Cycle-level intersection theory for toric varieties

2003/06/09 by Hugh Thomas, Thomas, Hugh
Computer Science · Mathematics · #14C17 #14M25 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:14C17 #msc:14M25

paper · pdf · doi:10.48550/arxiv.math/0306144

24 pages, 3 figures

arxiv created 2003/06/09 · openalex publication_date 2003/06/09 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper addresses the problem of constructing a cycle-level intersection theory for toric varieties. We show that by making one global choice, we can determine a cycle representative for the intersection of an equivariant Cartier divisor with an invariant cycle on a toric variety. For a toric variety defined by a fan in N, the choice consists of giving an inner product or a complete flag for M_Q=Hom(N,Q), or more generally giving for each cone sigma in the fan a linear subspace of M_Q complementary to the subspace of M_Q perpendicular to sigma, satisfying certain compatibility conditions. We show that these intersection cycles have properties analogous to the usual intersections modulo rational equivalence. If X is simplicial (for instance, if X is non-singular), we obtain a commutative ring structure on the invariant cycles of X with rational coefficients. This ring structure determines cycles representing certain characteristic classes of the toric variety. We also discuss how to define intersection cycles that require no choices, at the expense of increasing the size of the coefficient field.

Related

• Cycles representing the Todd class of a toric variety
• Combinatorial Cycle Classes in the Intersection Cohomology of Projective Toric Varieties
• A case study of intersections on blowups of the moduli of curves
• Toric intersections
• Intersection Theory on Toric Varieties