A graph is d-connected if it has at least d + 1 vertices, and the deletion of d − 1 or less vertices, however chosen, leaves it connected. (Or Menger’s theorem.) Balinski theorem. The graph (or equivalently, the dual graph) of every d-polytope is d-connected.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 21

Balinski’s theorem

From Neil’s talk this morning: A graph is d-connected if it has at least d + 1 vertices, and the deletion of d − 1 or less vertices, however chosen, leaves it connected. (Or Menger’s theorem.) Balinski theorem. The graph (or equivalently, the dual graph) of every d-polytope is d-connected. Proof idea. Choose the d − 1 vertices that have to go (green), and a “designated survivor” vertex x (red).

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 22

Balinski’s theorem

From Neil’s talk this morning: A graph is d-connected if it has at least d + 1 vertices, and the deletion of d − 1 or less vertices, however chosen, leaves it connected. (Or Menger’s theorem.) Balinski theorem. The graph (or equivalently, the dual graph) of every d-polytope is d-connected. Proof idea. Choose the d − 1 vertices that have to go (green), and a “designated survivor” vertex x (red). The hyperplane spanned by these d vertices chops the polytope into two polytopes, both containing x.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 23

Balinski’s theorem

From Neil’s talk this morning: A graph is d-connected if it has at least d + 1 vertices, and the deletion of d − 1 or less vertices, however chosen, leaves it connected. (Or Menger’s theorem.) Balinski theorem. The graph (or equivalently, the dual graph) of every d-polytope is d-connected. Proof idea. Choose the d − 1 vertices that have to go (green), and a “designated survivor” vertex x (red). The hyperplane spanned by these d vertices chops the polytope into two polytopes, both containing x. Apply the simplex method to both polytopes...

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 24

Balinski’s theorem

From Neil’s talk this morning: A graph is d-connected if it has at least d + 1 vertices, and the deletion of d − 1 or less vertices, however chosen, leaves it connected. (Or Menger’s theorem.) Balinski theorem. The graph (or equivalently, the dual graph) of every d-polytope is d-connected. Proof idea. Choose the d − 1 vertices that have to go (green), and a “designated survivor” vertex x (red). The hyperplane spanned by these d vertices chops the polytope into two polytopes, both containing x. Apply the simplex method to both polytopes...

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Plan for today

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Plan for today

Part I. Many Classes of Dual Graphs.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 27

Plan for today

Part I. Many Classes of Dual Graphs. Part II. Some Algebraic Machinery.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Plan for today

Part I. Many Classes of Dual Graphs. Part II. Some Algebraic Machinery. Part III. (time permitting) Arrangements of Curves.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Many Classes of Dual graphs

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 30

Many Classes of Dual graphs

Of (pure) simplicial complexes (e.g. polytope boundaries):

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 31

Many Classes of Dual graphs

Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 32

Many Classes of Dual graphs

Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves: (There’s also a “dual multigraph” model, keeping track on how many intersections, with multiple edges/loops.)

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Many Classes of Dual graphs

Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves: (There’s also a “dual multigraph” model, keeping track on how many intersections, with multiple edges/loops.) Of (equidimensional) subspace arrangements or algebraic varieties:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 34

Many Classes of Dual graphs

Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves: (There’s also a “dual multigraph” model, keeping track on how many intersections, with multiple edges/loops.) Of (equidimensional) subspace arrangements or algebraic varieties:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 35

Many Classes of Dual graphs

Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves: (There’s also a “dual multigraph” model, keeping track on how many intersections, with multiple edges/loops.) Of (equidimensional) subspace arrangements or algebraic varieties: Vertices correspond to the irreducible components C1, . . . , Cs.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 36

Many Classes of Dual graphs

Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves: (There’s also a “dual multigraph” model, keeping track on how many intersections, with multiple edges/loops.) Of (equidimensional) subspace arrangements or algebraic varieties: Vertices correspond to the irreducible components C1, . . . , Cs. (Equidimensional means, they all have same dimension.)

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 37

Many Classes of Dual graphs

Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves: (There’s also a “dual multigraph” model, keeping track on how many intersections, with multiple edges/loops.) Of (equidimensional) subspace arrangements or algebraic varieties: Vertices correspond to the irreducible components C1, . . . , Cs. (Equidimensional means, they all have same dimension.) We put an edge between two distinct vertices, if and only if the corresponding components intersect in dimension one less.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 38

Dual graphs of curves = dual graphs of varieties

By intersecting a d-dimensional object in Pn with a generic hyperplane, we get an object in Pn−1 with dimension d − 1, and same dual graph!

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 39

Dual graphs of curves = dual graphs of varieties

By intersecting a d-dimensional object in Pn with a generic hyperplane, we get an object in Pn−1 with dimension d − 1, and same dual graph! This way you can always reduce yourself to an (algebraic) curve arrangement with same dual graph.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 40

Dual graphs of curves = dual graphs of varieties

By intersecting a d-dimensional object in Pn with a generic hyperplane, we get an object in Pn−1 with dimension d − 1, and same dual graph! This way you can always reduce yourself to an (algebraic) curve arrangement with same dual graph.

picture from mathwarehouse.com Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 41

Dual graphs of curves = dual graphs of varieties

By intersecting a d-dimensional object in Pn with a generic hyperplane, we get an object in Pn−1 with dimension d − 1, and same dual graph! This way you can always reduce yourself to an (algebraic) curve arrangement with same dual graph.

picture from mathwarehouse.com

NOTE: If you started with an arrangement of hyperplanes (or of linear subspaces), you end up with an arrangement of lines.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 42

Dual graphs of curves = dual graphs of varieties

By intersecting a d-dimensional object in Pn with a generic hyperplane, we get an object in Pn−1 with dimension d − 1, and same dual graph! This way you can always reduce yourself to an (algebraic) curve arrangement with same dual graph.

picture from mathwarehouse.com

NOTE: If you started with an arrangement of hyperplanes (or of linear subspaces), you end up with an arrangement of lines.

   dual graphs

  • f subspace

arr’ts    = dual graphs

  • f lines
  • Bruno Benedetti (University of Miami)

Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 43

Dual graphs of curves = dual graphs of varieties

By intersecting a d-dimensional object in Pn with a generic hyperplane, we get an object in Pn−1 with dimension d − 1, and same dual graph! This way you can always reduce yourself to an (algebraic) curve arrangement with same dual graph.

picture from mathwarehouse.com

NOTE: If you started with an arrangement of hyperplanes (or of linear subspaces), you end up with an arrangement of lines.

   dual graphs

  • f subspace

arr’ts    = dual graphs

  • f lines
  • ⊂

dual graphs

  • f curves
  • Bruno Benedetti (University of Miami)

Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 44

Dual graphs of curves = dual graphs of varieties

By intersecting a d-dimensional object in Pn with a generic hyperplane, we get an object in Pn−1 with dimension d − 1, and same dual graph! This way you can always reduce yourself to an (algebraic) curve arrangement with same dual graph.

picture from mathwarehouse.com

NOTE: If you started with an arrangement of hyperplanes (or of linear subspaces), you end up with an arrangement of lines.

   dual graphs

  • f subspace

arr’ts    = dual graphs

  • f lines
  • ⊂

dual graphs

  • f curves
  • =

dual graphs

  • f varieties
  • .

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 45

Not all graphs are dual to a line arrangement

Attention!, graphs like G0 = {12, 34} ∪ {15, 25, 35, 45} ∪ {16, 26, 36, 46} ∪ {17, 27, 37, 47} are not dual to any Euclidean line arrangement!

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 46

Not all graphs are dual to a line arrangement

Attention!, graphs like G0 = {12, 34} ∪ {15, 25, 35, 45} ∪ {16, 26, 36, 46} ∪ {17, 27, 37, 47} are not dual to any Euclidean line arrangement! Try drawing it. Let P = r1 ∩ r2 and let Q = r3 ∩ r4.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 47

Not all graphs are dual to a line arrangement

Attention!, graphs like G0 = {12, 34} ∪ {15, 25, 35, 45} ∪ {16, 26, 36, 46} ∪ {17, 27, 37, 47} are not dual to any Euclidean line arrangement! Try drawing it. Let P = r1 ∩ r2 and let Q = r3 ∩ r4. Let p be the plane containing r1 ∪ r2, and let q be the plane containing r3 ∪ r4.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 48

Not all graphs are dual to a line arrangement

Attention!, graphs like G0 = {12, 34} ∪ {15, 25, 35, 45} ∪ {16, 26, 36, 46} ∪ {17, 27, 37, 47} are not dual to any Euclidean line arrangement! Try drawing it. Let P = r1 ∩ r2 and let Q = r3 ∩ r4. Let p be the plane containing r1 ∪ r2, and let q be the plane containing r3 ∪ r4. How can a line meet all four r1, r2, r3, r4?

