Degenerations and multiplicity-free formulas for products of $\psi$ and $\omega$ classes on $\overline{M}_{0,n}$

TL;DR

This paper develops a combinatorial and geometric framework for degenerating products of psi and omega classes on moduli spaces of stable rational curves, providing positive, multiplicity-free formulas and connections to permutation patterns.

Contribution

It introduces an inductive degeneration method and a 'slide labeling' algorithm to express products as boundary strata sums, and relates multidegrees to permutation pattern avoidance.

Findings

Constructed flat degenerations with boundary strata as special fibers.

Derived combinatorial formulas for products and kappa classes.

Linked multidegrees to permutation pattern avoidance and tournament interpretations.

Abstract

We consider products of $\psi$ classes and products of $\omega$ classes on $\overline{M}_{0,n+3}$. For each product, we construct a flat family of subschemes of $\overline{M}_{0,n+3}$ whose general fiber is a complete intersection representing the product, and whose special fiber is a generically reduced union of boundary strata. Our construction is built up inductively as a sequence of one-parameter degenerations, using an explicit parametrized collection of hyperplane sections. Combinatorially, our construction expresses each product as a positive, multiplicity-free sum of classes of boundary strata. These are given by a combinatorial algorithm on trees we call 'slide labeling'. As a corollary, we obtain a combinatorial formula for the $\kappa$ classes in terms of boundary strata. For degree-$n$ products of $\omega$ classes, the special fiber is a finite reduced union of (boundary) points, and its cardinality is one of the multidegrees of the corresponding embedding $\Omega_n: \overline{M}_{0,n+3}\to \mathbb{P}^1\times \cdots \times \mathbb{P}^n$. In the case of the product $\omega_1\cdots \omega_n$, these points exhibit a connection to permutation pattern avoidance. Finally, we show that in certain cases, a prior interpretation of the multidegrees via tournaments can also be obtained by degenerations.