Asymptotic directions in the moduli space of curves

In this paper we study asymptotic directions in the tangent bundle of the moduli space Mg 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 ≥ 4. We prove that if the rank d of a tangent direction ζ ∈ H¹(TC) (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.