Uniform Lower Bound for Intersection Numbers of $\psi$-Classes

Abstract

We approximate intersection numbers $⟨{\psi }_1^{d_1}\cdots {\psi }_n^{d_n}{⟩}_{g,n}$ on Deligne-Mumford's moduli space ${\overline{\mathcal{M}}}_{g,n}$ of genus $g$ stable complex curves with $n$ marked points by certain closed-form expressions in $d_1,\dots ,d_n$. Conjecturally, these approximations become asymptotically exact uniformly in $d_i$ when $g\to \infty$ and $n$ remains bounded or grows slowly. In this note we prove a lower bound for the intersection numbers in terms of the above-mentioned approximatingexpressions multiplied by an explicit factor $\lambda (g,n)$, which tends to $1$ when $g\to \infty$ and $d_1+\cdots +d_{n-2}=o(g)$.

Dedicated to D.B. Fuchs on the occasion of his 80th birthday