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 49

Not all graphs are dual to a line arrangement

Attention!, graphs like G0 = {12, 34} ∪ {15, 25, 35, 45} ∪ {16, 26, 36, 46} ∪ {17, 27, 37, 47} are not dual to any Euclidean line arrangement! Try drawing it. Let P = r1 ∩ r2 and let Q = r3 ∩ r4. Let p be the plane containing r1 ∪ r2, and let q be the plane containing r3 ∪ r4. How can a line meet all four r1, r2, r3, r4? There are only two chances (possibly coinciding): either it’s the line through P and Q,

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-50
SLIDE 50

Not all graphs are dual to a line arrangement

Attention!, graphs like G0 = {12, 34} ∪ {15, 25, 35, 45} ∪ {16, 26, 36, 46} ∪ {17, 27, 37, 47} are not dual to any Euclidean line arrangement! Try drawing it. Let P = r1 ∩ r2 and let Q = r3 ∩ r4. Let p be the plane containing r1 ∪ r2, and let q be the plane containing r3 ∪ r4. How can a line meet all four r1, r2, r3, r4? There are only two chances (possibly coinciding): either it’s the line through P and Q, or it’s the line of intersection of the planes p ∩ q

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-51
SLIDE 51

Not all graphs are dual to a line arrangement

Attention!, graphs like G0 = {12, 34} ∪ {15, 25, 35, 45} ∪ {16, 26, 36, 46} ∪ {17, 27, 37, 47} are not dual to any Euclidean line arrangement! Try drawing it. Let P = r1 ∩ r2 and let Q = r3 ∩ r4. Let p be the plane containing r1 ∪ r2, and let q be the plane containing r3 ∪ r4. How can a line meet all four r1, r2, r3, r4? There are only two chances (possibly coinciding): either it’s the line through P and Q, or it’s the line of intersection of the planes p ∩ q So two options!, not three.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-52
SLIDE 52

Not all graphs are dual to a line arrangement

Attention!, graphs like G0 = {12, 34} ∪ {15, 25, 35, 45} ∪ {16, 26, 36, 46} ∪ {17, 27, 37, 47} are not dual to any Euclidean line arrangement! Try drawing it. Let P = r1 ∩ r2 and let Q = r3 ∩ r4. Let p be the plane containing r1 ∪ r2, and let q be the plane containing r3 ∪ r4. How can a line meet all four r1, r2, r3, r4? There are only two chances (possibly coinciding): either it’s the line through P and Q, or it’s the line of intersection of the planes p ∩ q So two options!, not three. So some of the three lines r5, r6, r7 have to coincide.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-53
SLIDE 53

Not all graphs are dual to a line arrangement

Attention!, graphs like G0 = {12, 34} ∪ {15, 25, 35, 45} ∪ {16, 26, 36, 46} ∪ {17, 27, 37, 47} are not dual to any Euclidean line arrangement! Try drawing it. Let P = r1 ∩ r2 and let Q = r3 ∩ r4. Let p be the plane containing r1 ∪ r2, and let q be the plane containing r3 ∪ r4. How can a line meet all four r1, r2, r3, r4? There are only two chances (possibly coinciding): either it’s the line through P and Q, or it’s the line of intersection of the planes p ∩ q So two options!, not three. So some of the three lines r5, r6, r7 have to coincide. a contradiction

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 54

Dual graphs of curves = all graphs

Kollar 2012: every graph is dual to some arrangement of curves.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 55

Dual graphs of curves = all graphs

Kollar 2012: every graph is dual to some arrangement of curves.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-56
SLIDE 56

Dual graphs of curves = all graphs

Kollar 2012: every graph is dual to some arrangement of curves. IDEA: Start realizing Kn with n random lines in P2...

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-57
SLIDE 57

Dual graphs of curves = all graphs

Kollar 2012: every graph is dual to some arrangement of curves. IDEA: Start realizing Kn with n random lines in P2...

Kyle Jenkins, Urban Geometry #296, acrilic on canvas, 2010 Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-58
SLIDE 58

Dual graphs of curves = all graphs

Kollar 2012: every graph is dual to some arrangement of curves. IDEA: Start realizing Kn with n random lines in P2...

Kyle Jenkins, Urban Geometry #296, acrilic on canvas, 2010 Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-59
SLIDE 59

Dual graphs of curves = all graphs

Kollar 2012: every graph is dual to some arrangement of curves. IDEA: Start realizing Kn with n random lines in P2...

Kyle Jenkins, Urban Geometry #296, acrilic on canvas, 2010

...and then blowup “unwanted intersection points”.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-60
SLIDE 60

Dual graphs of curves = all graphs

Kollar 2012: every graph is dual to some arrangement of curves. IDEA: Start realizing Kn with n random lines in P2...

Kyle Jenkins, Urban Geometry #296, acrilic on canvas, 2010

...and then blowup “unwanted intersection points”. So,

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-61
SLIDE 61

Dual graphs of curves = all graphs

Kollar 2012: every graph is dual to some arrangement of curves. IDEA: Start realizing Kn with n random lines in P2...

Kyle Jenkins, Urban Geometry #296, acrilic on canvas, 2010

...and then blowup “unwanted intersection points”. So,

dual graphs

  • f lines
  • dual graphs
  • f curves
  • =

dual graphs

  • f varieties
  • = all graphs.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-62
SLIDE 62

Dual graphs of curves = all graphs

Kollar 2012: every graph is dual to some arrangement of curves. IDEA: Start realizing Kn with n random lines in P2...

Kyle Jenkins, Urban Geometry #296, acrilic on canvas, 2010

...and then blowup “unwanted intersection points”. So,

dual graphs

  • f lines
  • dual graphs
  • f curves
  • =

dual graphs

  • f varieties
  • = all graphs.

It remains to see how dual graphs of simpl. complexes fit the hierarchy.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-63
SLIDE 63

Simplicial Complexes, Seen as Varieties (Stanley-Reisner)

Definition by example:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-64
SLIDE 64

Simplicial Complexes, Seen as Varieties (Stanley-Reisner)

Definition by example: Consider the simplicial complex ∆ below.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-65
SLIDE 65

Simplicial Complexes, Seen as Varieties (Stanley-Reisner)

Definition by example: Consider the simplicial complex ∆ below. I∆ := (x4, x5, x6) ∩ (x1, x5, x6) ∩ (x1, x2, x6) ∩ (x1, x2, x3).

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-66
SLIDE 66

Simplicial Complexes, Seen as Varieties (Stanley-Reisner)

Definition by example: Consider the simplicial complex ∆ below. I∆ := (x4, x5, x6) ∩ (x1, x5, x6) ∩ (x1, x2, x6) ∩ (x1, x2, x3). (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet).

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-67
SLIDE 67

Simplicial Complexes, Seen as Varieties (Stanley-Reisner)

Definition by example: Consider the simplicial complex ∆ below. I∆ := (x4, x5, x6) ∩ (x1, x5, x6) ∩ (x1, x2, x6) ∩ (x1, x2, x3). (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). V (I∆) = x4 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x3 = 0

  • Bruno Benedetti (University of Miami)

Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 68

Simplicial Complexes, Seen as Varieties (Stanley-Reisner)

Definition by example: Consider the simplicial complex ∆ below. I∆ := (x4, x5, x6) ∩ (x1, x5, x6) ∩ (x1, x2, x6) ∩ (x1, x2, x3). (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). V (I∆) = x4 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x3 = 0

  • Dual graph of V (I∆)?

