Deformations of cones over hyperelliptic curves

Abstract

We determine the versal deformation of cones, in the simplest case: cones over hyperelliptic curves of high degree. In particular, we show that for degree $4g+4$, the highest degree for which interesting deformations exist, the number of smoothing components is $2^{2g+1}$ ($g\neq3$). We review in a general setting the relation of $T^1(-1)$ with Wahl's Gaussian map. We prove that $T^1(-1)$ vanishes for a general curve and an arbitrary embedding line bundle of degree at least $2g+11$. To find $T^2$ for hyperelliptic cones with the Main Lemma of [Behnke--Christophersen], we compute $T^1$ for the cone over $d$ points on a rational normal curve of degree $d-g-1$, using explicit equations. Actually, the equations for the cone over a hyperelliptic curve have a nice structure. We give an interpretation of $T^2(-2)$ in terms of this structure. Smoothing components are related to surfaces with $C$ as hyperplane section. An explicit description of the corresponding infinitesimal deformations allowss to conclude that the base space is a complete intersection of degree $2^{2g+1}$. We also consider smoothing data in the sense of [Looijenga--Wahl].