On submaximal curves in fake projective planes

TL;DR

This paper investigates the non-existence of specific submaximal curves in fake projective planes, providing new lower bounds on multipoint Seshadri constants for these surfaces.

Contribution

It establishes the non-existence of certain submaximal curves, leading to improved bounds on Seshadri constants in fake projective planes.

Findings

Certain submaximal curves do not exist in fake projective planes

Lower bounds on multipoint Seshadri constants are derived

Results contribute to understanding the geometry of fake projective planes

Abstract

The main purpose of the note is to exclude the existence of certain submaximal curves in fake projective planes. This will lead to lower bounds on multipoint Seshadri constants of `fake' $\mathcal{O}(1)$ on fake projective planes.