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

\[0\subset B_1\subset t^*F,\quad 0\subset A_1\subset A_2 \subset t^*E\]

denote the universal flag of subsheaves. As \(H^*(S,{\mathbb Q})\)-algebra, the cohomology ring of \(T\) is generated by the Chern roots

\[b_1=c_1(B_1),\; b_2=c_1(t^*F/B_1),\quad\quad a_1=c_1(A_1), a_2=c_1(A_2/A_1), a_3=c_1(t^*E/A_2).\]

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

\[B_1\to t^*F\to t^*E\otimes W\to (t^*E/A_1)\otimes W\text{ and }t^*F\to t^*E\otimes W\to (t^*E/A_2)\otimes W\]

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

\[c_8{\mathcal H}om(B_1,(t^*E/A_1)\otimes W)=0,\quad c_8{\mathcal H}om(t^*F, (t^*E/A_2)\otimes W)=0.\]

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,

\[0 \rightarrow\mathcal{O}_X\rightarrow End(E)\oplus End(F)\rightarrow SHom(F,E\otimes W)\rightarrow TX\rightarrow 0\]

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)
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
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)
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