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 69

Simplicial Complexes, Seen as Varieties (Stanley-Reisner)

Definition by example: Consider the simplicial complex ∆ below. I∆ := (x4, x5, x6) ∩ (x1, x5, x6) ∩ (x1, x2, x6) ∩ (x1, x2, x3). (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). V (I∆) = x4 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x3 = 0

  • Dual graph of V (I∆)? The intersection of the first 2 components is

{x : x4 = x5 = x6 = x1 = 0},

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 70

Simplicial Complexes, Seen as Varieties (Stanley-Reisner)

Definition by example: Consider the simplicial complex ∆ below. I∆ := (x4, x5, x6) ∩ (x1, x5, x6) ∩ (x1, x2, x6) ∩ (x1, x2, x3). (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). V (I∆) = x4 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x3 = 0

  • Dual graph of V (I∆)? The intersection of the first 2 components is

{x : x4 = x5 = x6 = x1 = 0}, which is 2-dimensional

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 71

Simplicial Complexes, Seen as Varieties (Stanley-Reisner)

Definition by example: Consider the simplicial complex ∆ below. I∆ := (x4, x5, x6) ∩ (x1, x5, x6) ∩ (x1, x2, x6) ∩ (x1, x2, x3). (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). V (I∆) = x4 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x3 = 0

  • Dual graph of V (I∆)? The intersection of the first 2 components is

{x : x4 = x5 = x6 = x1 = 0}, which is 2-dimensional ⇒ edge!

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 72

Simplicial Complexes, Seen as Varieties (Stanley-Reisner)

Definition by example: Consider the simplicial complex ∆ below. I∆ := (x4, x5, x6) ∩ (x1, x5, x6) ∩ (x1, x2, x6) ∩ (x1, x2, x3). (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). V (I∆) = x4 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x3 = 0

  • Dual graph of V (I∆)? The intersection of the first 2 components is

{x : x4 = x5 = x6 = x1 = 0}, which is 2-dimensional ⇒ edge! The intersection of the first and third component is {x : x4 = x5 = x6 = x1 = x2 = 0},

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 73

Simplicial Complexes, Seen as Varieties (Stanley-Reisner)

Definition by example: Consider the simplicial complex ∆ below. I∆ := (x4, x5, x6) ∩ (x1, x5, x6) ∩ (x1, x2, x6) ∩ (x1, x2, x3). (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). V (I∆) = x4 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x3 = 0

  • Dual graph of V (I∆)? The intersection of the first 2 components is

{x : x4 = x5 = x6 = x1 = 0}, which is 2-dimensional ⇒ edge! The intersection of the first and third component is {x : x4 = x5 = x6 = x1 = x2 = 0}, which is 1-dim.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 74

Simplicial Complexes, Seen as Varieties (Stanley-Reisner)

Definition by example: Consider the simplicial complex ∆ below. I∆ := (x4, x5, x6) ∩ (x1, x5, x6) ∩ (x1, x2, x6) ∩ (x1, x2, x3). (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). V (I∆) = x4 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x3 = 0

  • Dual graph of V (I∆)? The intersection of the first 2 components is

{x : x4 = x5 = x6 = x1 = 0}, which is 2-dimensional ⇒ edge! The intersection of the first and third component is {x : x4 = x5 = x6 = x1 = x2 = 0}, which is 1-dim. ⇒ no edge!

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 75

Simplicial Complexes, Seen as Varieties (Stanley-Reisner)

Definition by example: Consider the simplicial complex ∆ below. I∆ := (x4, x5, x6) ∩ (x1, x5, x6) ∩ (x1, x2, x6) ∩ (x1, x2, x3). (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). V (I∆) = x4 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x5 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x6 = 0

  • ∪

x1 = 0

x2 = 0 x3 = 0

  • Dual graph of V (I∆)? The intersection of the first 2 components is

{x : x4 = x5 = x6 = x1 = 0}, which is 2-dimensional ⇒ edge! The intersection of the first and third component is {x : x4 = x5 = x6 = x1 = x2 = 0}, which is 1-dim. ⇒ no edge! ... So dual graph of V (I∆) is same of ∆.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 76

{dual graphs of complexes} ⊂ {dual graphs of lines}

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 77

{dual graphs of complexes} ⊂ {dual graphs of lines}

Stanley-Reisner: simplicial complexes on n vertices are in bijection with radical monomial ideals in C[x1, . . . , xn].

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 78

{dual graphs of complexes} ⊂ {dual graphs of lines}

Stanley-Reisner: simplicial complexes on n vertices are in bijection with radical monomial ideals in C[x1, . . . , xn]. Zariski: radical ideals I in C[x1, . . . , xn] are in bijection with algebraic objects V (I) in An.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 79

{dual graphs of complexes} ⊂ {dual graphs of lines}

Stanley-Reisner: simplicial complexes on n vertices are in bijection with radical monomial ideals in C[x1, . . . , xn]. Zariski: radical ideals I in C[x1, . . . , xn] are in bijection with algebraic objects V (I) in An. Composing the two, from any complex ∆ we get an algebraic

  • bject V (I∆) ⊂ An.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 80

{dual graphs of complexes} ⊂ {dual graphs of lines}

Stanley-Reisner: simplicial complexes on n vertices are in bijection with radical monomial ideals in C[x1, . . . , xn]. Zariski: radical ideals I in C[x1, . . . , xn] are in bijection with algebraic objects V (I) in An. Composing the two, from any complex ∆ we get an algebraic

  • bject V (I∆) ⊂ An. A special variety (a coordinate subspace

arrangement):

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 81

{dual graphs of complexes} ⊂ {dual graphs of lines}

Stanley-Reisner: simplicial complexes on n vertices are in bijection with radical monomial ideals in C[x1, . . . , xn]. Zariski: radical ideals I in C[x1, . . . , xn] are in bijection with algebraic objects V (I) in An. Composing the two, from any complex ∆ we get an algebraic

  • bject V (I∆) ⊂ An. A special variety (a coordinate subspace

arrangement): So when we do generic hyperplane sections, we get an arrangement of lines.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 82

{dual graphs of complexes} ⊂ {dual graphs of lines}

Stanley-Reisner: simplicial complexes on n vertices are in bijection with radical monomial ideals in C[x1, . . . , xn]. Zariski: radical ideals I in C[x1, . . . , xn] are in bijection with algebraic objects V (I) in An. Composing the two, from any complex ∆ we get an algebraic

  • bject V (I∆) ⊂ An. A special variety (a coordinate subspace

arrangement): So when we do generic hyperplane sections, we get an arrangement of lines. FACT For any simplicial complex ∆, the dual graphs of ∆ and of V (I∆) are the same.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 83

{dual graphs of complexes} ⊂ {dual graphs of lines}

Stanley-Reisner: simplicial complexes on n vertices are in bijection with radical monomial ideals in C[x1, . . . , xn]. Zariski: radical ideals I in C[x1, . . . , xn] are in bijection with algebraic objects V (I) in An. Composing the two, from any complex ∆ we get an algebraic

  • bject V (I∆) ⊂ An. A special variety (a coordinate subspace

arrangement): So when we do generic hyperplane sections, we get an arrangement of lines. FACT For any simplicial complex ∆, the dual graphs of ∆ and of V (I∆) are the same.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 84

{dual graphs of complexes} ⊂ {dual graphs of lines}

Stanley-Reisner: simplicial complexes on n vertices are in bijection with radical monomial ideals in C[x1, . . . , xn]. Zariski: radical ideals I in C[x1, . . . , xn] are in bijection with algebraic objects V (I) in An. Composing the two, from any complex ∆ we get an algebraic

  • bject V (I∆) ⊂ An. A special variety (a coordinate subspace

arrangement): So when we do generic hyperplane sections, we get an arrangement of lines. FACT For any simplicial complex ∆, the dual graphs of ∆ and of V (I∆) are the same.

This implies

  • dual graphs of

simplicial complexes

  • ⊂

dual graphs

  • f lines
  • .

(Graphs like {12, 13, 15, 23, 24, 34, 45} show the containment is strict.)

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 85

Conclusions of Part I.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 86

Conclusions of Part I.

The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 87

Conclusions of Part I.

The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.)

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-88
SLIDE 88

Conclusions of Part I.

The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 89

Conclusions of Part I.

The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 90

Conclusions of Part I.

The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world: Example

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 91

Conclusions of Part I.

The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world: Example (from 3 slides forward - ignore obscure words for now) For any (d − 1)-sphere ∆, the variety V (I∆) is an arithmetically Gorenstein subspace arrangement of Castelnuovo–Mumford regularity d + 1.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 92

Conclusions of Part I.

The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world: Example (from 3 slides forward - ignore obscure words for now) For any (d − 1)-sphere ∆, the variety V (I∆) is an arithmetically Gorenstein subspace arrangement of Castelnuovo–Mumford regularity d + 1.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 93

Conclusions of Part I.

