Cohomology of moduli spaces of curves of genus three via point counts

Details Cohomology of moduli spaces of curves of genus three via point counts Cohomology of moduli spaces of curves of genus three via point counts

2006-11-27

arxiv.org/abs/math/0611815v2

Mathematics

Algebraic Geometry

25 pages, shortened version

Scientific paper

In this article we consider the moduli space of smooth $n$-pointed non-hyperelliptic curves of genus 3. In the pursuit of cohomological information about this space, we make $\mathbb{S}_n$-equivariant counts of its numbers of points defined over finite fields for $n \leq 7$. Combining this with results on the moduli spaces of smooth pointed curves of genus 0, 1 and 2, and the moduli space of smooth hyperelliptic curves of genus 3, we can determine the $\mathbb{S}_n$-equivariant Galois and Hodge structure of the ($\ell$-adic respectively Betti) cohomology of the moduli space of stable curves of genus 3 for $n \leq 5$ (to obtain $n \leq 7$ we would need counts of ``8-pointed curves of genus 2'').

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