K3 surfaces of Kummer type in characteristic two
We discuss K3 surfaces in characteristic two that contain the Kummer configuration formed by smooth rational curves on it.
math.AG↗
arXiv subjects
Igor V. Dolgachev
Publications and source records attributed to Igor V. Dolgachev.
Monoidal and submonoidal surfaces, and Cremona transformations
We study irreducible surfaces of degree d in $\mathbb{P}^3$ that contain a line of multiplicity d-1 (monoidal surfaces) or d-2 (submonoidal surfaces). We relate them to congruences of lines and Cremona transformations. Many of our results are not new and can be found in the classical literature, we give them modern proofs. In the last section, we extend some of our results to hypersurfaces of arbitrary dimension. We define two commuting Cremona involutions in the ambient space associated to a linear subspace of multiplicity d-2 contained in the hypersurface. Both leave the hypersurface invariant, but one acts as the identity on the hypersurface.
math.AG↗
Rational self-maps of moduli spaces
We discuss some examples of geometrically meaningful rational self-maps of moduli space of curves of low genus and homogeneous forms.
math.AG↗
A brief introduction to Enriques surfaces
This is a brief introduction to the theory of Enriques surfaces over arbitrary algebraically closed fields. Some new results about automorphism groups of Enriques surfaces are also included.
math.AG↗
Numerical trivial automorphisms of Enriques surfaces in arbitrary characteristic
We extend to arbitrary characteristic some known results about automorphisms of complex Enriques surfaces that act trivially on the cohomology or the cohomology modulo torsion.
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Finite subgroups of the plane Cremona group
We complete the classical and modern work on the classification of conjugacy classes of finite subgroups of the group of birational transformations of the complex projective plane.
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On elements of order p^s in the plane Cremona group over a field of characteristic p
We show that the plane Cremona group over a field of characteristic p > 0 does not contain elements of order of power of p larger than 2 and it does not contain elements of order p^2 unless p =2. Also we describe conjugacy classes of elements of order 4.
math.AG↗
On elements of prime order in the plane Cremona group over a perfect field
We show that the plane Cremona group over a perfect field $k$ of characteristic $p \ge 0$ contains an element of prime order $\ell\ge 7$ not equal to $p$ if and only if there exists a 2-dimensional algebraic torus $T$ over $k$ such that $T(k)$ contains an element of order $\ell$. If $p = 0$ and $k$ does not contain a primitive $\ell$-th root of unity, we show that there are no elements of prime order $\ell > 7$ in $\Cr_2(k)$ and all elements of order 7 are conjugate.
math.AG↗
McKay's correspondence for cocompact discrete subgroups of SU(1,1)
The classical McKay correspondence establishes an explicit link from the representation theory of a finite subgroup G of SU(2) and the geometry of the minimal resolution of the quotient of the affine plane by G. In this paper we discuss a possible generalization of the McKay correspondence to the case when G is replaced with a cocompact discrete subgroup of the universal cover of SU(1,1) such that its image in PSU(1,1) is a cocompact fuchsian group with quotient of genus 0. We establish a correspondence between a certain class of finite-dimensional unitary representations of G and vector bundles on an open algebraic surface with the trivial canonical class canonically associated to G.
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Reflection groups in algebraic geometry
This is a survey on appearances of reflection groups, real and complex, in algebraic geometry. We also include a brief introduction into the theory of reflection groups.
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Moduli spaces of K3 surfaces and complex ball quotients
These are lecture notes based on a series of talks given by the authors at the CIMPA Summer School on Algebraic Geometry and Hypergeometric Functions held in Istanbul in Summer of 2005. They provide an introduction to a recent work on the complex ball uniformization of the moduli spaces of Del Pezzo surfaces, K3 surfaces and algebraic curves of lower genus. We discuss the relationship of these constructions with the Deligne-Mostow theory of periods of hypergeometric differentianl forms. For convenience to a non-expert reader we include an introduction to the theory of periods of integrals on algebraic varieties with emphasis on abelian varieties and K3 surfaces.
math.AG↗
Logarithmic sheaves attached to arrangements of hyperplanes
A reduced divisor on a nonsingular variety defines the sheaf of logarithmic 1-forms. We introduce a certain coherent sheaf whose double dual coincides with this sheaf. It has some nice properties, for example, the residue exact sequence still holds even when the divisor is singular, and also it has a simple locally free resolution. We specialize to the case when the divisor is an arrangement of hyperplanes in projective space, and relate the properties of stability of the sheaf with the combinatorics of the arrangement. We also extend a Torelli type theorem to non generic arrangements which allows one to reconstruct an arrangement from the sheaf attached to it.
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Luigi Cremona and cubic surfaces
We discuss Luigi Cremona's contribution to the early development of the theory of cubic surfaces.
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On certain families of elliptic curves in projective space
We study families of elliptic curves of degree n+1 in $P^n$ containing a fixed set of m points. In the case m = n+3 we show that this family is birationally isomorphic to a smooth complete intersection of n-2 diagonal quadrics in $P^{n+2}$. We also describe an action of the 2-elementary abelian group $2^{n+2}$ on this family via Cremona transformations in $P^n$.
math.AG↗
Polar Cremona transformations
In this paper we address the following question arising from the work of P. Etingof, D. Kazhdan and A. Polishchuk (math.AG/0003009): given a homogeneous complex polynomial, when the rational map defined by its partials is of degree 1? We answer this question for reduced polynomials in three variables and also for reduced polynomials in four variables which factor into the product of linear polynomials.
math.AG↗
Birational automorphisms of quartic Hessian surfaces
We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16. The latter embeds naturally in the even unimodular lattice $II^{1,25}$ of rank 26 and signature $(1,25)$ as the orthogonal complement of a root sublattice of rank 10. Our generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of S. Kond$\bar {\roman o}$. We shall explain how our generators are related to the generators of the group of birational automorphisms of a general quartic Kummer surface which is birationally isomorphic to a special Hessian surface.
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Invariant stable bundles over modular curves X(p)
We study stable vector bundles over the modular curve X(p) corresponding to the principal congruence subgroup of the modular group of prime level p which are invariant with respect to its automorphism group.
alg-geom↗
Mirror symmetry for lattice polarized K3 surfaces
We establish a relationship between mirror symmetry for K3 surfaces and Arnold's strange duality for K3 surfaces. We compute various examples of mirror families. Among them the mirror moduli family for the moduli space of degree 2n polarized K3 surfaces. It turns out to be related to the moduli space of elliptic curves with level n.
alg-geom↗