A lower bound for $\chi ({\mathcal{O}}_S)$

Abstract

Let $(S,\mathcal{L})$ be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $\mathcal{L}$ of degree $d>25$. In this paper we prove that $\chi ({\mathcal{O}}_S)\ge -\frac{1}{8}d(d-6)$. The bound is sharp, and $\chi ({\mathcal{O}}_S)=-\frac{1}{8}d(d-6)$ if and only if $d$ is even, the linear system $|H^0(S,\mathcal{L})|$ embeds $S$ in a smooth rational normal scroll $T\subset {\mathbb{P}}^5$ of dimension $3$, and here, as a divisor, $S$ is linearly equivalent to $\frac{d}{2}Q$, where $Q$ is a quadric on $T$. Moreover, this is equivalent to the fact that the general hyperplane section $H\in |H^0(S,\mathcal{L})|$ of $S$ is the projection of a curve $C$ contained in the Veronese surface $V\subseteq {\mathbb{P}}^5$, from a point $x\in V\backslash C$.

7 pages