On algebraic surfaces of general type with negative c2

TL;DR

This paper establishes a lower bound on the Euler characteristic of algebraic surfaces of general type in characteristic p, confirming positivity and answering a question posed by Shepherd-Barron for primes p ≥ 3.

Contribution

It proves the existence of a positive constant relating the Euler characteristic and the square of the first Chern class for surfaces in characteristic p, extending known results.

Findings

hi(al_O_X)  \u03ba_p c_1^2 for all surfaces of general type in characteristic p

hi(al_O_X) > 0 in this setting

Answers Shepherd-Barron's question for p  3

Abstract

We prove that for any prime number $p\ge 3$, there exists a positive number $\kappa_p$ such that $\chi(\mathcal{O}_X)\ge \kappa_pc_1^2$ holds true for all algebraic surfaces $X$ of general type in characteristic $p$. In particular, $\chi(\mathcal{O}_X)>0$. This answers a question of N. Shepherd-Barron when $p\ge 3$.