Twisted Cubics¶
The variety of nets of quadrics defining twisted cubics relative to a base.
We briefly explain the algorithm of Ellingsrud and Strømme in order to compute the variety of nets of quadrics defining twisted cubics as well as the incidence variety relative to a base.
Let \(E_0\) and \(F_0\) denote vector spaces of dimension \(3\) and \(2\), respectively, considered as representations of the group \(G={\text{Gl}}_3\times{\text{Gl}}_2\) via the standard action of the left and right factor, respectively. If \(L=\det(E)\otimes \det(F)^*\), then the restriction of the representations \(E\otimes L^*\) and \(F\otimes L^*\) to the diagonal subgroup \(\Delta=\{(t I_3, tI_2)\;|\; t\in {\mathbb C}^*\}\) are trivial.
Let \(W\) denote a locally free sheaf on a scheme \(S\) and let \(U:={\text{SHom}}(E_0,F_0\otimes W)\) denote the \(S\)-scheme of linear maps \(A:E_0\to F_0\otimes _{\mathbb C} W_s\), \(s\in S\). The open subset \(U^{ss}\subset U\) of semistable points for the action of \(G\) on \(U\) consists of matrices \(A\) such that there are no non-zero subspaces \(F'\subset F_0\) and \(E'\subset E_0\) with \(A(F_0)\subset E_0\otimes W\) and \(\dim(E')/\dim(F')< \dim(E_0)/\dim(F_0)\). Let \(b:X_W\to S\) denote the GIT-quotient of \(U^{ss}\) by the action of \(G\).
The cohomology ring of \(X_W\) can be described as follows: The stabiliser subgroup of any point in \(U^{ss}\) is the diagonal group \(\Delta\). Since this group acts trivially on \({\mathcal O}_U\otimes (E_0\otimes L^*)\) and \({\mathcal O}_U\otimes (F_0\otimes L^*)\), these bundles descend to vector bundles \(E\) and \(F\) (of rank \(3\) and \(2\), resp.) on \(X_W\), and there is a tautological homomorphism \(a:F\to E\otimes b^*W\). The Chern classes \(e_1,e_2,e_3\) of \(E\) and \(f_1,f_2\) of \(F\) generate \(H^*(X_W,{\mathbb Q})\) as an \(H^*(S,{\mathbb Q})\)-algebra. One checks directly that \(\det(E)=\det(F)\) so that \(e_1=f_1\).
Let \(t:T\to X_W\) denote the variety of pairs of full flags of \(F\) and \(E\), and let
denote the universal flag of subsheaves. As \(H^*(S,{\mathbb Q})\)-algebra, the cohomology ring of \(T\) is generated by the Chern roots
In particular, the classes \(f_1,f_2\) and \(e_1,e_2,e_3\) are the elementary symmetric polynomials of the \(b_1,b_2\) and \(a_1,a_2,a_3\), respectively. The composite homomorphism
cannot vanish anywhere on \(T\) due to the semistability of the points on in \(X_W\). This implies that the top Chern class of the corresponding SHom-bundles must vanish, which yields the relations
The relations of the Chow-ring of \(X_W\) are now the coefficients of these Chern classes with respect to a basis of \(H^*(S,{\mathbb Q})[a_1,a_2,a_3,b_1,b_2]\) considered as a finite free module over \(H^*(S,{\mathbb Q})[f_1,f_2,e_1,e_1,e_3]\) plus the relation \(e_1=f_2\).
The tangent bundle is computed using the exact sequence,
This is implemented in variety_of_nets_of_quadrics()
.
The incidence variety can be obtained in a two-step process \(I_W={\mathbb P}(K_1^*)\xrightarrow{v} {\mathbb P}(W)\xrightarrow{u} S\), where \(0\to K_1\to u^*W\to L_1\to 0\) is the tautological sequence on \({\mathbb P}(W)\).
This is implemented in incidence_variety()
.
Also, let \(0\to K_2\to v^*K_1^*\to L_2\to 0\) be the tautological sequence on \(I_W\). Since the Chow ring of \(H^*(X_W)\) is generated by the Chern classes of \(E\) and \(F\), the embedding \(f:I_W\to X_W\) is determined as soon as one can identify \(f^*E\) and \(f^*F\). According to Ellingsrud and Strømme, these bundles are \(f^*E= K_1\otimes L_2^*\) and \(f^*F=K_2\otimes L_2^*\otimes \det(K_1)\).
Finally the morphism \(f\) is implemented in
map_incidence_to_nets_of_quadrics()
.
REFERENCE:
Ellingsrud, Geir and Strømme, Stein Arild: The number of twisted cubics on the general quintic threefold, Math. Scand. 76 (1995) 5-34
AUTHORS:
- Manfred Lehn (2013)
- Christoph Sorger (2013)
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sage.schemes.chow.library.twisted_cubics.
incidence_variety
(W, name=None, latex_name=None)¶ The Incidence variety I.
EXAMPLES:
sage: P = PointChowScheme sage: W = Bundle(P, 4, [1]) sage: I = incidence_variety(W) sage: I.dimension() 5 sage: I.euler_number() 12 sage: I.betti_numbers() [1, 2, 3, 3, 2, 1] sage: c5 = I.tangent_bundle().chern_classes()[5] sage: c5.integral() == I.euler_number() True sage: P = Grass(1, 5, 'w') sage: W = P.sheaves["universal_quotient"] sage: I = incidence_variety(W) sage: I.betti_numbers() [1, 3, 6, 9, 11, 11, 9, 6, 3, 1] sage: I.euler_number() 60 sage: top = I.tangent_bundle().chern_classes()[I.dimension()] sage: top.integral() == I.euler_number() True sage: G = Grass(6, 4, 'w') sage: W = G.sheaves["universal_quotient"] sage: I = incidence_variety(W) sage: I.betti_numbers() [1, 3, 7, 12, 18, 23, 26, 26, 23, 18, 12, 7, 3, 1] sage: I.euler_number() 180 sage: top = I.tangent_bundle().chern_classes()[I.dimension()] sage: top.integral() == I.euler_number() True
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sage.schemes.chow.library.twisted_cubics.
map_incidence_to_nets_of_quadrics
(W, domain_name=None, codomain_name=None, latex_domain_name=None, latex_codomain_name=None)¶ Returns the map \(f:I_S\rightarrow X_S\) from the incidence variety to the variety of nets of quadrics.
EXAMPLES:
sage: P = PointChowScheme sage: W = Bundle(P, 4, [1]) sage: f = map_incidence_to_nets_of_quadrics(W)
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sage.schemes.chow.library.twisted_cubics.
variety_of_nets_of_quadrics
(W, name=None, latex_name=None)¶ Returns the variety of nets of quadrics defining twisted cubics relative to \(S\) and a rank \(4\) vector bundle \(W\) of \(S\).
If S is given as parameter the variety will be relative to this instance of S.
EXAMPLES:
sage: P = PointChowScheme sage: W = Bundle(P, 4, [1]) sage: X = variety_of_nets_of_quadrics(W) sage: X.euler_number() 58 sage: top = X.tangent_bundle().chern_classes()[X.dimension()] sage: top.integral() == X.euler_number() True sage: P = Grass(1, 5, 'w') sage: W = P.sheaves["universal_quotient"] sage: X = variety_of_nets_of_quadrics(W) sage: X.euler_number() 290 sage: top = X.tangent_bundle().chern_classes()[X.dimension()] sage: top.integral() == X.euler_number() True