On algebraic surfaces of general type with negative

Abstract

We prove that for any prime number $p\geqslant 3$, there exists a positive number $\unicode[STIX]{x1D705}_{p}$ such that $\unicode[STIX]{x1D712}({\mathcal{O}}_{X})\geqslant \unicode[STIX]{x1D705}_{p}c_{1}^{2}$ holds true for all algebraic surfaces $X$ of general type in characteristic $p$. In particular, $\unicode[STIX]{x1D712}({\mathcal{O}}_{X})>0$. This answers a question of Shepherd-Barron when $p\geqslant 3$.