The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world: Example (from 3 slides forward - ignore obscure words for now) For any (d − 1)-sphere ∆, the variety V (I∆) is an arithmetically Gorenstein subspace arrangement of Castelnuovo–Mumford regularity d + 1. Maybe elementary facts like “the dual graph of any (d − 1)-sphere ∆ is d-connected” (Klee-Balinski) can be proven with algebraic methods?

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 94

Conclusions of Part I.

The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world: Example (from 3 slides forward - ignore obscure words for now) For any (d − 1)-sphere ∆, the variety V (I∆) is an arithmetically Gorenstein subspace arrangement of Castelnuovo–Mumford regularity d + 1. Maybe elementary facts like “the dual graph of any (d − 1)-sphere ∆ is d-connected” (Klee-Balinski) can be proven with algebraic methods?

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 95

Part II. The Algebraic Machinery (sketch).

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 96

Complete intersections

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 97

Complete intersections

Linear algebra: every k-dimensional subspace X of Pn can be described with exactly n − k linear equations.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 98

Complete intersections

Linear algebra: every k-dimensional subspace X of Pn can be described with exactly n − k linear equations. Non-Linear algebra: The best we can say about a variety X ⊂ Pn, is that we need at least n − k (polynomial) equations.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 99

Complete intersections

Linear algebra: every k-dimensional subspace X of Pn can be described with exactly n − k linear equations. Non-Linear algebra: The best we can say about a variety X ⊂ Pn, is that we need at least n − k (polynomial) equations.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 100

Complete intersections

Linear algebra: every k-dimensional subspace X of Pn can be described with exactly n − k linear equations. Non-Linear algebra: The best we can say about a variety X ⊂ Pn, is that we need at least n − k (polynomial) equations. Complete intersections are the varieties for which “=” holds.

The ”twisted cubic” (s3, s2t, st2, t3) of P3 is not a complete intersection: one needs at least three (hyper)surfaces to cut it out. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 101

Liaison theory and Gorenstein-ness

C.I. long studied.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 102

Liaison theory and Gorenstein-ness

C.I. long studied. If the union of two varieties A and B is a complete intersection, then there is some graded isomorphism in local cohomology between A and B H1

m(S/IA) ∼

= H1

m(S/IB)∨(2 − r).

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 103

Liaison theory and Gorenstein-ness

C.I. long studied. If the union of two varieties A and B is a complete intersection, then there is some graded isomorphism in local cohomology between A and B H1

m(S/IA) ∼

= H1

m(S/IB)∨(2 − r).

(Somewhat similar to Alexander duality in topology, when the union of two spaces is a sphere.)

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 104

Liaison theory and Gorenstein-ness

C.I. long studied. If the union of two varieties A and B is a complete intersection, then there is some graded isomorphism in local cohomology between A and B H1

m(S/IA) ∼

= H1

m(S/IB)∨(2 − r).

(Somewhat similar to Alexander duality in topology, when the union of two spaces is a sphere.) These studies go under the name liaison theory. Note: liaison theory (and the isomorphism above!) works also under a weaker assumption than “complete intersection”, called “Gorenstein”.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 105

Liaison theory and Gorenstein-ness

C.I. long studied. If the union of two varieties A and B is a complete intersection, then there is some graded isomorphism in local cohomology between A and B H1

m(S/IA) ∼

= H1

m(S/IB)∨(2 − r).

(Somewhat similar to Alexander duality in topology, when the union of two spaces is a sphere.) These studies go under the name liaison theory. Note: liaison theory (and the isomorphism above!) works also under a weaker assumption than “complete intersection”, called “Gorenstein”. Among Stanley-Reisner varieties, this Gorenstein property has been nicely explained by Stanley: “S/I∆ Gorenstein iff ∆ is the join of a homology sphere with a simplex”.

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SLIDE 106

Regularity for Algebraists

Given a projective scheme X in Pn, ∃! saturated homogeneous ideal IX ⊂ S := K[x0, . . . , xn] s. t. X = Proj(S/IX); one says X is aCM (resp. aG) if S/IX is Cohen–Macaulay (resp. Gorenstein).

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 107

Regularity for Algebraists

Given a projective scheme X in Pn, ∃! saturated homogeneous ideal IX ⊂ S := K[x0, . . . , xn] s. t. X = Proj(S/IX); one says X is aCM (resp. aG) if S/IX is Cohen–Macaulay (resp. Gorenstein). One sets reg X := reg IX, which is in turn defined as follows:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 108

Regularity for Algebraists

Given a projective scheme X in Pn, ∃! saturated homogeneous ideal IX ⊂ S := K[x0, . . . , xn] s. t. X = Proj(S/IX); one says X is aCM (resp. aG) if S/IX is Cohen–Macaulay (resp. Gorenstein). One sets reg X := reg IX, which is in turn defined as follows:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 109

Regularity for Algebraists

Given a projective scheme X in Pn, ∃! saturated homogeneous ideal IX ⊂ S := K[x0, . . . , xn] s. t. X = Proj(S/IX); one says X is aCM (resp. aG) if S/IX is Cohen–Macaulay (resp. Gorenstein). One sets reg X := reg IX, which is in turn defined as follows:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 110

Regularity for Algebraists

Given a projective scheme X in Pn, ∃! saturated homogeneous ideal IX ⊂ S := K[x0, . . . , xn] s. t. X = Proj(S/IX); one says X is aCM (resp. aG) if S/IX is Cohen–Macaulay (resp. Gorenstein). One sets reg X := reg IX, which is in turn defined as follows: Recall: regularity of an ideal Given a minimal graded free resolution · · · → Fj → · · · → F0 → I → 0, the Castelnuovo–Mumford regularity of I is the smallest r such that for each j, all minimal generators of Fj have degree ≤ r + j. Note for experts:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 111

Regularity for Algebraists

Given a projective scheme X in Pn, ∃! saturated homogeneous ideal IX ⊂ S := K[x0, . . . , xn] s. t. X = Proj(S/IX); one says X is aCM (resp. aG) if S/IX is Cohen–Macaulay (resp. Gorenstein). One sets reg X := reg IX, which is in turn defined as follows: Recall: regularity of an ideal Given a minimal graded free resolution · · · → Fj → · · · → F0 → I → 0, the Castelnuovo–Mumford regularity of I is the smallest r such that for each j, all minimal generators of Fj have degree ≤ r + j. Note for experts: There’s another way to define regularity if you like local cohomology, namely

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 112

Regularity for Algebraists

Given a projective scheme X in Pn, ∃! saturated homogeneous ideal IX ⊂ S := K[x0, . . . , xn] s. t. X = Proj(S/IX); one says X is aCM (resp. aG) if S/IX is Cohen–Macaulay (resp. Gorenstein). One sets reg X := reg IX, which is in turn defined as follows: Recall: regularity of an ideal Given a minimal graded free resolution · · · → Fj → · · · → F0 → I → 0, the Castelnuovo–Mumford regularity of I is the smallest r such that for each j, all minimal generators of Fj have degree ≤ r + j. Note for experts: There’s another way to define regularity if you like local cohomology, namely reg(S/I) := max{i + j : Hi

m(S/I)j = 0} and reg I = reg S/I + 1.

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SLIDE 113

Regularity for Algebraists - Examples

Example 1. If X is a line (or a hyperplane, or a linear subspace), it has regularity 1.

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SLIDE 114

Regularity for Algebraists - Examples

Example 1. If X is a line (or a hyperplane, or a linear subspace), it has regularity 1. Example 2. Moment curves, i.e. curves of type (t, t2, . . . , td), have regularity 2.

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SLIDE 115

Regularity for Algebraists - Examples

Example 1. If X is a line (or a hyperplane, or a linear subspace), it has regularity 1. Example 2. Moment curves, i.e. curves of type (t, t2, . . . , td), have regularity 2. Example 3. If a simplicial complex ∆ is a triangulated (d − 1)-sphere, X = V (I∆) is aG of regularity d + 1.

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SLIDE 116

Regularity for Algebraists - Examples

Example 1. If X is a line (or a hyperplane, or a linear subspace), it has regularity 1. Example 2. Moment curves, i.e. curves of type (t, t2, . . . , td), have regularity 2. Example 3. If a simplicial complex ∆ is a triangulated (d − 1)-sphere, X = V (I∆) is aG of regularity d + 1. Example 4. If IX = (g1, . . . , gs) is a complete intersection, then X is aG of regularity reg X = deg g1 + . . . + deg gs − s + 1.

