Pathology of formal locally-trivial deformations in positive characteristic

TL;DR

This paper constructs explicit algebraic examples in positive characteristic to demonstrate that locally trivial moduli functors can violate Schlessinger's condition, highlighting differences from characteristic zero.

Contribution

It provides the first explicit examples of algebraic varieties in positive characteristic where locally trivial moduli functors fail Schlessinger's condition (H_1).

Findings

Counterexamples include an algebraic curve and a rational surface.

Shows failure of Schlessinger's condition (H_1) in positive characteristic.

Contrasts behavior with characteristic zero cases.

Abstract

We construct explicit examples that are algebraic varieties in positive characteristic to show that locally trivial moduli functors do not always satisfy Schlessinger's condition $(H_1)$ in [3], in contrast to the complex/characteristic $0$ case. The first example is an algebraic curve, and the second is a normal rational projective surface with only one rational double point.