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Tr. Mat. Inst. Steklova, 2019, Volume 307, Pages 267–290 (Mi tm4042)  

The Tate–Oort Group Scheme $\mathbb {TO}_p$

Miles Reid

Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Abstract: Over an algebraically closed field of characteristic $p$, there are three group schemes of order $p$, namely the ordinary cyclic group $\mathbb Z/p$, the multiplicative group $\boldsymbol \mu _p\subset \mathbb G_\mathrm{m}$ and the additive group $\boldsymbol \alpha _p\subset \mathbb G_\mathrm{a}$. The Tate–Oort group scheme $\mathbb {TO}_p$ puts these into one happy family, together with the cyclic group of order $p$ in characteristic zero. This paper studies a simplified form of $\mathbb {TO}_p$, focusing on its representation theory and basic applications in geometry. A final section describes more substantial applications to varieties having $p$-torsion in $\mathrm {Pic}^\tau $, notably the $5$-torsion Godeaux surfaces and Calabi–Yau threefolds obtained from $\mathbb {TO}_5$-invariant quintics.

Funding Agency Grant Number
National Science Foundation 1440140
The bulk of this paper was written during a spring 2019 residence at MSRI, Berkeley, California, supported by NSF grant no. 1440140.


DOI: https://doi.org/10.4213/tm4042

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English version:
Proceedings of the Steklov Institute of Mathematics, 2019, 307, 245–266

Bibliographic databases:

UDC: 512.74
Received: May 25, 2019
Revised: June 27, 2019
Accepted: November 25, 2019

Citation: Miles Reid, “The Tate–Oort Group Scheme $\mathbb {TO}_p$”, Algebra, number theory, and algebraic geometry, Collected papers. Dedicated to the memory of Academician Igor Rostislavovich Shafarevich, Tr. Mat. Inst. Steklova, 307, Steklov Mathematical Institute of RAS, Moscow, 2019, 267–290; Proc. Steklov Inst. Math., 307 (2019), 245–266

Citation in format AMSBIB
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\by Miles~Reid
\paper The Tate--Oort Group Scheme $\mathbb {TO}_p$
\inbook Algebra, number theory, and algebraic geometry
\bookinfo Collected papers. Dedicated to the memory of Academician Igor Rostislavovich Shafarevich
\serial Tr. Mat. Inst. Steklova
\yr 2019
\vol 307
\pages 267--290
\publ Steklov Mathematical Institute of RAS
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4042}
\crossref{https://doi.org/10.4213/tm4042}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2019
\vol 307
\pages 245--266
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  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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