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SLIDE 117

Regularity for Poor Combinatorialists

Recall: A graph is d-regular if every vertex has exactly d neighbors.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Regularity for Poor Combinatorialists

Recall: A graph is d-regular if every vertex has exactly d neighbors. If a graph is d-regular and k-connected, necessarily k ≤ d.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Regularity for Poor Combinatorialists

Recall: A graph is d-regular if every vertex has exactly d neighbors. If a graph is d-regular and k-connected, necessarily k ≤ d. (If you kill all d neighbors of a vertex, you disconnect the graph, because now the vertex is isolated.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Regularity for Poor Combinatorialists

Recall: A graph is d-regular if every vertex has exactly d neighbors. If a graph is d-regular and k-connected, necessarily k ≤ d. (If you kill all d neighbors of a vertex, you disconnect the graph, because now the vertex is isolated. So a d-regular graph is not (d + 1)-connected.)

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Regularity for Poor Combinatorialists

Recall: A graph is d-regular if every vertex has exactly d neighbors. If a graph is d-regular and k-connected, necessarily k ≤ d. (If you kill all d neighbors of a vertex, you disconnect the graph, because now the vertex is isolated. So a d-regular graph is not (d + 1)-connected.) Balinski, Klee (1975) The dual graph of every (d − 1)-dimensional triangulated homology sphere (or manifold) is d-regular and d-connected.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Nomen est omen

Surprisingly, these two notions of regularity agree:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Nomen est omen

Surprisingly, these two notions of regularity agree: Theorem (B.–Di Marca–Varbaro, 2016+) Let X be an arithmetically-Gorenstein arrangement of projective

  • lines. Then the dual graph of X has connectivity ≥ reg X − 1.

If in addition no three lines meet in a common point, then the graph has connectivity = reg X − 1, and is (reg X − 1)-regular.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Nomen est omen

Surprisingly, these two notions of regularity agree: Theorem (B.–Di Marca–Varbaro, 2016+) Let X be an arithmetically-Gorenstein arrangement of projective

  • lines. Then the dual graph of X has connectivity ≥ reg X − 1.

If in addition no three lines meet in a common point, then the graph has connectivity = reg X − 1, and is (reg X − 1)-regular. (Since reg S/IX = reg X − 1, one can equivalently rephrase as “the Castelnuovo–Mumford regularity of S/IX and the regularity of the dual graph of X coincide”.)

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Nomen est omen

Surprisingly, these two notions of regularity agree: Theorem (B.–Di Marca–Varbaro, 2016+) Let X be an arithmetically-Gorenstein arrangement of projective

  • lines. Then the dual graph of X has connectivity ≥ reg X − 1.

If in addition no three lines meet in a common point, then the graph has connectivity = reg X − 1, and is (reg X − 1)-regular. (Since reg S/IX = reg X − 1, one can equivalently rephrase as “the Castelnuovo–Mumford regularity of S/IX and the regularity of the dual graph of X coincide”.) Special case 1: X is the Stanley–Reisner variety of a (d − 1)-sphere ∆.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Nomen est omen

Surprisingly, these two notions of regularity agree: Theorem (B.–Di Marca–Varbaro, 2016+) Let X be an arithmetically-Gorenstein arrangement of projective

  • lines. Then the dual graph of X has connectivity ≥ reg X − 1.

If in addition no three lines meet in a common point, then the graph has connectivity = reg X − 1, and is (reg X − 1)-regular. (Since reg S/IX = reg X − 1, one can equivalently rephrase as “the Castelnuovo–Mumford regularity of S/IX and the regularity of the dual graph of X coincide”.) Special case 1: X is the Stanley–Reisner variety of a (d − 1)-sphere ∆. Then reg X = d + 1,

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 127

Nomen est omen

Surprisingly, these two notions of regularity agree: Theorem (B.–Di Marca–Varbaro, 2016+) Let X be an arithmetically-Gorenstein arrangement of projective

  • lines. Then the dual graph of X has connectivity ≥ reg X − 1.

If in addition no three lines meet in a common point, then the graph has connectivity = reg X − 1, and is (reg X − 1)-regular. (Since reg S/IX = reg X − 1, one can equivalently rephrase as “the Castelnuovo–Mumford regularity of S/IX and the regularity of the dual graph of X coincide”.) Special case 1: X is the Stanley–Reisner variety of a (d − 1)-sphere ∆. Then reg X = d + 1, so the dual graph of X (= that of ∆!) is d-connected and d-regular.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 128

Nomen est omen

Surprisingly, these two notions of regularity agree: Theorem (B.–Di Marca–Varbaro, 2016+) Let X be an arithmetically-Gorenstein arrangement of projective

  • lines. Then the dual graph of X has connectivity ≥ reg X − 1.

If in addition no three lines meet in a common point, then the graph has connectivity = reg X − 1, and is (reg X − 1)-regular. (Since reg S/IX = reg X − 1, one can equivalently rephrase as “the Castelnuovo–Mumford regularity of S/IX and the regularity of the dual graph of X coincide”.) Special case 1: X is the Stanley–Reisner variety of a (d − 1)-sphere ∆. Then reg X = d + 1, so the dual graph of X (= that of ∆!) is d-connected and d-regular. Balinski-Klee!

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Nomen est omen

Surprisingly, these two notions of regularity agree: Theorem (B.–Di Marca–Varbaro, 2016+) Let X be an arithmetically-Gorenstein arrangement of projective

  • lines. Then the dual graph of X has connectivity ≥ reg X − 1.

If in addition no three lines meet in a common point, then the graph has connectivity = reg X − 1, and is (reg X − 1)-regular. (Since reg S/IX = reg X − 1, one can equivalently rephrase as “the Castelnuovo–Mumford regularity of S/IX and the regularity of the dual graph of X coincide”.) Special case 1: X is the Stanley–Reisner variety of a (d − 1)-sphere ∆. Then reg X = d + 1, so the dual graph of X (= that of ∆!) is d-connected and d-regular. Balinski-Klee! Special case 2: if X is a complete intersection. (reg X is the sum

  • f the degree of the components, minus their number, minus 1.)

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 1. The 27 lines on a cubic

Corollary Let X be an arrangement of lines in P3 that is a complete intersection of two surfaces, of degree a and b, say.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 1. The 27 lines on a cubic

Corollary Let X be an arrangement of lines in P3 that is a complete intersection of two surfaces, of degree a and b, say. Then each line

  • f the arrangement intersects at least a + b − 2 of the other lines.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 1. The 27 lines on a cubic

Corollary Let X be an arrangement of lines in P3 that is a complete intersection of two surfaces, of degree a and b, say. Then each line

  • f the arrangement intersects at least a + b − 2 of the other lines.

No three lines share a point?

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 1. The 27 lines on a cubic

Corollary Let X be an arrangement of lines in P3 that is a complete intersection of two surfaces, of degree a and b, say. Then each line

  • f the arrangement intersects at least a + b − 2 of the other lines.

No three lines share a point? ...⇒ exactly a + b − 2 other lines.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 1. The 27 lines on a cubic

Corollary Let X be an arrangement of lines in P3 that is a complete intersection of two surfaces, of degree a and b, say. Then each line

  • f the arrangement intersects at least a + b − 2 of the other lines.

No three lines share a point? ...⇒ exactly a + b − 2 other lines.

Greg Egan, The Clebsch cubic surface Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 1. The 27 lines on a cubic

Corollary Let X be an arrangement of lines in P3 that is a complete intersection of two surfaces, of degree a and b, say. Then each line

  • f the arrangement intersects at least a + b − 2 of the other lines.

No three lines share a point? ...⇒ exactly a + b − 2 other lines.

Greg Egan, The Clebsch cubic surface (in which all 27 lines are real, and there are triple points) Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 1. The 27 lines on a cubic

Corollary Let X be an arrangement of lines in P3 that is a complete intersection of two surfaces, of degree a and b, say. Then each line

  • f the arrangement intersects at least a + b − 2 of the other lines.

No three lines share a point? ...⇒ exactly a + b − 2 other lines.

Greg Egan, The Clebsch cubic surface (in which all 27 lines are real, and there are triple points)

Example 1. Any smooth cubic surface of P3 has 27 lines on it (if generic, no 3 share a point). The 27 lines are the complete int. of the cubic with a union of 9 planes.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 1. The 27 lines on a cubic

Corollary Let X be an arrangement of lines in P3 that is a complete intersection of two surfaces, of degree a and b, say. Then each line

  • f the arrangement intersects at least a + b − 2 of the other lines.

No three lines share a point? ...⇒ exactly a + b − 2 other lines.

Greg Egan, The Clebsch cubic surface (in which all 27 lines are real, and there are triple points)

Example 1. Any smooth cubic surface of P3 has 27 lines on it (if generic, no 3 share a point). The 27 lines are the complete int. of the cubic with a union of 9 planes. So a = 3, b = 9; each line intersects exactly 10 of the others.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6}

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6} where {ai, bj} is an edge iff i = j.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6} where {ai, bj} is an edge iff i = j. Then G is 5-regular, with diam G = 3.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6} where {ai, bj} is an edge iff i = j. Then G is 5-regular, with diam G = 3. Schl¨ afli’s double-six is a line arrangement X ⊆ P3 with dual graph G.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6} where {ai, bj} is an edge iff i = j. Then G is 5-regular, with diam G = 3. Schl¨ afli’s double-six is a line arrangement X ⊆ P3 with dual graph

  • G. It consists in 12 of the 27 lines on a smooth cubic Y ⊂ P3.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6} where {ai, bj} is an edge iff i = j. Then G is 5-regular, with diam G = 3. Schl¨ afli’s double-six is a line arrangement X ⊆ P3 with dual graph

  • G. It consists in 12 of the 27 lines on a smooth cubic Y ⊂ P3.

The intersection points of X are 30, Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6} where {ai, bj} is an edge iff i = j. Then G is 5-regular, with diam G = 3. Schl¨ afli’s double-six is a line arrangement X ⊆ P3 with dual graph

  • G. It consists in 12 of the 27 lines on a smooth cubic Y ⊂ P3.

The intersection points of X are 30, and the vector space of quartics of P3 has dimension 35. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6} where {ai, bj} is an edge iff i = j. Then G is 5-regular, with diam G = 3. Schl¨ afli’s double-six is a line arrangement X ⊆ P3 with dual graph

  • G. It consists in 12 of the 27 lines on a smooth cubic Y ⊂ P3.

The intersection points of X are 30, and the vector space of quartics of P3 has dimension 35. So there is a quartic Z ⊂ P3 passing through these 30 points. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6} where {ai, bj} is an edge iff i = j. Then G is 5-regular, with diam G = 3. Schl¨ afli’s double-six is a line arrangement X ⊆ P3 with dual graph

  • G. It consists in 12 of the 27 lines on a smooth cubic Y ⊂ P3.

The intersection points of X are 30, and the vector space of quartics of P3 has dimension 35. So there is a quartic Z ⊂ P3 passing through these 30 points. This quartic contains at least 5 points per line, so it contains each line! Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 147

Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6} where {ai, bj} is an edge iff i = j. Then G is 5-regular, with diam G = 3. Schl¨ afli’s double-six is a line arrangement X ⊆ P3 with dual graph

  • G. It consists in 12 of the 27 lines on a smooth cubic Y ⊂ P3.

The intersection points of X are 30, and the vector space of quartics of P3 has dimension 35. So there is a quartic Z ⊂ P3 passing through these 30 points. This quartic contains at least 5 points per line, so it contains each line! So X ⊂ Z. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6} where {ai, bj} is an edge iff i = j. Then G is 5-regular, with diam G = 3. Schl¨ afli’s double-six is a line arrangement X ⊆ P3 with dual graph

  • G. It consists in 12 of the 27 lines on a smooth cubic Y ⊂ P3.

The intersection points of X are 30, and the vector space of quartics of P3 has dimension 35. So there is a quartic Z ⊂ P3 passing through these 30 points. This quartic contains at least 5 points per line, so it contains each line! So X ⊂ Z. By picking other 4 points outside of Y and not co-planar, one can also choose Z not containing Y (because 35 > 30 + 4). Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6} where {ai, bj} is an edge iff i = j. Then G is 5-regular, with diam G = 3. Schl¨ afli’s double-six is a line arrangement X ⊆ P3 with dual graph

  • G. It consists in 12 of the 27 lines on a smooth cubic Y ⊂ P3.

The intersection points of X are 30, and the vector space of quartics of P3 has dimension 35. So there is a quartic Z ⊂ P3 passing through these 30 points. This quartic contains at least 5 points per line, so it contains each line! So X ⊂ Z. By picking other 4 points outside of Y and not co-planar, one can also choose Z not containing Y (because 35 > 30 + 4). So Y ∩ Z is a complete intersection containing X. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6} where {ai, bj} is an edge iff i = j. Then G is 5-regular, with diam G = 3. Schl¨ afli’s double-six is a line arrangement X ⊆ P3 with dual graph

  • G. It consists in 12 of the 27 lines on a smooth cubic Y ⊂ P3.

The intersection points of X are 30, and the vector space of quartics of P3 has dimension 35. So there is a quartic Z ⊂ P3 passing through these 30 points. This quartic contains at least 5 points per line, so it contains each line! So X ⊂ Z. By picking other 4 points outside of Y and not co-planar, one can also choose Z not containing Y (because 35 > 30 + 4). So Y ∩ Z is a complete intersection containing X. But 3 · 4 = 12, so X = Y ∩ Z. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s double-six

Let G be the bipartite graph on {a1, . . . , a6} ∪ {b1, . . . , b6} where {ai, bj} is an edge iff i = j. Then G is 5-regular, with diam G = 3. Schl¨ afli’s double-six is a line arrangement X ⊆ P3 with dual graph

  • G. It consists in 12 of the 27 lines on a smooth cubic Y ⊂ P3.

The intersection points of X are 30, and the vector space of quartics of P3 has dimension 35. So there is a quartic Z ⊂ P3 passing through these 30 points. This quartic contains at least 5 points per line, so it contains each line! So X ⊂ Z. By picking other 4 points outside of Y and not co-planar, one can also choose Z not containing Y (because 35 > 30 + 4). So Y ∩ Z is a complete intersection containing X. But 3 · 4 = 12, so X = Y ∩ Z.

We proved X is a complete intersection, with a = 3 and b = 4: as

  • ur Corollary claims, every line intersects exactly 5 other lines.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s distracted

Steal three of the 12 lines in Schl¨ afli’s arrangement. Can the remaining 9 lines be a complete intersection?

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s distracted

Steal three of the 12 lines in Schl¨ afli’s arrangement. Can the remaining 9 lines be a complete intersection?

  • A. No, the dual graph is not regular.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Example 2. Schl¨ afli’s distracted

Steal three of the 12 lines in Schl¨ afli’s arrangement. Can the remaining 9 lines be a complete intersection?

  • A. No, the dual graph is not regular.

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Part III. From Lines to Curves.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Balinski for curve arrangements

What about Gorenstein arrangements of curves?

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Balinski for curve arrangements

What about Gorenstein arrangements of curves? B.–Bolognese–Varbaro, 2015 Let X be an arithmetically-Gorenstein projective curve. Let R be the maximum of the regularities of the irreducible components

  • f X. Then the dual graph of X is ⌊ reg X+R−2

R

⌋-connected.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 158

Balinski for curve arrangements

What about Gorenstein arrangements of curves? B.–Bolognese–Varbaro, 2015 Let X be an arithmetically-Gorenstein projective curve. Let R be the maximum of the regularities of the irreducible components

  • f X. Then the dual graph of X is ⌊ reg X+R−2

R

⌋-connected. Line arrangements are the case R = 1.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 159

Balinski for curve arrangements

What about Gorenstein arrangements of curves? B.–Bolognese–Varbaro, 2015 Let X be an arithmetically-Gorenstein projective curve. Let R be the maximum of the regularities of the irreducible components

  • f X. Then the dual graph of X is ⌊ reg X+R−2

R

⌋-connected. Line arrangements are the case R = 1.

Proof idea: we need to show that removing k − 1 of the curves, the resulting object A is connected. Being connected can be expressed cohomologically;

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 160

Balinski for curve arrangements

What about Gorenstein arrangements of curves? B.–Bolognese–Varbaro, 2015 Let X be an arithmetically-Gorenstein projective curve. Let R be the maximum of the regularities of the irreducible components

  • f X. Then the dual graph of X is ⌊ reg X+R−2

R

⌋-connected. Line arrangements are the case R = 1.

Proof idea: we need to show that removing k − 1 of the curves, the resulting object A is connected. Being connected can be expressed cohomologically; but via liaison theory, the cohomology of A is related to that of its complement B (A ∪ B is Gorenstein!).

