Uniform Lower Bound for Intersection Numbers of 𝜓-Classes

Abstract

We approximate intersection numbers ⟨𝜓ᵈ¹₁⋯ 𝜓ᵈⁿₙ⟩𝑔,ₙ on Deligne-Mumford's moduli space M¯𝑔,ₙ of genus 𝑔 stable complex curves with n marked points by certain closed-form expressions in d₁, …, dₙ. Conjecturally, these approximations become asymptotically exact uniformly in 𝑑ᵢ when 𝑔 → ∞ and n remain bounded or grow slowly. In this note, we prove a lower bound for the intersection numbers in terms of the above-mentioned approximating expressions multiplied by an explicit factor λ(𝑔, 𝑛), which tends to 1 when 𝑔 → ∞ and d₁ +⋯+ dₙ₋₂ = o(𝑔).