Asymptotic directions in the moduli space of curves

Abstract In this paper, we study asymptotic directions in the tangent bundle of the moduli space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">M</m:mi> <m:mi>g</m:mi> </m:msub> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>≥</m:mo> <m:mn>4</m:mn> </m:mrow> </m:math> g\geq 4 . We prove that if the rank 𝑑 of a tangent direction <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ζ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>T</m:mi> <m:mi>C</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \zeta\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.