arXiv subjects
Publications and source records attributed to Yuri Prokhorov.
An extremal curve germ is the analytic germ of a threefold with terminal singularities along a reduced complete curve admitting a contraction whose fibers have dimension at most one. The aim of the present paper is to review the results concerning those contractions whose central fiber is irreducible and contains only one non-Gorenstein point.
We prove that automorphism groups of Inoue and primary Kodaira surfaces are Jordan.
We prove that $\mathbb{Q}$-Fano threefolds of Fano index $\ge 8$ are rational.
We classify del Pezzo surfaces of Picard number one with log canonical singularities admitting Q-Gorenstein smoothings.
We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: $\mathfrak{A}_5$, $\operatorname{PSL}_2(\mathbf{F}_7)$, $\mathfrak{A}_6$, $\operatorname{SL}_2(\mathbf{F}_8)$, $\mathfrak{A}_7$, $\operatorname{PSp}_4(\mathbf{F}_3)$, $\operatorname{SL}_2(\mathbf{F}_{7})$, $2.\mathfrak{A}_5$, $2.\mathfrak{A}_6$, $3.\mathfrak{A}_6$ or $6.\mathfrak{A}_6$. All of these groups with a possible exception of $2.\mathfrak{A}_6$ and $6.\mathfrak{A}_6$ indeed act on some rationally connected threefolds.
We give an explicit construction of prime Fano threefolds of genus 12 with a $G_m$-action, describe their isomorphism classes and automorphism groups.
This expository paper is concerned with the rationality problems for three-dimensional algebraic varieties with a conic bundle structure. We discuss the main methods of this theory. We sketch the proofs of certain principal results, and present some recent achievements. Many open problems are also stated.
It is known that the moduli space of smooth Fano-Mukai fourfolds $V_{18}$ of genus $10$ has dimension one. We show that any such fourfold is a completion of $\mathbb{C}^4$ in two different ways. Up to isomorphism, there is a unique fourfold $V_{18}^{\mathrm s}$ acted upon by $\operatorname{SL}_2(\mathbb{C})$. The group $\operatorname{Aut}(V_{18}^{\mathrm s})$ is a semidirect product $\operatorname{GL}_2(\mathbb{C})\rtimes(\mathbb{Z}/2\mathbb{Z})$. Furthermore, $V_{18}^{\mathrm s}$ is a $\operatorname{GL}_2(\mathbb{C})$-equivariant completion of $\mathbb{C}^4$, and as well of $\operatorname{GL}_2(\mathbb{C})$. The restriction of the $\operatorname{GL}_2(\mathbb{C})$-action on $V_{18}^{\mathrm s}$ to $\mathbb{C}^4\hookrightarrow V_{18}^{\mathrm s}$ yields a faithful representation with an open orbit. There is also a unique, up to isomorphism, fourfold $V_{18}^{\mathrm a}$ such that the group $\operatorname{Aut}(V_{18}^{\mathrm a})$ is a semidirect product $({\mathbb G}_{\mathrm{a}}\times{\mathbb G}_{\mathrm{m}})\rtimes (\mathbb{Z}/2\mathbb{Z})$. For a Fano-Mukai fourfold $V_{18}$ neither isomorphic to $V_{18}^{\mathrm s}$, nor to $V_{18}^{\mathrm a}$, one has $\operatorname{Aut}^0 (V_{18})\cong ({\mathbb G}_{\mathrm{m}})^2$, and $\operatorname{Aut}(V_{18})$ is a semidirect product of $\operatorname{Aut}^0(V_{18})$ and a finite cyclic group whose order is a factor of $6$.
We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds with Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold of index 1 and genus 10. We also show that the action of the automorphism group of a Fano threefold $X$ of index 2 (respectively, 1) on an irreducible component of its Hilbert scheme of lines (respectively, conics) is faithful if the anticanonical class of $X$ is very ample with a possible exception of several explicit cases. We use these faithfulness results to prove finiteness of the automorphism groups of most Fano threefolds and classify explicitly all Fano threefolds with infinite automorphism group. We also discuss a derived category point of view on the Hilbert schemes of lines and conics, and use this approach to identify some of them.
We prove that if $X$ is a rationally connected threefold and $G$ is a $p$-subgroup in the group of birational selfmaps of $X$, then $G$ is an abelian group generated by at most $3$ elements provided that $p\ge 17$. We also prove a similar result for $p\ge 11$ under an assumption that $G$ acts on a (Gorenstein) $G$-Fano threefold, and show that the same holds for $p\ge 5$ under an assumption that $G$ acts on a $G$-Mori fiber space.
Let $(X, C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible reduced complete curve $C$ with a contraction $f: (X, C)\to (Z, o)$ such that $C=f^{-1}(o)_{red}$ and $-K_X$ is ample. Assume that $(X, C)$ contains a point of type (IIA). This paper continues our study of such germs containing a point of type (IIA) started in our previous paper arXiv:1601.07671.
We determine three-dimensional algebraic varieties whose groups of birational selfmaps do not satisfy the Jordan property.
We give explicit bounds for Jordan constants of groups of birational automorphisms of rationally connected threefolds over fields of zero characteristic, in particular, for Cremona groups of ranks 2 and 3.
We classify quasi-simple finite groups of essential dimension 3.
The aim of this short note is to give a simple proof of the non-rationality of the double cover of the three-dimensional projective space branched over a sufficiently general quartic.
We show that, for a Q-Fano threefold X of Fano index 2, the inequality dim |-1/2K_X| <= 4 holds with a single well understood family of varieties having dim |-1/2K_X| = 4.
We show that, for a $\mathbb Q$-Fano threefold $X$ of Fano index 7, the inequality $\dim |-K_X| \ge 15$ implies that $X$ is isomorphic to one of the following varieties $\mathbb P (1^2,2,3)$, $X_6 \subset \mathbb P (1,2^2,3,5)$ or $X_6 \subset \mathbb P (1,2,3^2,4)$.