Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem The Weak Lefschetz for a Graded Module Zachary Flores 1/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Solomon Lefschetz 2/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Solomon Lefschetz 1 Lefschetz lost both his hands in an engineering accident and subsequently became a mathematician. 2/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Solomon Lefschetz 1 Lefschetz lost both his hands in an engineering accident and subsequently became a mathematician. 2 Lefschtez was an instructor at UNL from 1911-1913. 2/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Solomon Lefschetz 1 Lefschetz lost both his hands in an engineering accident and subsequently became a mathematician. 2 Lefschtez was an instructor at UNL from 1911-1913. 3 Lefschetz was a professor at KU from 1913-1924. 2/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Solomon Lefschetz 1 Lefschetz once received the following letter of recommendation for John Nash. 3/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Solomon Lefschetz 1 Lefschetz once received the following letter of recommendation for John Nash. 2 My roommate once held the door open for John Nash. 3/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Solomon Lefschetz 1 Lefschetz once received the following letter of recommendation for John Nash. 2 My roommate once held the door open for John Nash. 3 Conclusion: I have met Solomon Lefschetz. 3/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Outline 1 Basic Concepts 2 Semistable Bundles 3 Symmetric Hilbert Functions 4 Main Theorem 4/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem The Weak Lefschetz Property Let k be an algebraically closed field and S the polynomial ring k [ x 1 , . . . , x r ]. 5/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem The Weak Lefschetz Property Let k be an algebraically closed field and S the polynomial ring k [ x 1 , . . . , x r ]. Definition Given a graded S -module N of finite length, we say that N has the Weak Lefschetz Property if for any general linear form ℓ ∈ S 1 , the map × ℓ : N t → N t +1 has maximal rank for all t . 5/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem The Weak Lefschetz Property Let k be an algebraically closed field and S the polynomial ring k [ x 1 , . . . , x r ]. Definition Given a graded S -module N of finite length, we say that N has the Weak Lefschetz Property if for any general linear form ℓ ∈ S 1 , the map × ℓ : N t → N t +1 has maximal rank for all t . Question: Which graded S -modules of finite length have the Weak Lefschetz Property? 5/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem The Weak Lefschetz Property When N = S/I with I a homogeneous ideal of codimension r , the Weak Lefschetz Property has been studied extensively. 1 When r ≤ 2, S/I always has the Weak Lefschetz. 6/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem The Weak Lefschetz Property When N = S/I with I a homogeneous ideal of codimension r , the Weak Lefschetz Property has been studied extensively. 1 When r ≤ 2, S/I always has the Weak Lefschetz. 2 In [1] it is shown that if k has characteristic 0, r = 3 and I is a complete intersection generated in certain degrees, then S/I has the Weak Lefschetz. This was used to prove all complete intersections in three variables have the Weak Lefschetz. 6/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem The Weak Lefschetz Property When N = S/I with I a homogeneous ideal of codimension r , the Weak Lefschetz Property has been studied extensively. 1 When r ≤ 2, S/I always has the Weak Lefschetz. 2 In [1] it is shown that if k has characteristic 0, r = 3 and I is a complete intersection generated in certain degrees, then S/I has the Weak Lefschetz. This was used to prove all complete intersections in three variables have the Weak Lefschetz. What about characteristic p > 0? 6/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem An Example Suppose r = 3, k has characteristic 2 and I = ( x 2 1 , x 2 2 , x 2 3 ). If ℓ = ax 1 + bx 2 + cx 3 is a linear form, then a matrix for the map × ℓ : ( S/I ) 1 → ( S/I ) 2 is given by b a 0 A = c 0 a 0 c b 7/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem An Example Suppose r = 3, k has characteristic 2 and I = ( x 2 1 , x 2 2 , x 2 3 ). If ℓ = ax 1 + bx 2 + cx 3 is a linear form, then a matrix for the map × ℓ : ( S/I ) 1 → ( S/I ) 2 is given by b a 0 A = c 0 a 0 c b Then det( A ) = − 2 abc = 0, so × ℓ is not injective. Thus S/I does not have the Weak Lefschetz. 