Asymptotic directions in the moduli space of curves

TL;DR

The paper develops a framework to study asymptotic directions in the moduli space of genus $g$ curves via the second fundamental form $II$ of the Torelli map, reframing the problem through the Hodge Gaussian map and Dolbeault cohomology. It proves that deformations of rank $d$ below the Clifford index ${ m Cliff}(C)$ cannot be asymptotic and provides criteria for the borderline case $d={ m Cliff}(C)$, yielding a near-complete classification of rank-1 and rank-2 asymptotics. It provides concrete instances from trigonal and bielliptic curves, including explicit descriptions of rank-1 directions and nontrivial rank-2 directions, and it analyzes Maroni-special trigonal curves to produce explicit normal forms and loci where asymptotic directions occur beyond Schiffer-type variations. Overall, the work links base loci of a quadric system $II(I_2)$ to geometric properties of curves (Clifford index, trigonal/Maroni structure, biellipticity) and highlights potential applications to the geometry of the Torelli locus and related conjectures.

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 $ζ\in H^1(T_C)$ (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.

Theorems & Definitions (82)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Remark 3.1
• proof