Leading terms of generalized Pl\"ucker formulas

TL;DR

This paper analyzes generalized Pl"ucker formulas, which count tangent lines of hypersurfaces, and determines their leading terms using a recursive computational method, extending classical enumerative geometry results.

Contribution

It extends previous recursive methods to explicitly compute the leading terms of generalized Pl"ucker formulas for hypersurfaces.

Findings

Generalized Pl"ucker numbers are polynomial in degree d.

Leading terms of these polynomials are explicitly determined.

Method extends classical enumerative geometry results.

Abstract

Generalized Pl\"ucker numbers are defined to count certain types of tangent lines of generic degree $d$ complex projective hypersurfaces. They can be computed by identifying them as coefficients of GL(2)-equivariant cohomology classes of certain invariant subspaces, the so-called coincident root strata, of the vector space of homogeneous degree $d$ complex polynomials in two variables. In an earlier paper L\'aszl\'o M. Feh\'er and the author gave a new, recursive method for calculating these classes. Using this method, we showed that -- similarly to the classical Pl\"ucker formulas counting the bitangents and flex lines of a degree $d$ plane curve -- generalized Pl\"ucker numbers are polynomials in the degree $d$. In this paper, by further analyzing our recursive formula, we determine the leading terms of all the generalized Pl\"ucker formulas.