7/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem An Example Suppose r = 3, k has characteristic 2 and I = ( x 2 1 , x 2 2 , x 2 3 ). If ℓ = ax 1 + bx 2 + cx 3 is a linear form, then a matrix for the map × ℓ : ( S/I ) 1 → ( S/I ) 2 is given by b a 0 A = c 0 a 0 c b Then det( A ) = − 2 abc = 0, so × ℓ is not injective. Thus S/I does not have the Weak Lefschetz. In fact, if r ≥ 3 and k has characteristic p > 0, then S/ ( x p 1 , . . . , x p r ) does not have the Weak Lefschetz. 7/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Setup As the above results suggest, the characteristic of k plays a subtle role in determining when N has the Weak Lefschetz. 8/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Setup As the above results suggest, the characteristic of k plays a subtle role in determining when N has the Weak Lefschetz. 1 k has characteristic zero (and is still algebraically closed!). 8/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Setup As the above results suggest, the characteristic of k plays a subtle role in determining when N has the Weak Lefschetz. 1 k has characteristic zero (and is still algebraically closed!). 2 We set R = k [ x, y, z ]. 8/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Setup As the above results suggest, the characteristic of k plays a subtle role in determining when N has the Weak Lefschetz. 1 k has characteristic zero (and is still algebraically closed!). 2 We set R = k [ x, y, z ]. 3 n ≥ 1, ϕ : � n +2 j =1 R ( − b j ) → � n i =1 R ( − a i ) is an R -linear map with b j ≤ b j +1 and a i ≤ a i +1 . 8/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Setup As the above results suggest, the characteristic of k plays a subtle role in determining when N has the Weak Lefschetz. 1 k has characteristic zero (and is still algebraically closed!). 2 We set R = k [ x, y, z ]. 3 n ≥ 1, ϕ : � n +2 j =1 R ( − b j ) → � n i =1 R ( − a i ) is an R -linear map with b j ≤ b j +1 and a i ≤ a i +1 . 4 We set M = coker( ϕ ). 8/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Setup As the above results suggest, the characteristic of k plays a subtle role in determining when N has the Weak Lefschetz. 1 k has characteristic zero (and is still algebraically closed!). 2 We set R = k [ x, y, z ]. 3 n ≥ 1, ϕ : � n +2 j =1 R ( − b j ) → � n i =1 R ( − a i ) is an R -linear map with b j ≤ b j +1 and a i ≤ a i +1 . 4 We set M = coker( ϕ ). When M has finite length, we are interested in what numerical constraints we can place on the b j and a i so that M has the Weak Lefschetz. 8/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Some Algebraic Geometry If E is a vector bundle on P r , its slope µ ( E ) is the rational number c 1 ( E ) / rank( E ), where c 1 ( E ) is the first Chern class of E . 9/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Some Algebraic Geometry If E is a vector bundle on P r , its slope µ ( E ) is the rational number c 1 ( E ) / rank( E ), where c 1 ( E ) is the first Chern class of E . Definition We say that a vector bundle E on P r is semistable if µ ( E ′ ) ≤ µ ( E ) for every proper nonzero subbundle E ′ of E . 9/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Some Algebraic Geometry If E is a vector bundle on P r , its slope µ ( E ) is the rational number c 1 ( E ) / rank( E ), where c 1 ( E ) is the first Chern class of E . Definition We say that a vector bundle E on P r is semistable if µ ( E ′ ) ≤ µ ( E ) for every proper nonzero subbundle E ′ of E . For E with rank 2 and E normalized ( c 1 ( E ) ∈ {− 1 , 0 } ), E is semistable if and only if it has no sections. 9/24
Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem Some Algebraic Geometry By a theorem of Grothendieck, every vector bundle on P 1 splits as a sum of line bundles. Hence, if λ is general line in P 2 and E is a vector bundle on P 2 then E| λ = O λ ( e 1 ) ⊕ · · · ⊕ O λ ( e s ) 10/24
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