Self-intersection of the Torelli map
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916950877143040 |
|---|---|
| author | Drakengren, Lycka |
| author_facet | Drakengren, Lycka |
| contents | The Torelli map $t\colon \mathcal{M}^{ct}_g \to \mathcal{A}_g$ is far from an immersion for $g\geq 3$: the self-fiber product of the Torelli map for $g\geq 3$ has several components with nontrivial intersections. We give a stratification of the self-fiber product for arbitrary genus and describe how components in the fiber product intersect. In genus $4$, the Torelli fiber product is nonreduced, which we prove by analyzing the expansion of the period map near a nodal curve. We use the geometry of the Torelli fiber product to: Calculate the class of the pullback to $\mathcal{M}^{ct}_4$ of the Torelli cycle $t_*[\mathcal{M}^{ct}_4]$ on $\mathcal{A}_4$; Find the class $t_*[\overline{\mathcal{M}}_4]$ for suitable toroidal compactifications $\overline{\mathcal{A}}_4$; Calculate the class $t^*t_*[\mathcal{M}^{ct}_5]|_{\mathcal{M}_5}$. In the first appendix, we write down a calculation for finding the Chern classes of $\overline{\mathcal{M}}_{g,n}$. In the second, we give a formula for a coefficient occurring in an intersection of excess dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12449 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Self-intersection of the Torelli map Drakengren, Lycka Algebraic Geometry 14C17, 32G20, 14H10, 14K10 The Torelli map $t\colon \mathcal{M}^{ct}_g \to \mathcal{A}_g$ is far from an immersion for $g\geq 3$: the self-fiber product of the Torelli map for $g\geq 3$ has several components with nontrivial intersections. We give a stratification of the self-fiber product for arbitrary genus and describe how components in the fiber product intersect. In genus $4$, the Torelli fiber product is nonreduced, which we prove by analyzing the expansion of the period map near a nodal curve. We use the geometry of the Torelli fiber product to: Calculate the class of the pullback to $\mathcal{M}^{ct}_4$ of the Torelli cycle $t_*[\mathcal{M}^{ct}_4]$ on $\mathcal{A}_4$; Find the class $t_*[\overline{\mathcal{M}}_4]$ for suitable toroidal compactifications $\overline{\mathcal{A}}_4$; Calculate the class $t^*t_*[\mathcal{M}^{ct}_5]|_{\mathcal{M}_5}$. In the first appendix, we write down a calculation for finding the Chern classes of $\overline{\mathcal{M}}_{g,n}$. In the second, we give a formula for a coefficient occurring in an intersection of excess dimension. |
| title | Self-intersection of the Torelli map |
| topic | Algebraic Geometry 14C17, 32G20, 14H10, 14K10 |
| url | https://arxiv.org/abs/2509.12449 |