Universal polynomials for counts of secant planes to projective curves

TL;DR

This paper introduces a new method to derive explicit formulas for counting secant planes to projective curves using Segre classes and recursive techniques, advancing the algebraic geometry toolkit.

Contribution

It presents a novel recursive approach based on Segre classes and Hilbert schemes to compute secant plane counts for projective curves.

Findings

Derived explicit formulas for secant plane counts

Established a recursive computation method

Applied the approach to the projective line case

Abstract

In this article we provide another method for obtaining explicit formulas yielding counts of secant planes to a projective curve. We formulate the problem in terms of Segre classes of suitable bundles over the symmetric product of the curve and take advantage of the result of Ellingsrud, G\"ottsche, and Lehn concerning the structure of integrals of polynomials in Chern classes of tautological bundles over Hilbert schemes of points. We use this to set up a recursion starting from the easy to compute case of the projective line.