arXiv subjects
Publications and source records attributed to Igor Dolgachev.
The Cremona dimension of a group $G$ is the minimal $n$ such that $G$ is isomorphic to a subgroup of the Cremona group of birational transformations of an $n$-dimensional rational variety. In this survey article, we give many examples that gives evidence to the conjecture that the Cremona dimension of a finite group over the field of complex numbers is less than or equal to the essential dimension of the group.
We study K3 surfaces of degree 6 containing two sets of 12 skew lines such that each line from a set intersects exactly six lines from the other set. These surfaces arise as hyperplane sections of the cubic line complex associated with the pencil of desmic quartic surfaces introduced by George Humbert and recently studied by the second and third authors. We discuss alternative birational models of the surfaces, compute the Picard lattice and a group of projective automorphisms, and describe rational curves of low degree on the general surface.
A desmic quartic surface is a birational model of the Kummer surface of the self-product of an elliptic curve. We recall the classical geometry of these surfaces and study their analogs in arbitrary characteristic. Moreover, we discuss the cubic line complex $\frakG$ associated with the desmic tetrahedra introduced by G. Humbert. We prove that $\frakG$ is a rational Fano threefold with $34$ nodes. The number $34$ is the maximum number of nodes on a Fano threefold of degree 6 in $\bbP^5$, and the group of projective automorphisms is isomorphic to $\frakS_4\wr 2 = (\frakS_4\times \frakS_4)\rtimes 2$.
We study the congruence of bitangent lines of an irreducible surface in the 3-dimensional projective space in arbitrary characteristic, with special attention to quartic surfaces with rational double points and, in particular, Kummer quartic surfaces.
We complete the classification of automorphism groups of del Pezzo surfaces over algebraically closed fields of odd positive characteristic.
We classify the automorphism groups of del Pezzo surfaces of degrees one and two over an algebraically closed field of characteristic two. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics.
We find normal forms for del Pezzo surfaces of degree $2$ over algebraically closed fields of characteristic $2$. For each normal form, we describe the structure of the group of automorphisms of the surface. In particular, we classify all finite groups that can act on such del Pezzo surfaces.
We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such surfaces that admit non-trivial automorphisms that act trivially on the Picard group. As an application, we classify classical Enriques surfaces in characteristic $2$ that admit non-trivial numerically trivial automorphisms.
After recalling some basic facts about F-wound commutative unipotent algebraic groups over an imperfect field F we study their regular integral models over Dedekind schemes of positive characteristic and compute the group of isomorphisms classes of torsors of one-dimensional groups.
Using the theory of rational elliptic fibrations, we construct and discuss a one parameter family of configurations of $12$ conics and $9$ points in the projective plane that realizes an abstract configuration $(12_6,9_8)$. This is analogous to the famous Hesse configuration of $12$ lines and $9$ points forming an abstract configuration $(12_3,9_4)$. We also show that any Halphen elliptic fibration of index $2$ with four triangular singular fibers arises from such configuration of conics.
An automorphism of an algebraic surface $S$ is called cohomologically (numerically) trivial if it acts identically on the second $l$-adic cohomology group (this group modulo torsion subgroup). Extending the results of S. Mukai and Y. Namikawa to arbitrary characteristic $p > 0$, we prove that the group of cohomologically trivial automorphisms $\rm{Aut}_{\rm{ct}}(S)$ of an Enriques surface $S$ is of order $\leq 2$ if $S$ is not supersingular. If $p = 2$ and $S$ is supersingular, we show that $\rm{Aut}_{\rm{ct}}(S)$ is a cyclic group of odd order $n\in \{1,2,3,5,7,11\}$ or the quaternion group $Q_8$ of order $8$ and we describe explicitly all the exceptional cases. If $K_S \neq 0$, we also prove that the group $\rm{Aut}_{\rm{nt}}(S)$ of numerically trivial automorphisms is a subgroup of a cyclic group of order $\leq 4$ unless $p = 2$, where $\rm{Aut}_{\rm{nt}}(S)$ is a subgroup of a $2$-elementary group of rank $\leq 2$.
We describe a set of generators and defining relations for the group of birational automorphisms of a general 15-nodal quartic surface in the complex projective 3-dimensional space.
This is a brief history of discovery and later study of Kummer surfaces. The article is based on the author's Oliver Club talk at Cornell University on October 10, 2019 delivered exactly 101 years since the first talk at the club given by John Hutchinson.
We explain a classical construction of a del Pezzo surface of degree d = 4 or 5 as a smooth order two congruence of lines in 3-space whose focal surface is a quartic surface $X_{20-d}$ with 20-d ordinary double points. We also show that $X_{15}$ can be realized as a hyperplane section of the Castelnuovo-Richmond-Igusa quartic hypersurface. This leads to the proof of rationality of the moduli space of 15-nodal quartic surfaces. We discuss some other birational models of $X_{15}$: quartic symmetroids, 5-nodal quartic surfaces, 10-nodal sextic surfaces in $P^4$ and nonsingular surfaces of degree 10 in $P^6$. Finally we study some birational involutions of a 15-nodal quartic surface which, as it is shown in Part 2 of the paper jointly with I. Shimada, belong to a finite set of generators of the group of birational automorphisms of a general 15-nodal quartic surface.
The Fano models of Enriques surfaces produce a family of tens of mutually intersecting planes in $\mathbf P^5$ with a $10$-dimensional moduli space. The latter is linked to several 10-dimensional moduli spaces parametrizing other types of objects: a) cubic fourfolds containing the tens of planes, b) Beauville--Donagi holomorphically symplectic fourfolds, and c) double EPW sextics. The varieties in b) parametrize lines on cubic fourfolds from a). The double EPW sextics are associated, via O'Grady's construction, to Lagrangian subspaces of the Pl\"ucker space of the Grassmannian $Gr(2,\mathbf P^5)$ spanned by 10 mutually intersecting planes in $\mathbf P^5$. These links imply the irreducibility of the moduli space of supermarked Enriques surfaces, where a supermarking is a choice of a minimal generating system of the Picard group of the surface. Also some results are obtained on the variety of tens of mutually intersecting planes, not necessarily associated to Fano models of Enriques surfaces.
We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero.
Consider a cubic surface satisfying the mild condition that it may be described in Sylvester's pentahedral form. There is a well-known Enriques or Coble surface S with K3 cover birationally isomorphic to the Hessian surface of this cubic surface. We describe the nef cone and the (-2)-curves of S. In the case of pentahedral parameters (1, 1, 1, 1, nonzero t) we compute the automorphism group of S. For t not 1 it is the semidirect product of the free product (Z/2)*(Z/2)*(Z/2)*(Z/2) by the symmetric group S4. In the special case t=1/16 we study the action of Aut(S) on an invariant smooth rational curve C on the Coble surface S. We describe the action and its image, both geometrically and arithmetically. In particular, we prove that Aut(S)-->Aut(C) is injective in characteristic 0 and we identify its image with the subgroup of PGL2 coming from the symmetries of a regular tetrahedron and the reflections across its facets.
In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edge's work was based on an 1895 paper of A. Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel [FL], we consider $\mathcal{C}$ from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.