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

Andrea Fanelli, Stefan Schröer

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.

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 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.

Paper Structure

This paper contains 7 sections, 38 theorems, 52 equations.

Table of Contents

  1. The maximal finite unipotent quotient
  2. Picard scheme and Bockstein operators
  3. The case of surfaces
  4. Enriques surfaces and del Pezzo surfaces
  5. Exceptional Enriques surfaces
  6. Cones and Fano varieties
  7. Fano threefolds with unusual torsion

Key Result

Theorem 1

(see Thm. hasse--witt) We have $\Upsilon^0_{Y/k}=0$ provided the Frobenius map on $H^2(Y,\mathscr{O}_Y)$ has maximal Hasse--Witt rank.

Theorems & Definitions (38)

  • Theorem
  • Theorem
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  • Lemma \oldthetheorem
  • Proposition \oldthetheorem
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  • ...and 28 more