The classic construction of 5-torsion Godeaux surfaces and Calabi-Yau 3-folds is toric in nature. The regular representation of the multiplicative group mu_5 is 1/5(0,1,2,3,4), and invariant quintic form F5 in PP^4(x0,x1,x2,x3,x4) (for example the Fermat) gives equivariant hypersurfaces and their fixed-point free quotients. A quotient by mu_5 has the dual group ZZ/5 in its Picard group.
Over a field of characteristic 5, Godeaux surfaces X exist with Picard scheme Pic X containing each of ZZ/5, mu_5 or al_5 (Greek α_5) – that is, each of the possible order 5 group schemes. My construction in [TOp] of the elementary split form of the Tate--Oort group TO_5 gives an irreducible family over a regular base of mixed characteristic, having fibres with each of these three cases together with the classic Godeaux surfaces in characteristic zero. (See [TOp, Section 6].)
Over a field of characteristic 3, Godeaux surfaces X exist with Picard scheme Pic X containing each of ZZ/3, mu_3 or al_3, that is, all possible order 3 group schemes. I build an irreducible family over a regular base of mixed characteristic, having fibres with each of these three cases together with Godeaux surfaces in characteristic zero with 3-torsion. As well as being a showcase for unprojection methods applied to Gorenstein graded rings, this construction serves as a extended primer on working with the Tate–Oort group TOp of [TO] and [TOp] and its representation theory (with p = 3). This goes some way towards extending Kummer’s cyclotomic methods to mixed characteristic.
[TOp] The Tate-Oort group scheme TOp, Proc. Steklov Inst. Math. 307 (2019), 245--266, get from
https://mreid.warwick.ac.uk/TOp/