A lower bound for $\chi (\mathcal O_S)$

TL;DR

This paper establishes a sharp lower bound for the holomorphic Euler characteristic of certain complex surfaces based on their polarization degree, characterizing cases of equality with geometric descriptions involving rational normal scrolls and Veronese surfaces.

Contribution

It proves a new lower bound for hi(\u02c8_S) in terms of the polarization degree and characterizes the extremal cases with explicit geometric conditions.

Findings

hi(_S)  -^{-1}d(d-6) for surfaces with degree d > 25.

Equality holds iff the surface embeds in a rational normal scroll with specific divisor class.

The hyperplane section relates to projections of curves on the Veronese surface.

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 $\chi (\mathcal O_S)\geq -\frac{1}{8}d(d-6)$. 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 $x\in V\backslash C$.