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Multigraded Hurwitz forms

Elizabeth Pratt, Luca Sodomaco, Bernd Sturmfels

Abstract

The Hurwitz form of a projective variety characterizes linear spaces of complementary dimension which meet the variety non-transversally. We extend this notion to varieties in a product of projective spaces. This parallels the multigraded Chow forms due to Osserman and Trager. We study the degrees of multigraded Hurwitz forms. An explicit degree formula is given for complete intersections. This offers a new tool for elimination theory that has many applications, ranging from Nash equilibria to Feynman integrals.

Multigraded Hurwitz forms

Abstract

The Hurwitz form of a projective variety characterizes linear spaces of complementary dimension which meet the variety non-transversally. We extend this notion to varieties in a product of projective spaces. This parallels the multigraded Chow forms due to Osserman and Trager. We study the degrees of multigraded Hurwitz forms. An explicit degree formula is given for complete intersections. This offers a new tool for elimination theory that has many applications, ranging from Nash equilibria to Feynman integrals.
Paper Structure (6 sections, 7 theorems, 79 equations, 6 figures)

This paper contains 6 sections, 7 theorems, 79 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Foundations and Examples
  3. Degree Formula
  4. Complete Intersections and Software
  5. Polytopes and Nash Equilibria
  6. Grassmannians and Particle Physics

Key Result

Proposition 2.7

The ideal that defines the multigraded Hurwitz incidence $\Phi^\alpha_X$ equals The ideal of the Hurwitz locus $\mathcal{H}_X^\alpha$ is obtained from (eq:idealPhi) by eliminating all the variables $x_{i,j}$ in $R$. If this elimination ideal is principal then its generator is the Hurwitz form $\mathrm{Hu}^\alpha_X$.

Figures (6)

  • Figure 1: The resultants (left) and discriminants (right) attached to a fixed variety $X$.
  • Figure 2: Commands for multiGenera when $X$ is presented by a degree matrix $B$.
  • Figure 3: Commands for multiGenera when $X$ is presented by its ideal $I_X$.
  • Figure 4: The function multidegHurwitz for the two input formats in Figures \ref{['fig:multiGenera1']} and \ref{['fig:multiGenera2']}.
  • Figure 5: The commands from Figures \ref{['fig:multiGenera1']} and \ref{['fig:multiGenera3']} when $X = U_G$ for a graph $G$.
  • ...and 1 more figures

Theorems & Definitions (30)

  • Example 2.1: Toric $4$-fold
  • Example 2.2: Plane curves
  • Remark 2.3
  • Example 2.4: $\ell=1$
  • Example 2.5: Mixed discriminants
  • Example 2.6: Semi-mixed discriminants
  • Proposition 2.7
  • proof
  • Example 2.8
  • Example 3.1
  • ...and 20 more