Global sections of twisted normal bundles of K3 surfaces and their hyperplane sections

Abstract

Let $S\subset {\mathbb{P}}^g$ be a smooth $K3$ surface of degree $2g-2$, $g\ge 3$. We classify all the cases for which $h^0({\mathcal{N}}_{S/{\mathbb{P}}^g}(-2))\ne 0$ and the cases for which $h^0({\mathcal{N}}_{S/{\mathbb{P}}^g}(-2))<h^0({\mathcal{N}}_{C/{\mathbb{P}}^{g-1}}(-2))$ for $C\subset {\mathbb{P}}^{g-1}$ a general canonical curve section of $S$.