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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.
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, Trudy Mat. Inst. Steklova, 307, Steklov Mathematical Institute of RAS, Moscow, 2019, 267–290; Proc. Steklov Inst. Math., 307 (2019), 245–266
Linking options:
https://www.mathnet.ru/eng/tm4042https://doi.org/10.4213/tm4042 https://www.mathnet.ru/eng/tm/v307/p267
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Abstract page: | 257 | Full-text PDF : | 46 | References: | 21 | First page: | 8 |
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