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Towards large genus asymptotics of intersection numbers on moduli spaces of curves. (English) Zbl 1327.14136
Let \(V_{g,n}\) be the Weil-Petterson volume of the moduli space \(\mathcal{M}_{g,n}\) of complex algebraic curves of genus \(g\) with \(n\) punctures. The paper under review studies the asymptotic behaviour of \(V_{g,n}\) as \(g \rightarrow \infty\). The main result is theorem 1.2, which states that there exists a constant \(C \in (0,\infty)\), such that for any given \(k \geq 1, n \geq 0\),
\[ V_{g,n} = C \frac {(2g-3+n)! (4 \pi^2)^{2g-3+n}} {\sqrt g} \left(1 + \sum_{i=1}^{k} \frac{c_n^{(i)}} {g^i} + O\left(\frac {1} {g^{k+1}}\right)\right) \]
as \(g \rightarrow \infty\). Each \(c_n^{(i)}\) is a polynomial in \(n\) of degree \(2i\) with coefficients in \(\mathbb{Q}[\pi ^{-2}, \pi ^2]\), effectively computable.
This result is consistent with the conjecture made by the second author in [“On the large genus asymptotics of Weil-Peterson volumes”, Preprint (2008) arXiv:0812.0544]. In fact, the conjecture being true, the constant \(C\) would be \(\frac {1}{\sqrt \pi}\).
The proof of the result is made by studying the quotient \(\frac {V_{g+1,n}} {V_{g,n}}\) for a given \(n\), and \(g \rightarrow \infty\). Estimates for this ratio were obtained by the first author in [J. Differ. Geom. 94, No. 2, 267–300 (2013; Zbl 1270.30014)].
In the final section of the paper, the authors consider the behaviour of \(V_{g,n(g)}\) when \(n(g) \rightarrow \infty\) as \(g \rightarrow \infty\) but \(n(g)^2/g \rightarrow 0\).

MSC:
14H10 Families, moduli of curves (algebraic)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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