On the number of moduli of extendable canonical curves

Abstract

Let $C \subset \mathbb{P}^{g-1}$ be a canonical curve of genus $g$. In this article we study the problem of extendability of $C$, that is when there is a surface $S\subset \mathbb{P}^g$ different from a cone and having $C$ as hyperplane section. Using the work of Epema we give a bound on the number of moduli of extendable canonical curves. This for example implies that a family of large dimension of curves that are cover of another curve has general member nonextendable. Using a theorem of Wahl we prove the surjectivity of the Wahl map for the general k-gonal curve of genus $g$ when $k = 5, g \geq 15$ or $k = 6, g \geq 13$ or $k \geq 7, g \geq 12$.