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

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.