Group schemes provide a refined notion of symmetry in positive characteristic: they detect infinitesimal structure invisible to the discrete automorphism group. Classical examples such as mu_p or alpha_p equip the trivial topological space with a non-trivial algebraic structure.
In this talk I will explain how this perspective can be used to classify weak and RDP del Pezzo surfaces admitting global vector fields, and how phenomena unique to small characteristic – such as non-lifting vector fields on rational double point singularities (RDPs) – can be illuminated using the group-scheme framework.
If time permits, I will outline applications and ongoing projects: towards higher-dimensional Fano varieties with infinite automorphism groups, (equivariant) compactifications of the affine plane, group scheme torsors and their smoothness behavior.