On non-vanishing of cohomologies of generalized Raynaud polarized surfaces

TL;DR

This paper investigates the non-vanishing of certain cohomology groups on generalized Raynaud surfaces in positive characteristic, extending known results and identifying new polarizations with similar properties.

Contribution

It introduces a family of extended Raynaud surfaces with specific polarizations and analyzes their cohomology, including non-Mumford-Szpiro types, revealing new non-vanishing phenomena.

Findings

Computed cohomologies H^i(X, Z^n) for the surfaces.

Identified conditions for non-vanishing of H^1(X, Z^{-1}).

Presented a large family of polarizations with Kodaira non-vanishing.

Abstract

We consider a family of slightly extended version of the Raynaud's surfaces X over the field of positive characteristic with Mumford-Szpiro type polarizations Z, which have Kodaira non-vanishing H^1(X, Z^{-1})\ne 0. The surfaces are at least normal but smooth under a special condition. We compute the cohomologies H^i(X, Z^n), for intergers i and n, and study their (non-)vanishing. Finally, we give a fairly large family of non Mumford-Szpiro type polarizations Z_{a,b} with Kodaira non-vanishing.