Asymptotic directions in the moduli space of curves
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913830198575104 |
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| author | Colombo, Elisabetta Frediani, Paola Pirola, Gian Pietro |
| author_facet | Colombo, Elisabetta Frediani, Paola Pirola, Gian Pietro |
| contents | In this paper we study asymptotic directions in the tangent bundle of the moduli space ${\mathcal M}_g$ of curves of genus $g$, namely those tangent directions that are annihilated by the second fundamental form of the Torelli map. We give examples of asymptotic directions for any $g \geq 4$. We prove that if the rank $d$ of a tangent direction $ζ\in H^1(T_C)$ (with respect to the infinitesimal deformation map) is less than the Clifford index of the curve $C$, then $ζ$ is not asymptotic. If the rank of $ζ$ is equal to the Clifford index of the curve, we give sufficient conditions ensuring that the infinitesimal deformation $ζ$ is not asymptotic. Then we determine all asymptotic directions of rank 1 and we give an almost complete description of asymptotic directions of rank 2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic directions in the moduli space of curves Colombo, Elisabetta Frediani, Paola Pirola, Gian Pietro Algebraic Geometry 14H10, 14H15, 32G20 In this paper we study asymptotic directions in the tangent bundle of the moduli space ${\mathcal M}_g$ of curves of genus $g$, namely those tangent directions that are annihilated by the second fundamental form of the Torelli map. We give examples of asymptotic directions for any $g \geq 4$. We prove that if the rank $d$ of a tangent direction $ζ\in H^1(T_C)$ (with respect to the infinitesimal deformation map) is less than the Clifford index of the curve $C$, then $ζ$ is not asymptotic. If the rank of $ζ$ is equal to the Clifford index of the curve, we give sufficient conditions ensuring that the infinitesimal deformation $ζ$ is not asymptotic. Then we determine all asymptotic directions of rank 1 and we give an almost complete description of asymptotic directions of rank 2. |
| title | Asymptotic directions in the moduli space of curves |
| topic | Algebraic Geometry 14H10, 14H15, 32G20 |
| url | https://arxiv.org/abs/2405.08641 |