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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
##### Keywords:
Weil-Petersson volume; moduli space; algebraic curve
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##### References:
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