Del Pezzo surfaces with global vector fields

par Claudia Stadlmayr

jeudi 17 avr. 2025, 14:00 → 16:00 Europe/Paris

If X is a del Pezzo surface (or a weak del Pezzo, or an RDP del Pezzo), then its automorphism scheme Aut_X is a, possibly non-reduced, affine group scheme of finite type. In particular, X has infinitely many automorphisms if and only if Aut_X is positive-dimensional and then X admits global vector fields (since the space of global vector fields on X is the tangent space to the automorphism scheme). The last implication is an equivalence in characteristic 0, but its converse can fail in positive characteristic. Over the complex numbers, a del Pezzo surface with rational double point singularities admits global vector fields if and only if its minimal resolution, the corresponding weak del Pezzo surface, does. In small characteristics, one implication of this equivalence breaks down due to the existence of non-lifting vector fields on rational double points. I will explain how to overcome these obstacles in order to classify weak and RDP (if p \neq 2) del Pezzo surfaces with global vector fields. Further, I will show examples of such surfaces displaying interesting behaviour in small characteristics. This is joint work with G. Martin.