Asymptotic directions in the moduli space of curves

TL;DR

This paper investigates the asymptotic directions in the tangent bundle of the moduli space of curves, providing examples, conditions for non-asymptotic directions based on rank and Clifford index, and characterizing low-rank asymptotic directions.

Contribution

It introduces new criteria for identifying asymptotic directions in the moduli space of curves based on rank and Clifford index, and classifies low-rank asymptotic directions.

Findings

Examples of asymptotic directions for all g ≥ 4

Asymptotic directions with rank less than Clifford index are not asymptotic

Complete description of rank 1 asymptotic directions

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.