A lower bound for $K^2_S$

TL;DR

This paper establishes a sharp lower bound for the self-intersection number of the canonical divisor on certain complex surfaces, linking it to the degree of the polarization and characterizing the extremal cases.

Contribution

It proves a new lower bound for $K^2_S$ in terms of the degree $d$, and characterizes the surfaces achieving equality as embedded in rational normal scrolls.

Findings

Proves $K^2_S geq -d(d-6)$ for polarized surfaces with $d>35$.

Characterizes the extremal surfaces as those embedded in rational normal scrolls.

The bound is sharp and equality cases are explicitly described.

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 $K^2_S\geq -d(d-6)$. The bound is sharp, and $K^2_S=-d(d-6)$ 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 $T$.