Explicit surfaces in characteristic p with non-reduced Picard.

In this notes of Jesse Kass, an example due to Igusa is discussed. One takes an ordinary elliptic curve $E$ over characteristic $2$ with an actual $2$-torsion point $P$, and form the quotient of $E \times E'$ by a (fixed point free) action of $\mathbb{Z}/2 = \langle \sigma \rangle$ where the automorphism $\sigma$ acts by $\sigma(x,y) = (x+P,-y)$ and here $E'$ is another elliptic curve (could be the same as $E$). The point is that this quotient is then a smooth projective surface in characteristic $2$ with non-reduced Picard scheme.

One can simply make similar examples in other characteristics too. For instance, in characteristic $3$, take ordinary $E$ with an actual $3$-torsion point. Then take $E'$ to be the supersingular elliptic curve with an automorphism by “ $\omega$“. Do the similar construction above, and run the same argument, you can prove that the Picard scheme has tangent space of dimension $2$ whereas the Albanese has dimension $1$ (use $\omega^2 + \omega + 1 = 0$).

As for characteristic $p \geq 5$, take ordinary $E$ with an actual $p$-torsion as usual. Then let $C$ be the hyperelliptic curve given by $y^2 = x^p - x$, which admits the automorphism which fixes $y$ and sends $x$ to $x+1$. This is an automorphism of order $p$. So do the similar construction, by Hochschild–Serre spectral sequence, you will find that the tangent space of the Picard scheme is at least $2$. However, its Albanese has dimension $1$ again, this is because that our automorphism acts on $H^1_{crys}(C)[\frac{1}{p}]$ via those non-trivial $p-1$ characters of order $p$, hence there is no invariants.

So I think these are concrete examples of elliptic surfaces with wild fibres, as discussed in Liedtke’s notes.