# A note on non-reduced Picard schemes - Mathematics > Algebraic Geometry

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 theEnriques-Kodaira classification can have non-reduced Picard schemes and whetherthere are restrictions on the characteristic of the ground field. It turns outthat non-reduced Picard schemes are uncommon in Kodaira dimension$\kappa\leq0$, that this phenomenon can be bounded for $\kappa=2$ generaltype and that it is as bad as can be in $\kappa=1$.

Author: Christian Liedtke

Source: https://arxiv.org/