Canonical surfaces of higher degree

TL;DR

This paper studies a specific family of algebraic surfaces with ample canonical bundles, proving their canonical systems embed them into projective space, and discusses related open problems in the classification of such surfaces.

Contribution

It establishes that for these surfaces, the canonical system is base point free and induces an embedding into projective space, answering a question by Kapustka and colleagues.

Findings

Canonical system is base point free for these surfaces.

Canonical map yields an embedding into -dimensional projective space.

Addresses open problems for surfaces with p_g=5.

Abstract

We consider a family of surfaces of general type $S$ with $K_S$ ample, having $K^2_S = 24, p_g (S) = 6, q(S)=0$. We prove that for these surfaces the canonical system is base point free and yields an embedding $\Phi_1 : S \rightarrow \mathbb{P}^5$. This result answers a question posed by G. and M. Kapustka. We discuss some related open problems, concerning also the case $p_g(S) = 5$, where one requires the canonical map to be birational onto its image.