Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
The rational cohomology of $\overline{\mathcal{M}}_4$
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-6128OAI: oai:DiVA.org:kth-6128DiVA: diva2:10751
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

Open Access in DiVA

No full text

Search in DiVA

By author/editor
Bergström, JonasTommasi, Orsola
By organisation
Mathematics (Dept.)
Mathematics

Search outside of DiVA

GoogleGoogle Scholar

urn-nbn

Altmetric score

urn-nbn
Total: 112 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf