The maximal unipotent finite quotient, unusual torsion in Fano threefolds, and exceptional Enriques surfaces

Abstract

We introduce and study the maximal unipotent finite quotient for algebraic group schemes in positive characteristics. Applied to Picard schemes, this quotient encodes exotic torsion. We construct integral Fano threefolds where such exotic torsion actually appears. The existence of such threefolds is surprising, because the exotic torsion vanishes for del Pezzo surfaces. Our construction relies on the theory of exceptional Enriques surfaces, as developed by Ekedahl and Shepherd-Barron.