On surfaces with zero vanishing cycles

TL;DR

This paper presents a simplified proof of Zak's classification of smooth projective surfaces with zero vanishing cycles, extending the result to finite characteristic fields using an idea from Van de Ven.

Contribution

It introduces a new, streamlined proof technique that broadens the applicability of Zak's theorem to positive characteristic settings.

Findings

Simplified proof of Zak's classification

Extension to finite characteristic fields

Broader applicability of vanishing cycle results

Abstract

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak's classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak's theorem to the case of finite characteristic.