Asymptotic directions in the moduli space of curves
Year: 2025
Type: article
Abstract: In this paper, we study asymptotic directions in the tangent bundle of the moduli space M g \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 ≥ 4 g\geq 4 . We prove that if the rank 𝑑 of a tangent direction ζ ∈ H 1 ( T C ) \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.
Institutions: University of Milan, University of Pavia
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,712
OpenAlex ID: W4410885209