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Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2009 (English)In: Documenta Mathematica, ISSN 1431-0643, E-ISSN 1431-0635, Vol. 14, 259-296 p.Article in journal (Refereed) Published
Abstract [en]

We consider the moduli space H-g,H-n of n-pointed smooth hyperelliptic curves of genus g. In order to get cohomological information we wish to make S-n-equivariant counts of the numbers of points defined over finite fields of this moduli space. We find recurrence relations in the genus that these numbers fulfill. Thus, if we can make S-n-equivariant counts of H-g,H-n for low genus, then we can do this for every genus. Information about curves of genus 0 and 1 is then found to be sufficient to compute the answers for H-g,H-n for all g and for n <= 7. These results are applied to the moduli spaces of stable curves of genus 2 with up to 7 points, and this gives us the S-n-equivariant Galois (resp. Hodge) structure of their l-adic (resp. Betti) cohomology.

Place, publisher, year, edition, pages
2009. Vol. 14, 259-296 p.
Keyword [en]
Cohomology of moduli spaces of curves; curves over finite fields
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-6129ISI: 000268811200001OAI: oai:DiVA.org:kth-6129DiVA: diva2:10752
Note
QC 20100701Available from: 2006-09-18 Created: 2006-09-18 Last updated: 2010-07-01Bibliographically approved
In thesis
1. Point counts and the cohomology of moduli spaces of curves
Open this publication in new window or tab >>Point counts and the cohomology of moduli spaces of curves
2006 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

n this thesis we count the number of points defined over finite fields of certain moduli spaces of pointed curves. The aim is primarily to gain cohomological information.

Paper I is joint work with Orsola Tommasi. Here we present details of the method of finding cohomological information on moduli spaces of curves by counting points. Another method of determining the cohomology of moduli spaces of curves is also presented. It is by stratifying them into pieces that are quotients of complements of discriminants in complex vector spaces. Results obtained by these two methods allow us to compute the Hodge structure of the cohomology of $\overline{\mathcal{M}}_4$.

In Paper II we consider the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyper-elliptic curves of genus $g$. We find that there are recursion formulas in the genus that the numbers of points of $\mathcal{H}_{g,n}$ fulfill. Thus, if we can make $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus zero and one is then found to be sufficient to compute the answers for hyperelliptic curves of all genera and with up to seven points. These results are applied to $\overline{\mathcal{M}}_{2,n}$ for $n$ up to seven, and give us the $\mathbb{S}_n$-equivariant Hodge structure of their cohomology. Moreover, we find that the $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ depend upon whether the characteristic is even or odd, where the first instance of this dependence is for six-pointed curves of genus three.

In Paper III we consider the moduli space $\mathcal{Q}_{n}$ of smooth $n$-pointed nonhyperelliptic curves of genus three. Using the canonical embedding of these curves as plane quartics, we make $\mathbb{S}_n$-equivariant counts of the numbers of points of $\mathcal{Q}_{n}$ for $n$ up to seven. We also count pointed plane cubics. This gives us $\mathbb{S}_n$-equivariant counts of the moduli space $\mathcal{M}_{1,n}$ for $n$ up to ten. We can then determine the $\mathbb{S}_n$-equivariant Hodge structure of the cohomology of $\overline{\mathcal{M}}_{3,n}$ for $n$ up to five.

n this thesis we count the number of points defined over finite fields of certain moduli spaces of pointed curves. The aim is primarily to gain cohomological information.

Paper I is joint work with Orsola Tommasi. Here we present details of the method of finding cohomological information on moduli spaces of curves by counting points. Another method of determining the cohomology of moduli spaces of curves is also presented. It is by stratifying them into pieces that are quotients of complements of discriminants in complex vector spaces. Results obtained by these two methods allow us to compute the Hodge structure of the cohomology of $\overline{\mathcal{M}}_4$.

In Paper II we consider the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyper-elliptic curves of genus $g$. We find that there are recursion formulas in the genus that the numbers of points of $\mathcal{H}_{g,n}$ fulfill. Thus, if we can make $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus zero and one is then found to be sufficient to compute the answers for hyperelliptic curves of all genera and with up to seven points. These results are applied to $\overline{\mathcal{M}}_{2,n}$ for $n$ up to seven, and give us the $\mathbb{S}_n$-equivariant Hodge structure of their cohomology. Moreover, we find that the $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ depend upon whether the characteristic is even or odd, where the first instance of this dependence is for six-pointed curves of genus three.

In Paper III we consider the moduli space $\mathcal{Q}_{n}$ of smooth $n$-pointed nonhyperelliptic curves of genus three. Using the canonical embedding of these curves as plane quartics, we make $\mathbb{S}_n$-equivariant counts of the numbers of points of $\mathcal{Q}_{n}$ for $n$ up to seven. We also count pointed plane cubics. This gives us $\mathbb{S}_n$-equivariant counts of the moduli space $\mathcal{M}_{1,n}$ for $n$ up to ten. We can then determine the $\mathbb{S}_n$-equivariant Hodge structure of the cohomology of $\overline{\mathcal{M}}_{3,n}$ for $n$ up to five.

n this thesis we count the number of points defined over finite fields of certain moduli spaces of pointed curves. The aim is primarily to gain cohomological information.

Paper I is joint work with Orsola Tommasi. Here we present details of the method of finding cohomological information on moduli spaces of curves by counting points. Another method of determining the cohomology of moduli spaces of curves is also presented. It is by stratifying them into pieces that are quotients of complements of discriminants in complex vector spaces. Results obtained by these two methods allow us to compute the Hodge structure of the cohomology of $\overline{\mathcal{M}}_4$.

In Paper II we consider the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyper-elliptic curves of genus $g$. We find that there are recursion formulas in the genus that the numbers of points of $\mathcal{H}_{g,n}$ fulfill. Thus, if we can make $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus zero and one is then found to be sufficient to compute the answers for hyperelliptic curves of all genera and with up to seven points. These results are applied to $\overline{\mathcal{M}}_{2,n}$ for $n$ up to seven, and give us the $\mathbb{S}_n$-equivariant Hodge structure of their cohomology. Moreover, we find that the $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ depend upon whether the characteristic is even or odd, where the first instance of this dependence is for six-pointed curves of genus three.

In Paper III we consider the moduli space $\mathcal{Q}_{n}$ of smooth $n$-pointed nonhyperelliptic curves of genus three. Using the canonical embedding of these curves as plane quartics, we make $\mathbb{S}_n$-equivariant counts of the numbers of points of $\mathcal{Q}_{n}$ for $n$ up to seven. We also count pointed plane cubics. This gives us $\mathbb{S}_n$-equivariant counts of the moduli space $\mathcal{M}_{1,n}$ for $n$ up to ten. We can then determine the $\mathbb{S}_n$-equivariant Hodge structure of the cohomology of $\overline{\mathcal{M}}_{3,n}$ for $n$ up to five.

Place, publisher, year, edition, pages
Stockholm: KTH, 2006
Series
Trita-MAT, ISSN 1401-2286
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-4105 (URN)91-7178-447-0 (ISBN)
Public defence
2006-09-29, F3, Lindstedtsvägen 26, Stockholm, 14:00
Opponent
Supervisors
Note
QC 20100701Available from: 2006-09-18 Created: 2006-09-18 Last updated: 2010-07-01Bibliographically approved

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