Pedro
Montero
Universidad Técnica Valparaíso
Additive structures on quintic del Pezzo varieties
Abstract.
A classical problem of F. Hirzebruch concerns the classification of compactifications of affine space into smooth projective varieties with Picard rank one. It turns out that any such compactification must be a Fano manifold, i.e., it has an ample anti-canonical divisor. After reviewing some known results, I will focus on the specific case of equivariant compactifications of affine space (i.e., of the "vector group" \(\mathcal{G}_a^n\)), particularly in the case of del Pezzo varieties.<br>We will recall that del Pezzo varieties are a natural higher-dimensional generalization of classical del Pezzo surfaces. Over the field of complex numbers, these varieties were extensively studied by T. Fujita in the 1980s, who classified them by their degree.<br>I will present a result on the existence and uniqueness of "additive structures" on del Pezzo quintic varieties. Specifically, we determine when and how many distinct ways they can be obtained as equivariant compactifications of the commutative unipotent group. As an application, we obtain results on the k-forms of quintic del Pezzo varieties over an arbitrary field k of characteristic zero, as well as for singular quintic varieties. This is a joint work with Adrien Dubouloz and Takashi Kishimoto.