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 α_p ⊂ \(\mathbb{G}_{a}\). 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 Pic^τ, notably the 5-torsion Godeaux surfaces and Calabi-Yau threefolds obtained from \(\mathbb{TO}_5\)-invariant quintics.