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

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.