Abstract
Over an algebraically closed field of characteristic p, there are three group schemes of order p, namely the ordinary cyclic group ℤ/p, the multiplicative group \(\mu_{p}\subset\mathbb{G}_{m}\), and the additive group αp ⊂ \(\mathbb{G}_{a}\). 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 Picτ, notably the 5-torsion Godeaux surfaces and Calabi-Yau threefolds obtained from \(\mathbb{TO}_5\)-invariant quintics.
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Acknowledgments
The context for this work includes p-closed vector fields and inseparable covers, much of which I learned from Shafarevich while he was working on his paper on K3 surfaces [11] with Rudakov. It is a pleasure to acknowledge this debt by dedicating the paper to his memory.
My involvement with group schemes of order p started during a visiting professorship at Sogang University, Seoul, on a Korean government grant, in connection with Soonyoung Kim’s 2014 PhD thesis [5] on Godeaux surfaces. I am extremely grateful to Yongnam Lee for his work setting up and administering the grant, for his generous hospitality, and for setting the thesis problem. Different aspects of my work benefited from extended stays at Sogang University and at KIAS over many years.
Funding
The bulk of this paper was written during a spring 2019 residence at MSRI, Berkeley, California, supported by NSF grant no. 1440140.
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In memory of Igor Rostislavovich Shafarevich, from whom we have all learned so much
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 307, pp. 267–290.
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Reid, M. The Tate-Oort Group Scheme \(\mathbb{TO}_p\). Proc. Steklov Inst. Math. 307, 245–266 (2019). https://doi.org/10.1134/S0081543819060154
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DOI: https://doi.org/10.1134/S0081543819060154