An Ehrhart Theory For Tautological Intersection Numbers

TL;DR

This paper reveals that tautological intersection numbers on moduli spaces of stable curves can be expressed as evaluations of Ehrhart polynomials, connecting algebraic geometry with combinatorial polytope theory.

Contribution

It establishes a novel link between tautological intersection numbers and Ehrhart polynomials, using Virasoro constraints and classification theorems for partial polytopal complexes.

Findings

Tautological intersection numbers are evaluations of Ehrhart polynomials.

The Virasoro constraints translate into recursions for integer-valued polynomials.

In low dimensions, the associated polytopal complexes are inside-out polytopes.

Abstract

We discover that tautological intersection numbers on $\bar{\mathcal{M}}_{g, n}$, the moduli space of stable genus $g$ curves with $n$ marked points, are evaluations of Ehrhart polynomials of partial polytopal complexes. In order to prove this, we realize the Virasoro constraints for tautological intersection numbers as a recursion for integer-valued polynomials. Then we apply a theorem of Breuer that classifies Ehrhart polynomials of partial polytopal complexes by the nonnegativity of their $f^*$-vector. In dimensions 1 and 2, we show that the polytopal complexes that arise are \emph{inside-out polytopes} i.e. polytopes that are dissected by a hyperplane arrangement.