On the projective normality of double coverings over a rational surface

TL;DR

This paper investigates the projective normality of certain double coverings over rational surfaces, including Horikawa surfaces, demonstrating that specific adjoint divisors are normally generated under certain conditions.

Contribution

It establishes the normal generation of $bZ_2$-invariant adjoint divisors on double coverings over rational surfaces, extending understanding of projective normality in this context.

Findings

$bZ_2$-invariant adjoint divisors $K_X+r ext{pi}^*A$ are normally generated for $r extgreater= 3$

Results apply to Horikawa surfaces, minimal surfaces of general type with specific invariants

Provides conditions under which projective normality holds for double coverings over rational surfaces

Abstract

We study the projective normality of a minimal surface $X$ which is a ramified double covering over a rational surface $S$ with $\dim|-K_S|\ge 1$. In particular Horikawa surfaces, the minimal surfaces of general type with $K^2_X=2p_g(X)-4$, are of this type, up to resolution of singularities. Let $\pi$ be the covering map from $X$ to $S$. We show that the $\mathbb{Z}_2$-invariant adjoint divisors $K_X+r\pi^*A$ are normally generated, where the integer $r\ge 3$ and $A$ is an ample divisor on $S$.