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

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.