Surfaces with canonical map of odd degree

TL;DR

This paper investigates complex algebraic surfaces with odd-degree canonical maps, establishing bounds on invariants and classifying cases for certain degrees, revealing potential boundedness of such surfaces' properties.

Contribution

It provides new bounds and classifications for surfaces with odd-degree canonical maps, refining previous results and exploring the boundedness of invariants.

Findings

$p_g ext{ is at most } d+2$ for the surfaces studied.

$ ext{Surface } ext{Sigma} ext{ is a cone over a rational normal curve}$.

$d=3,9,11$ are the only degrees where $p_g=d+2$ occurs.

Abstract

Let $S$ be a smooth complex minimal surface of general type with $p_g:=h^0(K_S)\ge 4$ whose canonical map is generically finite of odd degree $d>1$ onto a surface $\Sigma$. We assume that the general canonical curve of $S$ is smooth and that $\Sigma$ is ruled by lines, and we prove: - $p_g\le d+2$ - $\Sigma$ is a cone over the rational normal curve of degree $p_g-2$ in ${\mathbb P}^{p_g-1}$ - $p_g=d+2$ can occur only for $d=3,9,11$. As a byproduct, we refine previous results by Beauville and Xiao by proving that if one drops the assumption that $\Sigma$ is ruled by lines then $d\le 5$ if $p_g\ge 112$. The case $d=3$ being completely classified by the first two named authors, we focus on $d=5$, showing that $p_g\le 5$ and that for $p_g=5$ the surface $S$ has a pencil $|C|$ with $C^2=1$ and $K_SC=5$. These results suggest that the answer to the question whether the surfaces with canonical map of odd degree $d>1$ have bounded invariants could be positive, in sharp contrast with the case of even degree.