A lower bound for $K_S^2$

Abstract

Let $(S,\mathcal{L})$ be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $\mathcal{L}$ of degree $d>35$. In this paper we prove that $K_S^2\ge -d(d-6)$. The bound is sharp, and $K_S^2=-d(d-6)$ 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 $T$.

12 pages, Dedicated to Philippe Ellia on his sixtieth birthday