A note on non-reduced Picard schemes

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 κ≤0κ≤0, that this phenomenon can be bounded for κ=2κ=2 (general type) and that it is as bad as can be for κ=1κ=1.