Segre Class Computation and Practical Applications

TL;DR

This paper presents new effective methods for computing Segre classes and intersection products in projective varieties, enabling algebraic multiplicity and containment testing without Groebner bases, thus improving computational algebraic geometry tools.

Contribution

It introduces algorithms for Segre class computation and applications that avoid Groebner bases, enhancing efficiency and practicality in algebraic geometry computations.

Findings

Computed Segre classes as classes in the Chow group of toric varieties.

Provided algorithms for intersection product computation and algebraic multiplicity determination.

Enabled containment testing of subvarieties without local ring calculations.

Abstract

Let $X \subset Y$ be closed (possibly singular) subschemes of a smooth projective toric variety $T$. We show how to compute the Segre class $s(X,Y)$ as a class in the Chow group of $T$. Building on this, we give effective methods to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of $T$. Our methods may be implemented without using Groebner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used.