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Dual Variety Singularities of dual varieties Cusp Component Node Component Singularities of Dual Varieties Associated to Exterior Representations Emre S ¸EN Northeastern University November 20, 2017 Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Outline Dual Variety 1 Singularities of dual varieties 2 Cusp Component 3 Node Component 4 Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Projectivization Let V be a vector space over C , V ∗ be its dual. The set of one dimensional subspaces of V is called projectivization of V and denoted by P ( V ) . For each point in P ( V ) we can associate a hyperplane. After regarding those hyperplanes as points, dual projective space P ( V ) ∗ ∼ = P ( V ∗ ) is obtained. Picture of Duality P 2 � ∗ P 2 � ∗ � � Point in Line in Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Dual Variety P N � ∗ is Let X ⊂ P N be a projective variety. Dual variety X ∨ ⊂ � defined as the closure of the set of all tangent hyperplanes to X . Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Dual Variety P N � ∗ is Let X ⊂ P N be a projective variety. Dual variety X ∨ ⊂ � defined as the closure of the set of all tangent hyperplanes to X . Examples (1) Let � Ax, x � = 0 be a plane conic, where A is 3 × 3 nondegenerate symmetric matrix. Then, dual curve is given by � A − 1 ζ, ζ � . Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Dual Variety P N � ∗ is Let X ⊂ P N be a projective variety. Dual variety X ∨ ⊂ � defined as the closure of the set of all tangent hyperplanes to X . Examples (1) Let � Ax, x � = 0 be a plane conic, where A is 3 × 3 nondegenerate symmetric matrix. Then, dual curve is given by � A − 1 ζ, ζ � . (2) y η x ξ 4 ξ 3 = 27 η 2 y = x 3 Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Determinant Consider Segre embedding: P 1 × P 1 − → P 3 [ x 0 : x 1 ] × [ y 0 : y 1 ] �→ [ x 0 y 0 : x 0 y 1 : x 1 y 0 : x 1 y 1 ] Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Determinant Consider Segre embedding: P 1 × P 1 − → P 3 [ x 0 : x 1 ] × [ y 0 : y 1 ] �→ [ x 0 y 0 : x 0 y 1 : x 1 y 0 : x 1 y 1 ] Multilinear form f = ax 0 y 0 + bx 0 y 1 + cx 1 y 0 + dx 1 y 1 Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Determinant Consider Segre embedding: P 1 × P 1 − → P 3 [ x 0 : x 1 ] × [ y 0 : y 1 ] �→ [ x 0 y 0 : x 0 y 1 : x 1 y 0 : x 1 y 1 ] Multilinear form f = ax 0 y 0 + bx 0 y 1 + cx 1 y 0 + dx 1 y 1 ∂f ∂f ∂x 0 = ay 0 + by 1 = 0 , ∂x 1 = cy 0 + dy 1 = 0 ∂f ∂f ∂y 0 = ax 0 + cx 1 = 0 , ∂y 1 = bx 0 + dx 1 = 0 Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Determinant Consider Segre embedding: P 1 × P 1 − → P 3 [ x 0 : x 1 ] × [ y 0 : y 1 ] �→ [ x 0 y 0 : x 0 y 1 : x 1 y 0 : x 1 y 1 ] Multilinear form f = ax 0 y 0 + bx 0 y 1 + cx 1 y 0 + dx 1 y 1 ∂f ∂f ∂x 0 = ay 0 + by 1 = 0 , ∂x 1 = cy 0 + dy 1 = 0 ∂f ∂f ∂y 0 = ax 0 + cx 1 = 0 , ∂y 1 = bx 0 + dx 1 = 0 System of equations have a nontrivial solution if and only if ad − bc = 0 Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Hyperdeterminant Segre Embedding Consider the Segre embedding: X = P k 1 × . . . × P k r ֒ → P ( k 1 +1) ... ( k r +1) − 1 j = C k j +1 . If X ∨ is a where each P k j is projectivization of V ∗ hypersurface then its defining equation is called hyperdeterminant which is a homogeneous polynomial function on V 1 ⊗ . . . ⊗ V r . Examples If r = 2 , k 1 = k 2 then hyperdeterminant is classical determinant. The first nontrivial was case founded by Cayley, when r = 3 , k i = 1 : ∆ (det | Ax + By | ) where A, B are 2 × 2 matrices, x, y are variables to take discriminant. Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Coordinate System Choose a coordinate system x j = � � x j 0 , . . . , x j on each V ∗ j , then k j F ∈ V 1 ⊗ . . . ⊗ V r is represented after restriction on X by a multilinear form: x 1 , . . . , x r � i 1 ,...,i r a i 1 ,...,i r x 1 � i 1 · · · x r F = � i r F ∈ X ∨ ⇔ system of equations F ( x ) = ∂F ( x ) = 0 ∂x j i x 1 , . . . , x r � � ( for all i,j ) has a nontrivial solution for some x = . Remark Hyperdeterminant of format ( k 1 , . . . , k r ) exists iff k j ≤ � i � = j k i . Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Dual Grassmannian Let X be Grassmanian of k dimensional subspaces of n dimensional vector space V . Consider the Pl¨ ucker embedding: �� k V � G ( k, V ) ֒ → P . After choosing coordinate matrix: x 1 x 1 x 1   1 0 · · · 0 · · · k +1 k +2 n x 2 x 2 x 2 0 1 · · · 0 · · ·   k +1 k +2 n K =  .  . . . . . .  ... ... . . . . . .   . . . . . .  x k x k x k 0 0 · · · 1 · · · n k +1 k +2 F ( A, K ) = � a i 1 ...i k η i 1 ...i k 1 ≤ i 1 <i 2 <...<i k ≤ n where η i 1 ...i k is the minor of K indexed by ( i 1 , . . . , i k ) . F ∈ G ( k, n ) ∨ ⇔ system of equations F ( x ) = ∂F ( x ) = 0 ∂x j i ( for all i,j ) has a nontrivial solution for some x . Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Segre-Pl¨ ucker Embedding �� k 1 C N 1 � � � � � � k r C N r � k 1 C N 1 ⊗ . . . ⊗ � k r C N r X = P × . . . × P �→ P N i ≥ 2 k i . For each component we have Pl¨ ucker embedding like above. Then take the Segre embedding. Generic form becomes: F = � a I 1 ; ... ; I r η 1 I 1 · · · η r I r where I j is the index set of � k j C N j of size k j . Again F ∈ X ∨ ⇔ system of equations F ( x ) = ∂F ( x ) = 0 ∂x j i ( for all i,j ) has a nontrivial solution for some x . Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component For the analysis of singularities the key tool is Hessian matrix. Definition Given form F , we define Hessian matrix at point p ∈ X ie. matrix of double partial derivatives H ( F ) p = � ∂ 2 F � p ∂ i ′ j ′ ∂ i j evaluated at p for all possible indices i, i ′ , j, j ′ , and ∂ i j = ∂x i j . Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Definition The cusp component is the subvariety of X ∨ such that determinant of Hessian matrix vanishes. Formally: X cusp := { F | ∃ p ∈ X s.t P T p X ⊂ F and det H ( F ) | p = 0 } Definition The node component is the subvariety of X ∨ which is the set of forms such that F ( p ) = F ( q ) = 0 for two distinct points p, q ∈ X . Formally: X node := { F | ∃ p, q ∈ X such that P T p X, P T q X ⊂ F } Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations

Dual Variety Singularities of dual varieties Cusp Component Node Component Summary of Results Representation Cusp Node Jth Node Hyperdeterminant WZ WZ WZ C k 1 ⊗ . . . ⊗ C k r Dual Grassmannian M,S H,M,S S � k C N � k C N ⊗ C M S S S � k 1 C N 1 ⊗ . . . ⊗ � k r C N r partial S partial WZ: Weyman, Zelevinsky 1996 H: Holweck 2011, M: Maeda 2001 S: Sen Emre S ¸EN Northeastern University Singularities of dual varieties associated to exterior representations