In this notes of Jesse Kass, an example due to Igusa is discussed. One takes an ordinary elliptic curve over characteristic with an actual -torsion point , and form the quotient of by a (fixed point free) action of where the automorphism acts by and here is another elliptic curve (could be the same as ). The point is that this quotient is then a smooth projective surface in characteristic with non-reduced Picard scheme.

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

As for characteristic , take ordinary with an actual -torsion as usual. Then let be the hyperelliptic curve given by , which admits the automorphism which fixes and sends to . This is an automorphism of order . So do the similar construction, by Hochschild–Serre spectral sequence, you will find that the tangent space of the Picard scheme is at least . However, its Albanese has dimension again, this is because that our automorphism acts on via those non-trivial characters of order , hence there is no invariants.

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

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