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Plücker formulas using equivariant cohomology of coincident root strata

László M. Fehér, András P. Juhász

TL;DR

The paper develops an equivariant cohomology framework to study coincident root strata and uses it to prove that Plücker numbers counting tangent structures to generic hypersurfaces vary polynomially in the degree $d$. Central to the method is a recursive formula for the equivariant class $[\overline{Y}_\lambda(d)]$ that reduces to lower-length partitions via a divided-difference operator, enabling explicit leading-term and degree analyses. By translating to non-equivariant resolutions, the authors connect these polynomials to classical enumerative geometry, derive closed formulas for large families of Plücker numbers, and relate hyperflex and line-count problems (including Zagier-type results) to Kostka numbers and Schubert calculus. The work provides a unifying, computational approach to a broad class of incidence problems in projective geometry and offers tools for studying asymptotics, linear systems, and incidence varieties. Overall, it demonstrates how equivariant methods can yield precise, computable enumerative data for higher-order tangency problems in algebraic geometry.

Abstract

We give a new method to calculate the universal cohomology classes of coincident root loci. We show a polynomial behavior of them and apply this result to prove that generalized Plücker formulas are polynomials in the degree, just as the classical Plücker formulas counting the bitangents and flexes of a degree $d$ generic plane curve. We establish an upper bound for the degrees of these polynomials, and we calculate the leading terms of those whose degrees reach this upper bound. We believe that the paper is understandable without detailed knowledge of equivariant cohomology. It may serve as a demonstration of the use of equivariant cohomology in enumerative geometry through the examples of coincident root strata. We also explain how the equivariant method can be "translated" into the traditional non-equivariant method of resolutions.

Plücker formulas using equivariant cohomology of coincident root strata

TL;DR

The paper develops an equivariant cohomology framework to study coincident root strata and uses it to prove that Plücker numbers counting tangent structures to generic hypersurfaces vary polynomially in the degree . Central to the method is a recursive formula for the equivariant class that reduces to lower-length partitions via a divided-difference operator, enabling explicit leading-term and degree analyses. By translating to non-equivariant resolutions, the authors connect these polynomials to classical enumerative geometry, derive closed formulas for large families of Plücker numbers, and relate hyperflex and line-count problems (including Zagier-type results) to Kostka numbers and Schubert calculus. The work provides a unifying, computational approach to a broad class of incidence problems in projective geometry and offers tools for studying asymptotics, linear systems, and incidence varieties. Overall, it demonstrates how equivariant methods can yield precise, computable enumerative data for higher-order tangency problems in algebraic geometry.

Abstract

We give a new method to calculate the universal cohomology classes of coincident root loci. We show a polynomial behavior of them and apply this result to prove that generalized Plücker formulas are polynomials in the degree, just as the classical Plücker formulas counting the bitangents and flexes of a degree generic plane curve. We establish an upper bound for the degrees of these polynomials, and we calculate the leading terms of those whose degrees reach this upper bound. We believe that the paper is understandable without detailed knowledge of equivariant cohomology. It may serve as a demonstration of the use of equivariant cohomology in enumerative geometry through the examples of coincident root strata. We also explain how the equivariant method can be "translated" into the traditional non-equivariant method of resolutions.
Paper Structure (33 sections, 33 theorems, 150 equations)

This paper contains 33 sections, 33 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Plücker numbers
  3. Calculating Plücker numbers
  4. Results and structure of the paper
  5. Cohomological calculations
  6. The variety of tangent lines as a coincident root locus
  7. Calculating the cohomology class
  8. Equivariant cohomology classes of invariant subvarieties
  9. A recursive formula for [Ylambda(d)]
  10. Earlier formulas
  11. Proof of the recursion formula
  12. The construction of the covering space: twisting with a line bundle
  13. Conventions
  14. The twisted class
  15. The pushforward map pi!
  16. ...and 18 more sections

Key Result

Proposition 1.9

Let $\lambda=(2^{e_2},\dots,m^{e_m})$ be a partition without $1$'s, $n\geq |\tilde{\lambda}|+2$ and $f\in \mathop{\mathrm{Pol}}\nolimits^d({\mathbb C}^n)$ generic. Then where $t=\lfloor|\tilde{\lambda}|/2\rfloor$.

Theorems & Definitions (76)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Definition 1.4
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Remark 1.8
  • Proposition 1.9
  • Remark 1.10
  • ...and 66 more