arXiv subjects
Publications and source records attributed to Matilde Maccan.
We investigate horospherical homogeneous spaces--a class of spherical homogeneous spaces encompassing both flag varieties and algebraic tori--over algebraically closed fields of characteristic p>0, and establish their complete classification for p>2.
We consider rational projective homogeneous varieties over an algebraically closed field of positive characteristic, namely quotients of a semi-simple group by a possibly non-reduced parabolic subgroup. We determine the group scheme structure of the neutral component of their automorphism group, generalizing the classical result of Demazure on automorphism groups of flag varieties.
We study relatively minimal surfaces equipped with a strongly isotrivial elliptic fibration in positive characteristic by means of the notion of equivariantly normal curves introduced and developed recently by Brion. Such surfaces are isomorphic to a contracted product $E\times^G X$, where $E$ is an elliptic curve, $G$ is a finite subgroup scheme of $E$ and $X$ is a $G$-normal curve. Using this description, we compute their Betti numbers to determine their birational classes. This allow us to complete the classification of maximal automorphism groups of surfaces in any characteristic. When $G$ is diagonalizable, we compute additional invariants to study the structure of their Picard schemes.
This text brings to an end the classification of non-reduced parabolic subgroups in positive characteristic, especially two and three: they are all obtained as intersections of parabolics having maximal reduced part. We prove this result and deduce a few geometric consequences on rational projective homogeneous varieties.
We extend to characteristic $2$ and $3$ the classification of projective homogeneous varieties of Picard group isomorphic to $\mathbf{Z}$, corresponding to parabolic subgroup schemes with maximal reduced subgroup. The latter are all obtained as product of a maximal reduced parabolic with the kernel of a purely inseparable isogeny. This fails in type $G_2$ and characteristic $2$, for which we exhibit an explicit counterexample and show it is the only one, thus completing the classification. We then construct new examples of projective homogeneous varieties of Picard rank two.