Semi-stable fibration of generic p-rank 0

TL;DR

This paper investigates the behavior of semi-stable fibrations of smooth surfaces over curves in positive characteristic, revealing that Frobenius base changes can violate semi-positivity when the generic fiber has p-rank 0, with implications for p-rank distributions in moduli spaces.

Contribution

It proves that Frobenius base changes break semi-positivity in certain fibrations with p-rank 0 fibers, offering new insights into p-rank behavior in algebraic geometry.

Findings

Frobenius base change can violate semi-positivity for p-rank 0 fibers.

The p-rank of the generic fiber influences the fibration's properties.

Implications for p-rank distribution in moduli spaces over number fields.

Abstract

In this paper, we proved that, for a semi-stable fibration of a proper smooth surface to a proper smooth curve over a filed of positive characteristic, if the p-rank of the generic fiber is 0, then the base change of the fibration by a sufficiently many iterative Frobenius morphism of the base curve violates the semi-positivity theorem. As an application, we suggest a statement on a distribution of p-ranks of reductions for a certain non-closed point in the moduli space over a number field.