Chern Numbers of Smooth Varieties via Homotopy Continuation and Intersection Theory

TL;DR

This paper introduces a numerical method combining homotopy continuation and intersection theory to compute Chern numbers of smooth projective varieties, demonstrated through practical examples.

Contribution

It develops a novel approach that integrates numerical and theoretical tools for calculating Chern numbers of smooth varieties.

Findings

Successfully computes Chern numbers for several examples

Demonstrates the effectiveness of combining homotopy continuation with intersection theory

Provides a practical framework for numerical algebraic geometry

Abstract

Homotopy continuation provides a numerical tool for computing the equivalence of a smooth variety in an intersection product. Intersection theory provides a theoretical tool for relating the equivalence of a smooth variety in an intersection product to the degrees of the Chern classes of the variety. A combination of these tools leads to a numerical method for computing the degrees of Chern classes of smooth projective varieties in P^n. We illustrate the approach through several worked examples.