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

## TL;DR

This paper introduces the maximal unipotent finite quotient for algebraic group schemes in positive characteristic, revealing unusual torsion in Fano threefolds through the use of exceptional Enriques surfaces, contrasting with del Pezzo surfaces.

## Contribution

It constructs explicit Fano threefolds exhibiting unusual torsion, utilizing the theory of exceptional Enriques surfaces, and advances understanding of torsion phenomena in algebraic geometry.

## Key findings

- Unusual torsion appears in certain Fano threefolds
- Maximal unipotent finite quotient encodes torsion in Picard schemes
- Construction relies on 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 unusual torsion. We construct integral Fano threefolds where such unusual torsion actually appears. The existence of such threefolds is surprising, because the torsion vanishes for del Pezzo surfaces. Our construction relies on the theory of exceptional Enriques surfaces, as developed by Ekedahl and Shepherd-Barron.