Asymptotic directions in the moduli space of curves
Abstract
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 $\zeta \in H^1(T_C)$ (with respect to the infinitesimal deformation map) is less than the Clifford index of the curve $C$, then $\zeta$ is not asymptotic. If the rank of $\zeta$ is equal to the Clifford index of the curve, we give sufficient conditions ensuring that the infinitesimal deformation $\zeta$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.08641
- arXiv:
- arXiv:2405.08641
- Bibcode:
- 2024arXiv240508641C
- Keywords:
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- Mathematics - Algebraic Geometry;
- 14H10;
- 14H15;
- 32G20