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 161

Balinski for curve arrangements

What about Gorenstein arrangements of curves? B.–Bolognese–Varbaro, 2015 Let X be an arithmetically-Gorenstein projective curve. Let R be the maximum of the regularities of the irreducible components

  • f X. Then the dual graph of X is ⌊ reg X+R−2

R

⌋-connected. Line arrangements are the case R = 1.

Proof idea: we need to show that removing k − 1 of the curves, the resulting object A is connected. Being connected can be expressed cohomologically; but via liaison theory, the cohomology of A is related to that of its complement B (A ∪ B is Gorenstein!). By a known cohomological characterization of regularity, it suffices to bound from above the regularity of B.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 162

Balinski for curve arrangements

What about Gorenstein arrangements of curves? B.–Bolognese–Varbaro, 2015 Let X be an arithmetically-Gorenstein projective curve. Let R be the maximum of the regularities of the irreducible components

  • f X. Then the dual graph of X is ⌊ reg X+R−2

R

⌋-connected. Line arrangements are the case R = 1.

Proof idea: we need to show that removing k − 1 of the curves, the resulting object A is connected. Being connected can be expressed cohomologically; but via liaison theory, the cohomology of A is related to that of its complement B (A ∪ B is Gorenstein!). By a known cohomological characterization of regularity, it suffices to bound from above the regularity of B. But B consists of exactly k − 1 curves, each of regularity ≤ R:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 163

Balinski for curve arrangements

What about Gorenstein arrangements of curves? B.–Bolognese–Varbaro, 2015 Let X be an arithmetically-Gorenstein projective curve. Let R be the maximum of the regularities of the irreducible components

  • f X. Then the dual graph of X is ⌊ reg X+R−2

R

⌋-connected. Line arrangements are the case R = 1.

Proof idea: we need to show that removing k − 1 of the curves, the resulting object A is connected. Being connected can be expressed cohomologically; but via liaison theory, the cohomology of A is related to that of its complement B (A ∪ B is Gorenstein!). By a known cohomological characterization of regularity, it suffices to bound from above the regularity of B. But B consists of exactly k − 1 curves, each of regularity ≤ R: We prove a bound of the type (k − 1) · R, by first proving that the regularity of curve arrangements is subadditive (new!).

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Balinski for curve arrangements

What about Gorenstein arrangements of curves? B.–Bolognese–Varbaro, 2015 Let X be an arithmetically-Gorenstein projective curve. Let R be the maximum of the regularities of the irreducible components

  • f X. Then the dual graph of X is ⌊ reg X+R−2

R

⌋-connected. Line arrangements are the case R = 1.

Proof idea: we need to show that removing k − 1 of the curves, the resulting object A is connected. Being connected can be expressed cohomologically; but via liaison theory, the cohomology of A is related to that of its complement B (A ∪ B is Gorenstein!). By a known cohomological characterization of regularity, it suffices to bound from above the regularity of B. But B consists of exactly k − 1 curves, each of regularity ≤ R: We prove a bound of the type (k − 1) · R, by first proving that the regularity of curve arrangements is subadditive (new!).

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Bonus slides

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Bonus slides

Theorem (Hartshorne, 1962) If X is an arithmetically Cohen–Macaulay (aCM) curve, the dual graph of X is connected.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Bonus slides

Theorem (Hartshorne, 1962) If X is an arithmetically Cohen–Macaulay (aCM) curve, the dual graph of X is connected. Let G be a connected graph. Can we find an aCM curve X with dual graph G?

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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Bonus slides

Theorem (Hartshorne, 1962) If X is an arithmetically Cohen–Macaulay (aCM) curve, the dual graph of X is connected. Let G be a connected graph. Can we find an aCM curve X with dual graph G? (Genericity arguments do not work.)

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 169

Bonus slides

Theorem (Hartshorne, 1962) If X is an arithmetically Cohen–Macaulay (aCM) curve, the dual graph of X is connected. Let G be a connected graph. Can we find an aCM curve X with dual graph G? (Genericity arguments do not work.)

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 170

Bonus slides

Theorem (Hartshorne, 1962) If X is an arithmetically Cohen–Macaulay (aCM) curve, the dual graph of X is connected. Let G be a connected graph. Can we find an aCM curve X with dual graph G? (Genericity arguments do not work.) Good news! (B.–Bolognese–Varbaro, 2015) For any connected graph G, one can canonically construct an aCM curve XG with dual graph G, with three “bonus” features:

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 171

Bonus slides

Theorem (Hartshorne, 1962) If X is an arithmetically Cohen–Macaulay (aCM) curve, the dual graph of X is connected. Let G be a connected graph. Can we find an aCM curve X with dual graph G? (Genericity arguments do not work.) Good news! (B.–Bolognese–Varbaro, 2015) For any connected graph G, one can canonically construct an aCM curve XG with dual graph G, with three “bonus” features: reg XG ≤ 3

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-172
SLIDE 172

Bonus slides

Theorem (Hartshorne, 1962) If X is an arithmetically Cohen–Macaulay (aCM) curve, the dual graph of X is connected. Let G be a connected graph. Can we find an aCM curve X with dual graph G? (Genericity arguments do not work.) Good news! (B.–Bolognese–Varbaro, 2015) For any connected graph G, one can canonically construct an aCM curve XG with dual graph G, with three “bonus” features: reg XG ≤ 3 (smallest possible, can do 2 only if G a tree).

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 173

Bonus slides

Theorem (Hartshorne, 1962) If X is an arithmetically Cohen–Macaulay (aCM) curve, the dual graph of X is connected. Let G be a connected graph. Can we find an aCM curve X with dual graph G? (Genericity arguments do not work.) Good news! (B.–Bolognese–Varbaro, 2015) For any connected graph G, one can canonically construct an aCM curve XG with dual graph G, with three “bonus” features: reg XG ≤ 3 (smallest possible, can do 2 only if G a tree). the components of XG have regularity ≤ 2 (smallest possible - regularity 1 means “line”), they’re all rational normal curves.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 174

Bonus slides

Theorem (Hartshorne, 1962) If X is an arithmetically Cohen–Macaulay (aCM) curve, the dual graph of X is connected. Let G be a connected graph. Can we find an aCM curve X with dual graph G? (Genericity arguments do not work.) Good news! (B.–Bolognese–Varbaro, 2015) For any connected graph G, one can canonically construct an aCM curve XG with dual graph G, with three “bonus” features: reg XG ≤ 3 (smallest possible, can do 2 only if G a tree). the components of XG have regularity ≤ 2 (smallest possible - regularity 1 means “line”), they’re all rational normal curves. no three components of XG meet at a same point.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-175
SLIDE 175

Bonus slides

Theorem (Hartshorne, 1962) If X is an arithmetically Cohen–Macaulay (aCM) curve, the dual graph of X is connected. Let G be a connected graph. Can we find an aCM curve X with dual graph G? (Genericity arguments do not work.) Good news! (B.–Bolognese–Varbaro, 2015) For any connected graph G, one can canonically construct an aCM curve XG with dual graph G, with three “bonus” features: reg XG ≤ 3 (smallest possible, can do 2 only if G a tree). the components of XG have regularity ≤ 2 (smallest possible - regularity 1 means “line”), they’re all rational normal curves. no three components of XG meet at a same point.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 176

Bonus slides

Theorem (Hartshorne, 1962) If X is an arithmetically Cohen–Macaulay (aCM) curve, the dual graph of X is connected. Let G be a connected graph. Can we find an aCM curve X with dual graph G? (Genericity arguments do not work.) Good news! (B.–Bolognese–Varbaro, 2015) For any connected graph G, one can canonically construct an aCM curve XG with dual graph G, with three “bonus” features: reg XG ≤ 3 (smallest possible, can do 2 only if G a tree). the components of XG have regularity ≤ 2 (smallest possible - regularity 1 means “line”), they’re all rational normal curves. no three components of XG meet at a same point. The recipe for constructing XG is computationally hard, but it is

  • nly a few lines long, and explicit...

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 177

Bonus slides

Theorem (Hartshorne, 1962) If X is an arithmetically Cohen–Macaulay (aCM) curve, the dual graph of X is connected. Let G be a connected graph. Can we find an aCM curve X with dual graph G? (Genericity arguments do not work.) Good news! (B.–Bolognese–Varbaro, 2015) For any connected graph G, one can canonically construct an aCM curve XG with dual graph G, with three “bonus” features: reg XG ≤ 3 (smallest possible, can do 2 only if G a tree). the components of XG have regularity ≤ 2 (smallest possible - regularity 1 means “line”), they’re all rational normal curves. no three components of XG meet at a same point. The recipe for constructing XG is computationally hard, but it is

  • nly a few lines long, and explicit...

