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PencilsOfQuadrics :: PencilsOfQuadrics

PencilsOfQuadrics -- Clifford Algebra of a Pencil of quadratic forms on PP^(2g+1)

Description

The Clifford algebra forms a link between the intersection of two quadrics X and a hyperelliptic curve E. For example, one can recover the coordinate ring of the hyperelliptic curve as the center of the even Clifford algebra. Using a maximal linear subspace contained in the intersection, we get a Morita bundle that connects graded modules over the coordinate ring of the hyperelliptic curve and modules over the even Clifford algebra. This leads to a proof of Reid's theorem which identifies the set of maximal isotropic subspaces in the complete intersection of two quadrics to the set of degree 0 line bundles on E. This approach was taken in an unpublished manuscript of Ragnar-Olaf Buchweitz and Frank-Olaf Schreyer. The package allows a computational approach to the result of Bondal and Orlov which showed that the Kuznetsov component of X and the derived category of E are equivalent by a Fourier-Mukai transformation (see Section 2 of [A. Bondal, D. Orlov, arXiv:alg-geom/9506012], or Section 6 of [A. Bondal, D. Orlov, Proceedings of ICM, Vol. II (Beijing, 2002)]).

We demonstrate this, over finite fields, with the constructions of further random linear spaces on the intersection of two quadrics, and random Ulrich modules of lowest possible rank on the complete intersection of two quadrics for small g.

Types

Basic Construction of the Clifford Algebra

Vector Bundles

Clifford Modules

Computations using Clifford Algebras

Authors

Version

This documentation describes version 0.9 of PencilsOfQuadrics.

Source code

The source code from which this documentation is derived is in the file PencilsOfQuadrics.m2.

Exports

  • Types
  • Functions and commands
  • Methods
    • cliffordModuleToCIResolution(CliffordModule,Ring,Ring), see cliffordModuleToCIResolution -- transforms a Clifford module to a resolution over a complete intersection ring
    • cliffordModuleToMatrixFactorization(CliffordModule,Ring), see cliffordModuleToMatrixFactorization -- reads off a matrix factorization from a Clifford module
    • degOnE(VectorBundleOnE), see degOnE -- degree of a vector bundle on E
    • orderInPic(VectorBundleOnE), see orderInPic -- order of a line bundle of degree 0 in Pic(E)
    • randomExtension(VectorBundleOnE,VectorBundleOnE), see randomExtension -- a random extension of a vector bundle on E by another vector bundle
    • randomIsotropicSubspace(CliffordModule,PolynomialRing), see randomIsotropicSubspace -- choose a random isotropic subspace
    • searchUlrich(CliffordModule,Ring), see searchUlrich -- searching an Ulrich module of smallest possible rank
    • tensorProduct(CliffordModule,VectorBundleOnE), see tensorProduct -- tensor product of sheaves on the elliptic curve or sheaf times CliffordModule
    • tensorProduct(VectorBundleOnE,VectorBundleOnE), see tensorProduct -- tensor product of sheaves on the elliptic curve or sheaf times CliffordModule
    • translateIsotropicSubspace(CliffordModule,VectorBundleOnE,PolynomialRing), see translateIsotropicSubspace -- choose a random isotropic subspace
  • Symbols