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Uniform lower bound for intersection numbers of \(\psi\)-classes. (English) Zbl 1457.14012

Let \(\overline{\mathcal{M}}_{g,n}\) be the Deligne-Mumford moduli space of genus \(g\) complex stable algebraic curves (possibly with nodes), with \(n>0\) distinct labeled marked points. The aim of this paper is to approximate intersection numbers \(\langle\psi_1^{d_1}\dots\psi_n^{d_n}\rangle_{g,n}\) on \(\overline{\mathcal{M}}_{g,n}\) with \(n\) marked points by certain closedform expressions in \(d_1,\dots,d_n\). Conjecturally, these approximations become asymptotically exact uniformly in \(d_i\) when \(g\rightarrow\infty\) and \(n\) remains bounded or grows slowly. The authors prove a lower bound for the intersection numbers in terms of the above-mentioned approximating expressions multiplied by an explicit factor \(\lambda(g,n)\), which tends to \(1\) when \(g\rightarrow\infty\) and \(d_1+ \cdots+d_{n-2}=o(g)\).

MSC:

14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14H70 Relationships between algebraic curves and integrable systems
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