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 178

Construction

Say G is a graph with s vertices.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 179

Construction

Say G is a graph with s vertices. Pick s lines in P2, given by equations ℓi = 0, so that no three lines have a common point.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 180

Construction

Say G is a graph with s vertices. Pick s lines in P2, given by equations ℓi = 0, so that no three lines have a common point. (‘Generic’ works perfectly.)

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 181

Construction

Say G is a graph with s vertices. Pick s lines in P2, given by equations ℓi = 0, so that no three lines have a common point. (‘Generic’ works perfectly.) Set I =

{i,j}/ ∈E(G)(ℓi, ℓj).

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 182

Construction

Say G is a graph with s vertices. Pick s lines in P2, given by equations ℓi = 0, so that no three lines have a common point. (‘Generic’ works perfectly.) Set I =

{i,j}/ ∈E(G)(ℓi, ℓj).

Let R[d] be the subalgebra of the polynomial ring generated by the degree-d elements of I.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 183

Construction

Say G is a graph with s vertices. Pick s lines in P2, given by equations ℓi = 0, so that no three lines have a common point. (‘Generic’ works perfectly.) Set I =

{i,j}/ ∈E(G)(ℓi, ℓj).

Let R[d] be the subalgebra of the polynomial ring generated by the degree-d elements of I. Set A[d] = R[d] (ℓ1ℓ2 · · · ℓs) ∩ R[d].

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 184

Construction

Say G is a graph with s vertices. Pick s lines in P2, given by equations ℓi = 0, so that no three lines have a common point. (‘Generic’ works perfectly.) Set I =

{i,j}/ ∈E(G)(ℓi, ℓj).

Let R[d] be the subalgebra of the polynomial ring generated by the degree-d elements of I. Set A[d] = R[d] (ℓ1ℓ2 · · · ℓs) ∩ R[d]. The dual graph of A[d] is G; maybe A[d] is not CM, but this can be fixed taking reg A[d] many Veronese.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 185

Construction

Say G is a graph with s vertices. Pick s lines in P2, given by equations ℓi = 0, so that no three lines have a common point. (‘Generic’ works perfectly.) Set I =

{i,j}/ ∈E(G)(ℓi, ℓj).

Let R[d] be the subalgebra of the polynomial ring generated by the degree-d elements of I. Set A[d] = R[d] (ℓ1ℓ2 · · · ℓs) ∩ R[d]. The dual graph of A[d] is G; maybe A[d] is not CM, but this can be fixed taking reg A[d] many Veronese.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 186

Construction

Say G is a graph with s vertices. Pick s lines in P2, given by equations ℓi = 0, so that no three lines have a common point. (‘Generic’ works perfectly.) Set I =

{i,j}/ ∈E(G)(ℓi, ℓj).

Let R[d] be the subalgebra of the polynomial ring generated by the degree-d elements of I. Set A[d] = R[d] (ℓ1ℓ2 · · · ℓs) ∩ R[d]. The dual graph of A[d] is G; maybe A[d] is not CM, but this can be fixed taking reg A[d] many Veronese. Example: G = K4 minus the edge 12. Let us choose ℓ1 = x, ℓ2 = y, ℓ3 = z, ℓ4 = x + y + z;

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 187

Construction

Say G is a graph with s vertices. Pick s lines in P2, given by equations ℓi = 0, so that no three lines have a common point. (‘Generic’ works perfectly.) Set I =

{i,j}/ ∈E(G)(ℓi, ℓj).

Let R[d] be the subalgebra of the polynomial ring generated by the degree-d elements of I. Set A[d] = R[d] (ℓ1ℓ2 · · · ℓs) ∩ R[d]. The dual graph of A[d] is G; maybe A[d] is not CM, but this can be fixed taking reg A[d] many Veronese. Example: G = K4 minus the edge 12. Let us choose ℓ1 = x, ℓ2 = y, ℓ3 = z, ℓ4 = x + y + z; so I = (x, y). Then A[3] = C[x3, x2y, x2z, xy2, xyz, xz2, y3, y2z, yz2] (xyz(x + y + z)) .

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 188

Construction

Say G is a graph with s vertices. Pick s lines in P2, given by equations ℓi = 0, so that no three lines have a common point. (‘Generic’ works perfectly.) Set I =

{i,j}/ ∈E(G)(ℓi, ℓj).

Let R[d] be the subalgebra of the polynomial ring generated by the degree-d elements of I. Set A[d] = R[d] (ℓ1ℓ2 · · · ℓs) ∩ R[d]. The dual graph of A[d] is G; maybe A[d] is not CM, but this can be fixed taking reg A[d] many Veronese. Example: G = K4 minus the edge 12. Let us choose ℓ1 = x, ℓ2 = y, ℓ3 = z, ℓ4 = x + y + z; so I = (x, y). Then A[3] = C[x3, x2y, x2z, xy2, xyz, xz2, y3, y2z, yz2] (xyz(x + y + z)) .

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 189

Extra frame: Proof details

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 190

Extra frame: Proof details

Regularity can be characterized using Grothendieck duality as follows: reg(S/I) = max{i + j : Hi

m(S/I)j = 0},

where Hi

m stands for local cohomology with support in the maximal ideal m = (x1, . . . , xn).

Order of the prime ideals as you wish. Let IB = p1 ∩ . . . pr−1 and IA = pr ∩ . . . ∩ ps. Want to prove that G(IA) is connected. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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SLIDE 191

Extra frame: Proof details

Regularity can be characterized using Grothendieck duality as follows: reg(S/I) = max{i + j : Hi

m(S/I)j = 0},

where Hi

m stands for local cohomology with support in the maximal ideal m = (x1, . . . , xn).

Order of the prime ideals as you wish. Let IB = p1 ∩ . . . pr−1 and IA = pr ∩ . . . ∩ ps. Want to prove that G(IA) is connected. 1 CA and CB are geometrically linked by C = Proj(S/I) which is arit. Gorenstein; so by Migliore’s theory, we have a graded isomorphism H1

m(S/IA) ∼

= H1

m(S/IB )∨(2 − r).

2 By Derksen–Sidman, reg(IB ) ≤ r − 1, so reg(S/IB ) ≤ r − 2. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-192
SLIDE 192

Extra frame: Proof details

Regularity can be characterized using Grothendieck duality as follows: reg(S/I) = max{i + j : Hi

m(S/I)j = 0},

where Hi

m stands for local cohomology with support in the maximal ideal m = (x1, . . . , xn).

Order of the prime ideals as you wish. Let IB = p1 ∩ . . . pr−1 and IA = pr ∩ . . . ∩ ps. Want to prove that G(IA) is connected. 1 CA and CB are geometrically linked by C = Proj(S/I) which is arit. Gorenstein; so by Migliore’s theory, we have a graded isomorphism H1

m(S/IA) ∼

= H1

m(S/IB )∨(2 − r).

2 By Derksen–Sidman, reg(IB ) ≤ r − 1, so reg(S/IB ) ≤ r − 2. 3 By definition of regularity, reg(S/IB ) ≤ r − 2 implies that H1

m(S/IB )r−2 = 0.

Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

slide-193
SLIDE 193

Extra frame: Proof details

Regularity can be characterized using Grothendieck duality as follows: reg(S/I) = max{i + j : Hi

m(S/I)j = 0},

where Hi

m stands for local cohomology with support in the maximal ideal m = (x1, . . . , xn).

Order of the prime ideals as you wish. Let IB = p1 ∩ . . . pr−1 and IA = pr ∩ . . . ∩ ps. Want to prove that G(IA) is connected. 1 CA and CB are geometrically linked by C = Proj(S/I) which is arit. Gorenstein; so by Migliore’s theory, we have a graded isomorphism H1

m(S/IA) ∼

= H1

m(S/IB )∨(2 − r).

2 By Derksen–Sidman, reg(IB ) ≤ r − 1, so reg(S/IB ) ≤ r − 2. 3 By definition of regularity, reg(S/IB ) ≤ r − 2 implies that H1

m(S/IB )r−2 = 0.

4 So H1

m(S/IA)0 = 0. This implies that H0(CA, OCA ) ∼

= K, which in turn implies that CA is a connected curve. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements