Multigraded Hurwitz forms

1. Problem Statement

The paper addresses a gap in elimination theory regarding the characterization of linear spaces that intersect a projective variety non-transversally.

• Background: The classical Chow form (1937) describes linear spaces that intersect a variety $X$ non-trivially. The Hurwitz form (introduced in 2017) describes linear spaces that intersect $X$ non-transversally (i.e., tangentially or with higher multiplicity).
• The Gap: While multigraded versions of Chow forms exist for varieties in products of projective spaces ($\mathbb{P}^{n_1} \times \dots \times \mathbb{P}^{n_\ell}$), a corresponding theory for multigraded Hurwitz forms was missing.
• Objective: To define multigraded Hurwitz forms for varieties in a product of projective spaces, derive explicit degree formulas, and apply these tools to problems in algebraic game theory (Nash equilibria) and particle physics (Feynman integrals).

2. Methodology

The authors employ a combination of algebraic geometry, combinatorics, and computational algebra.

• Geometric Definition:

  • Let $X \subset \mathbb{P} = \mathbb{P}^{n_1} \times \dots \times \mathbb{P}^{n_\ell}$ be an irreducible variety of codimension $c$.
  • They define a multigraded Hurwitz incidence variety $\Phi_X^\alpha$ consisting of pairs $(p, L)$ where $L = L_1 \times \dots \times L_\ell$ is a product of linear spaces (with $L_i \in G(\alpha_i, n_i)$) such that $L$ intersects $X$ non-transversally at $p$, but transversally on all proper sub-products.
  • The multigraded Hurwitz form $Hu_X^\alpha$ is the defining polynomial of the projection of $\Phi_X^\alpha$ onto the product of Grassmannians $G_\alpha$.

• Algebraic Construction (Proposition 2.7):

  • The authors provide an algorithm to compute the ideal of the Hurwitz incidence. This involves:
    1. Constructing the ideal of the incidence variety $W$ (points $p \in L \cap X$).
    2. Using Jacobian matrices to detect non-transversality (singularities of the intersection).
    3. Using saturation and elimination to remove singular loci and project onto the Grassmannian coordinates (Plücker coordinates).

• Degree Analysis:

  • The authors introduce the multisectional genus $g(X)$, a polynomial encoding the geometric genus of generic curve sections of $X$.
  • They define polynodal varieties, a class where generic curve sections have at most nodal singularities (generalizing the condition for classical Hurwitz forms).
  • They utilize Khovanskii's theorem and volume polynomials of Newton polytopes to compute invariants for toric varieties.

• Computational Implementation:

  • The authors implemented their formulas in Macaulay2 (package MultiHurwitz.m2), allowing for the computation of multidegrees and multisectional genera for both complete intersections and general varieties defined by ideals.

3. Key Contributions

A. Theoretical Framework

• Definition: Formalized the concept of multigraded Hurwitz forms $Hu_X^\alpha$ for varieties in $\mathbb{P}^{n_1} \times \dots \times \mathbb{P}^{n_\ell}$.
• Degree Formula (Theorem 3.4): Established an upper bound for the degree of the Hurwitz form:
  $$u_i \leq 2(g_{\alpha+e_i} + \delta_\alpha - 1)$$
  where $\delta_\alpha$ is the coefficient of the multidegree and $g_{\alpha+e_i}$ is the multisectional genus. Equality holds if $X$ is polynodal.
• Complete Intersections (Theorem 4.1): Derived an explicit combinatorial formula for the degree of Hurwitz forms for generic complete intersections defined by a matrix of degrees $B = (b_{ij})$.

B. Applications

• Algebraic Game Theory (Section 5):
  • Characterized Nash equilibria in totally mixed games using multigraded Hurwitz forms.
  • The Nash discriminant (identifying degenerate equilibria) is shown to be a multigraded Hurwitz form of a specific toric variety.
  • Provided degree formulas for Nash discriminants in games with arbitrary numbers of players and strategies.
• Particle Physics (Section 6):
  • Connected the theory to Landau analysis for Feynman integrals.
  • Identified LS discriminants (leading singularities) as multigraded Hurwitz forms of incidence varieties of lines in $\mathbb{P}^3$ (embedded in Grassmannians $G(1,3)$).
  • Demonstrated how graph structures in Feynman diagrams correspond to the geometry of these incidence varieties.

C. Computational Tools

• Released the software package MultiHurwitz.m2 on Zenodo, providing functions to compute multidegrees, multisectional genera, and Hurwitz degrees for both ideal-based and polytope-based inputs.

4. Key Results

1. Explicit Degree Formula for Complete Intersections:
   For a generic complete intersection defined by degrees $b_{ij}$, the degree $u_i$ of the Hurwitz form in the $i$-th Grassmannian factor is:
   $$u_i = 2\delta_\alpha + \sum_{j} \delta_{\alpha+e_i-e_j} \left( -\alpha_j - 1 - \delta_{ij} + \sum_{p=1}^c b_{pj} \right)$$
   This allows for the prediction of the complexity of the discriminant without explicit construction.

2. Toric Varieties and Mixed Discriminants:
   For toric varieties associated with Newton polytopes, the multidegree and multisectional genus can be derived from volume polynomials and lattice point counting (Khovanskii's theorem). This links the Hurwitz degree directly to the combinatorics of the underlying polytopes.

3. Case Study: The 4-fold in $\mathbb{P}^3 \times \mathbb{P}^3$:
   The paper analyzes a specific toric 4-fold defined by $2 \times 2$ minors.

   • Multidegree: $[X] = 2T_1^2 + 4T_1T_2 + 2T_2^2$.
   • Expected Degree: Based on the formula, the Hurwitz form $Hu_X^{(1,1)}$ was expected to have degree $(8,8)$.
   • Actual Degree: The explicit calculation yielded $(6,6)$.
   • Explanation: The variety is not polynodal; the generic curve section has cusps rather than just nodes. The degree formula must be adjusted by subtracting contributions from non-nodal singularities (specifically, cusps reduce the degree by 1 per cusp in the relevant projection).

4. Nash Equilibria:
   For a game with $\ell$ players and binary choices, the degree of the Nash discriminant is bounded by a combinatorial sum involving derangements. For $\ell=4$, the degree is $(20, 20, 20, 20)$.

5. Significance

• Unification of Elimination Theory: The paper completes the "square" of elimination theory by adding the multigraded Hurwitz form to the existing framework of Chow forms and classical discriminants.
• New Tool for Applied Mathematics: It provides a rigorous algebraic framework for analyzing degeneracies in:
  • Game Theory: Identifying when Nash equilibria bifurcate or merge.
  • Physics: Analyzing the singularities of Feynman integrals (Landau singularities), which are crucial for understanding particle scattering amplitudes.
• Algorithmic Advancement: The explicit degree formulas allow researchers to estimate the complexity of discriminants (which can be massive, e.g., 10,156 terms in the introduction) before attempting to compute them, guiding the design of efficient algorithms.
• Geometric Insight: The distinction between polynodal and non-polynodal varieties highlights the subtle role of singularity types (nodes vs. cusps) in determining the degree of discriminants in multigraded settings.

In summary, this work extends classical algebraic geometry into the multigraded setting, providing both theoretical formulas and practical computational tools for solving complex problems in physics and economics.