Outline Preliminaries Determination of VCIs A Virtually Complete Classification of Virtual Complete Intersections in P 1 × P 1 Jiyang Gao, Yutong Li, Amal Mattoo University of Minnesota - Twin Cities REU 2018 1 August 2018 VCIs in P 1 × P 1 Gao, Li, Mattoo 1 / 25
Outline Preliminaries Determination of VCIs 1 Preliminaries Projective Space and Varieties Free and Virtual Resolutions Virtual Complete Intersections (VCIs) 2 Determination of VCIs Overview VCI Existence Cases VCI Non-Existence Conditions on VCIs Conclusion VCIs in P 1 × P 1 Gao, Li, Mattoo 2 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) The Projective Space P n Definition A projective space P n over the field C is the set of one-dimensional subspaces of the vector space C n +1 . • The coordinate ring of P n is S = C [ x 0 , x 1 , . . . , x n ]. • Grading: Constants have degree 0. Each x i has degree 1. VCIs in P 1 × P 1 Gao, Li, Mattoo 3 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) The Projective Space P n Definition A projective space P n over the field C is the set of one-dimensional subspaces of the vector space C n +1 . • The coordinate ring of P n is S = C [ x 0 , x 1 , . . . , x n ]. • Grading: Constants have degree 0. Each x i has degree 1. [0 : 1] [1 : 1] [2 : 1] [3 : 1] [4 : 1] P 1 O VCIs in P 1 × P 1 Gao, Li, Mattoo 3 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) The Projective Space P n Definition A projective space P n over the field C is the set of one-dimensional subspaces of the vector space C n +1 . • The coordinate ring of P n is S = C [ x 0 , x 1 , . . . , x n ]. • Grading: Constants have degree 0. Each x i has degree 1. Definition A projective variety X ⊂ P n is the zero locus of a collection of homogeneous polynomials f α ∈ C [ x 0 , x 1 , . . . , x n ]. VCIs in P 1 × P 1 Gao, Li, Mattoo 3 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) The Biprojective Space P 1 × P 1 Definition The biprojective space P 1 × P 1 is the set of equivalence classes: P 1 × P 1 := { (( a 0 , a 1 ) , ( b 0 , b 1 )) ∈ C 2 × C 2 � ( a 0 ,a 1 ) � =(0 , 0) and ( b 0 ,b 1 ) � =(0 , 0) } / ∼ � ⇒ x = λy , where x, y ∈ P 1 , λ ∈ C ∗ x ∼ y ⇐ • Varieties ↔ zero locus of bihomogenous f ∈ C [ x 0 , x 1 , y 0 , y 1 ] • Multigrading: deg( x i ) = (1 , 0) , deg( y i ) = (0 , 1) ex. x 2 0 y 0 + x 1 x 2 y 1 has degree (2 , 1). VCIs in P 1 × P 1 Gao, Li, Mattoo 4 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) The Biprojective Space P 1 × P 1 Definition The biprojective space P 1 × P 1 is the set of equivalence classes: P 1 × P 1 := { (( a 0 , a 1 ) , ( b 0 , b 1 )) ∈ C 2 × C 2 � ( a 0 ,a 1 ) � =(0 , 0) and ( b 0 ,b 1 ) � =(0 , 0) } / ∼ � ⇒ x = λy , where x, y ∈ P 1 , λ ∈ C ∗ x ∼ y ⇐ • Varieties ↔ zero locus of bihomogenous f ∈ C [ x 0 , x 1 , y 0 , y 1 ] • Multigrading: deg( x i ) = (1 , 0) , deg( y i ) = (0 , 1) ex. x 2 0 y 0 + x 1 x 2 y 1 has degree (2 , 1). • Irrelevant ideal: B = � x 0 , x 1 � ∩ � y 0 , y 1 � ↔ V ( B ) = ∅ • Saturation: I : B ∞ = { s ∈ S | sB n ⊂ I for some n } VCIs in P 1 × P 1 Gao, Li, Mattoo 4 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) The Nullstellensatz The Nullstellensatz establishes a correspondence between ideals and varieties: Theorem Let X be a non-empty variety with the coordinate ring S and irrelevant ideal B . If I ⊆ S is a homogeneous ideal, then there is an inclusion-reversing bijective correspondence: I { V ( I ) � = ∅} V { radical homogeneous B -saturated ideals ⊂ S } − → ← − • V ( I ) := zero locus of all f ∈ I √ • I ( V ( I )) = I VCIs in P 1 × P 1 Gao, Li, Mattoo 5 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) Varieties in P 1 × P 1 Definition P 1 × P 1 := { (( a 0 , a 1 ) , ( b 0 , b 1 )) ∈ C 2 × C 2 � � ( a 0 ,a 1 ) � =(0 , 0) ( b 0 ,b 1 ) � =(0 , 0) } / ∼ [1 : 0] [0 : 1] [1 : 1] [1 : 2] [1 : 3] [1 : 0] [0 : 1] [1 : 1] [1 : 2] [1 : 3] VCIs in P 1 × P 1 Gao, Li, Mattoo 6 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) Varieties in P 1 × P 1 Definition P 1 × P 1 := { (( a 0 , a 1 ) , ( b 0 , b 1 )) ∈ C 2 × C 2 � � ( a 0 ,a 1 ) � =(0 , 0) ( b 0 ,b 1 ) � =(0 , 0) } / ∼ [1 : 0] [0 : 1] [1 : 1] [1 : 2] [1 : 3] [1 : 0] [0 : 1] X = ([0 : 1] , [0 : 1]) [1 : 1] I = � x 0 , y 0 � [1 : 2] [1 : 3] VCIs in P 1 × P 1 Gao, Li, Mattoo 6 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) Varieties in P 1 × P 1 Definition P 1 × P 1 := { (( a 0 , a 1 ) , ( b 0 , b 1 )) ∈ C 2 × C 2 � � ( a 0 ,a 1 ) � =(0 , 0) ( b 0 ,b 1 ) � =(0 , 0) } / ∼ [1 : 0] [0 : 1] [1 : 1] [1 : 2] [1 : 3] [1 : 0] [0 : 1] X = ([0 : 1] , [0 : 1]) ∪ ([1 : 1] , [1 : 1]) [1 : 1] I = � x 0 , y 0 � ∩� x 0 − x 1 , y 0 − y 1 � [1 : 2] [1 : 3] VCIs in P 1 × P 1 Gao, Li, Mattoo 6 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) Free Resolution Definition A free resolution of a module M is an exact sequence of homomorphisms: ϕ 0 ϕ 1 ϕ 2 0 ← − M ← − F 0 ← − F 1 ← − F 2 ← − · · · , • im ϕ i +1 = ker ϕ i at each step • every F i ∼ = R r i is a free module VCIs in P 1 × P 1 Gao, Li, Mattoo 7 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) Minimal Free Resolution Definition A free resolution is minimal if for every ℓ ≥ 1, the nonzero entries of the graded matrix of ϕ ℓ have positive degree. • For each ℓ > 0, ϕ ℓ takes the standard basis of F ℓ to a minimal generating set of im( ϕ ℓ ). • Unique up to isomorphism. • Depends on geometry of points (configuration/cross ratios) VCIs in P 1 × P 1 Gao, Li, Mattoo 8 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) Virtual Resolution Definition A virtual resolution for an ideal I in the biprojective space P 1 × P 1 is a free complex: ϕ 0 ϕ 1 ϕ 2 ϕ 3 0 ← − I ← − S ← − F 1 ← − F 2 ← − · · · such that • F i are free modules for i ≥ 0 � ker( ϕ i ) ⊇ B l • ann � im( ϕ i +1 ) • im( ϕ 1 ) : B ∞ = I : B ∞ . VCIs in P 1 × P 1 Gao, Li, Mattoo 9 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) Complete and Virtual Complete Intersection • X is a complete intersection if I ( X ) has 2 generators. ([0 : 1] , [1 : 0]) , V ( x 0 x 1 ) ([1 : 0] , [1 : 0]) , X = ([0 : 1] , [0 : 1]) , V ( y 0 y 1 ) ([1 : 0] , [0 : 1]) = ⇒ I ( X ) = � x 0 x 1 , y 0 y 1 � VCIs in P 1 × P 1 Gao, Li, Mattoo 10 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) Complete and Virtual Complete Intersection • X is a complete intersection if I ( X ) has 2 generators. ([0 : 1] , [1 : 0]) , V ( x 0 x 1 ) ([1 : 0] , [1 : 0]) , X = ([0 : 1] , [0 : 1]) , V ( y 0 y 1 ) ([1 : 0] , [0 : 1]) = ⇒ I ( X ) = � x 0 x 1 , y 0 y 1 � • Complete intersection ⇐ ⇒ min. free resolution is Koszul: S 1 ← S 2 ← S 1 ← 0 VCIs in P 1 × P 1 Gao, Li, Mattoo 10 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) Complete and Virtual Complete Intersection • X is a complete intersection if I ( X ) has 2 generators. ([0 : 1] , [1 : 0]) , V ( x 0 x 1 ) ([1 : 0] , [1 : 0]) , X = ([0 : 1] , [0 : 1]) , V ( y 0 y 1 ) ([1 : 0] , [0 : 1]) = ⇒ I ( X ) = � x 0 x 1 , y 0 y 1 � • Complete intersection ⇐ ⇒ min. free resolution is Koszul: S 1 ← S 2 ← S 1 ← 0 Definition An ideal I of points in P 1 × P 1 is a virtual complete intersection (VCI) if I has a short virtual resolution that is Koszul. In particular, V ( I ) = V ( f ) ∩ V ( g ). VCIs in P 1 × P 1 Gao, Li, Mattoo 10 / 25
Outline Projective Space and Varieties Preliminaries Free and Virtual Resolutions Determination of VCIs Virtual Complete Intersections (VCIs) VCI Examples S 1 ← S 2 ← S 1 ← 0 S 1 ← S 6 ← S 8 ← S 3 ← 0 = ⇒ Complete intersection = ⇒ Not complete intersection VCIs in P 1 × P 1 Gao, Li, Mattoo 11 / 25
Recommend
More recommend
