Leading terms of generalized Pl\"ucker formulas

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.