A note on non-reduced Picard schemes

TL;DR

This paper investigates the conditions under which surfaces, classified by Enriques-Kodaira, can have non-reduced Picard schemes, revealing that such cases are rare in certain classes and more prevalent in others.

Contribution

It characterizes the occurrence of non-reduced Picard schemes across different classes of surfaces and discusses restrictions related to the characteristic of the ground field.

Findings

Non-reduced Picard schemes are rare for Kodaira dimension ≤ 0.

The phenomenon is bounded for surfaces of general type (κ=2).

Non-reduced Picard schemes are most common in κ=1 surfaces.

Abstract

The Picard scheme of a smooth curve and a smooth complex variety is reduced. In this note we discuss which classes of surfaces in terms of the Enriques-Kodaira classification can have non-reduced Picard schemes and whether there are restrictions on the characteristic of the ground field. It turns out that non-reduced Picard schemes are uncommon in Kodaira dimension $\kappa\leq0$, that this phenomenon can be bounded for $\kappa=2$ (general type) and that it is as bad as can be in $\kappa=1$.