arXiv subjects
Publications and source records attributed to Emre Alp Özavcı.
We prove that globally $+$-regular varieties are rationally chain connected in dimension three and mixed characteristic with residue field characteristic $p>5$. We also introduce a notion of strongly globally $+$-regular, and show that varieties of arbitrary dimension which are strongly globally $+$-regular over a dense open subset of $\mathrm{Spec}(\mathbb{Z})$ are rationally chain connected.
We study smooth projective varieties whose Frobenius-trace kernel, also known as the Tschirnhausen bundle associated with the Frobenius morphism, is ample. We show that any non-constant morphism out of such a variety must be finite onto its image, providing strong evidence that these varieties must be Fano varieties of Picard rank 1. Using infinitesimal representation theory and by constructing special Frobenius splittings, we show that generalized Grassmannian of classical type and type $\mathrm{G}_2$ have ample Frobenius-trace kernel, with the exception of certain low characteristic examples which are explained by the existence of exotic isogenies of the associated algebraic groups.
We give a geometric description of the positivity of the Frobenius-trace kernel on a $\mathbb{Q}$-factorial projective toric variety. To do so, we define its Frobenius support as well as the notions of $F$-effectiveness for divisors and $1$-cycles. As it turns out, the interaction of the corresponding cone of $F$-effective curves with the Mori cone of curves reflects the type of extremal Mori contractions that the variety can undergo. As a corollary, we obtain that the Frobenius-trace kernel is ample if and only if the Picard rank is $1$.