The minimal and next minimal volumes of normal KSBA stable surfaces with $p_g\ge 2$

TL;DR

This paper determines the minimal and next minimal volumes of normal KSBA stable surfaces with geometric genus at least 2, providing explicit formulas and characterizations based on the properties of the canonical linear system.

Contribution

It explicitly computes the minimal and next minimal volumes of these surfaces and characterizes the surfaces that attain the minimal volumes, depending on the canonical linear system.

Findings

Minimal volume when |K_X| not composed with a pencil: 2p_g-4 and 2p_g-4+1/3.

Minimal volume when |K_X| composed with a pencil: (p_g-1)/(p_g+1)*(p_g-1).

Characterization of surfaces achieving minimal volumes.

Abstract

In this paper we investigate the minimal and the next minimal volumes of normal KSBA stable surfaces with $p_g\ge 2$. We show that in case of $|K_X|$ not composed with a pencil, the minimal and next minimal volumes are $2p_g-4$ and $2p_g-4+\frac{1}{3}$. In case of $|K_X|$ composed with a pencil, the minimal and next minimal volumes are $\frac{p_g-1}{p_g+1}(p_g-1)$ and $\mathrm{min}\{\frac{2p_g-2}{2p_g+1}(p_g-1), \frac{(3p_g-2)p_g-4}{3(p_g+2)}\}$. We also characterize the surfaces achieving the minimal volumes.