# 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in the complete intersection of two quadrics to the set of degree 0 line bundles on E. This approach was taken in an unpublished manuscript of Ragnar-Olaf Buchweitz and Frank-Olaf Schreyer.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.

## Vector Bundles

• randomLineBundle -- a random line bundle on the hyperelliptic curve
• vectorBundleOnE -- creates a VectorBundleOnE, represented as a matrix factorization
• yAction -- defines a vector bundle on E
• tensorProduct -- tensor product of sheaves on the elliptic curve or sheaf times CliffordModule
• degOnE -- degree of a vector bundle on E
• orderInPic -- order of a line bundle of degree 0 in Pic(E)
• randomExtension -- a random extension of a vector bundle on E by another vector bundle

## 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
• "centers(List,List)" -- see centers -- even and odd action of the center of the even Clifford algebra
• "ciModuleToCliffordModule(Module)" -- see ciModuleToCliffordModule -- transforms a module over a complete intersection of 2 quadrics into a Clifford Module.
• "ciModuleToMatrixFactorization(Module)" -- see ciModuleToMatrixFactorization -- transforms a module over a complete intersection of 2 quadrics into a matrix factorization
• "cliffordModule(List,List)" -- see cliffordModule -- makes a clifford Module
• "cliffordModule(Matrix,Matrix,Ring)" -- see cliffordModule -- makes a clifford Module
• "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
• "cliffordOperators(Matrix,Matrix,Ring)" -- see cliffordOperators -- Generators for a Clifford Algebra
• "degOnE(Matrix)" -- see degOnE -- degree of a vector bundle on E
• "degOnE(VectorBundleOnE)" -- see degOnE -- degree of a vector bundle on E
• "matrixFactorizationK(Matrix,Matrix)" -- see matrixFactorizationK -- Knoerrer matrix factorization from a bilinear form X*transpose Y
• "orderInPic(Matrix)" -- see orderInPic -- order of a line bundle of degree 0 in Pic(E)
• "orderInPic(VectorBundleOnE)" -- see orderInPic -- order of a line bundle of degree 0 in Pic(E)
• "randNicePencil(Ring,ZZ)" -- see randNicePencil -- sets up a random pencil of quadrics, and returns a hash table of the type RandomNicePencil.
• "randomExtension(Matrix,Matrix)" -- see randomExtension -- a random extension of a vector bundle on E by another vector bundle
• "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
• "randomLineBundle(RingElement)" -- see randomLineBundle -- a random line bundle on the hyperelliptic curve
• "randomLineBundle(ZZ,RingElement)" -- see randomLineBundle -- a random line bundle on the hyperelliptic curve
• "randomNicePencil(Ring,ZZ)" -- see randomNicePencil -- sets up a random example to construct Clifford algebra and representation
• "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(Matrix,Matrix)" -- 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
• "vectorBundleOnE(Matrix)" -- see vectorBundleOnE -- creates a VectorBundleOnE, represented as a matrix factorization
• Symbols

## For the programmer

The object PencilsOfQuadrics is .