arXiv subjects
Publications and source records attributed to Artem Avilov.
We study the coregularity of smooth Fano threefolds. We prove that for 100 out of 105 families of smooth Fano threefolds, a general member in the family has coregularity 0; moreover, for 92 families out of these 100, any member in the family has coregularity 0; for the remaining 5 families, we obtain some partial results. In particular, we show that there exist families of smooth Fano threefolds whose general elements have positive coregularity.
In this article we study forms of the Segre cubic over non-algebraically closed fields, their automorphism groups and equivariant birational rigidity. In particular, we show that all forms of the Segre cubic are cubic hypersurfaces and all of them have a point.
In this paper we classify all potentially G-birationally rigid del Pezzo threefolds of degree 4 and their automorphism groups and prove the G-birational rigidity of one of them
Three-dimensional del Pezzo varieties of degree 2 are double covers of projective space $\mathbb{P}^{3}$ branced in a quadric. In this paper we prove that if a del Pezzo variety of degree 2 has exactly 15 nodes then the corresponding quadric is a hyperplane section of the Igusa quartic or, equivalently, all such del Pezzo varieties are members of one particular linear system on the Coble fourfold. Their automorphism groups are induced from the automorphism group of Coble fourfold. Also we classify all $G$-birationally rigid varieties of such type.
In this paper we classify three-dimensional singular cubic hypersurfaces with an action of a finite group $G$, which are not $G$-rational, are not $G$-birationally isomorphic to a quadric and have no birational structure of $G$-Mori fiber space with the base of positive dimension. Also we prove the $A_{5}$ -birational superrigidity of the Segre cubic.
In 1982 V.G. Sarkisov proved the existense of standard models of conic fibrations over algebraically closed fields of $\operatorname{char}\neq 2$. In this paper we will prove the analogous result for three-dimensional conic fibrations over arbitrary fields of characteristic zero with a finite group action.