Cylindricity of weighted singular del Pezzo surfaces defined over fields of characteristic zero

The study of cylinders in normal projective varieties is of significant interest due to their intrinsic link to unipotent group actions on affine algebraic varieties. Over a field $\mathbb{k}$ of characteristic zero, it is known that cylindricity in lower-dimensional varieties (appearing as generic fibers) implies the existence of vertical cylinders in higher-dimensional fibered spaces. In this talk, we give a criterion for the cylindricity of $\mathbb{k}$-forms of a singular del Pezzo surface obtained as a blow-up of a weighted projective plane. Moreover, using the aforementioned criterion, we construct a Fano 3-fold which contains a vertical cylinder. This is joint work with In-Kyun Kim and Masatomo Sawahara.