Torification and factorization of birational maps

Building on work of the fourth author and Morelli's work, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field of characteristic zero is a composite of blowings up and blowings down with nonsingular centers.

     

Deformation of canonical morphisms and the moduli of surfaces of general type

In this article we study the deformation of finite maps and show how to use this deformation theory to construct varieties with given invariants in a projective space. Among other things, we prove a criterion that determines when a finite map can be deformed to a one-to-one map. We use this criterion to construct new surfaces of general type with birational canonical map, for different c₁²c_{1}^{2} and χ (the canonical map of the surfaces we construct is in fact a finite, birational morphism). Our general results enable us to describe some new components of the moduli of surfaces of general type. We also find infinitely many moduli spaces having one component whose general point corresponds to a canonically embedded surface and another component whose general point corresponds to a surface whose canonical map is a degree 2 morphism.     

Birationally rigid del Pezzo fibrations

In this paper we study del Pezzo fibrations z:X¹ of degree 1 and 2 such that X is smooth, rk Pic(X)=2 and These are examples of smooth birationally rigid 3-fold Mori fibre spaces. We describe all birational transformations of the 3-fold X into elliptic fibrations, fibrations of surfaces of Kodaira dimension zero, and canonical Fano 3-folds.     

Desargues Configurations via Polar Conics of Plane Cubics

Using polar conics of plane cubics we define a rational map from the moduli space of stable binary sextics into the moduli space of Desargues configurations. We show that this map is the inverse of a birational map defined via the von Staudt conic. In particular Ψ _m is a birational map.     

Super-cocharacters,star-cocharacters and multiplicities bounded by one

Let A be an algebra over a field of characteristic zero with an additional structure of superalgebra or algebra with involution. The ordinary representation theory of the hyperoctahedral group is exploited in order to study the super-identities or the star-identities of A. One associates to A a sequence -characters and one of the main objective of the theory is to determine their decomposition into irreducibles. Here we classify the super-identities and the star-identities in case the corresponding multiplicities are bounded by one. This is strictly related to the varieties of algebras whose lattice of subvarieties is distributive. A. Giambruno was partially supported by MIUR of Italy. S. Mishchenko was partially supported by RFBR grant 07-01-00080.     

Sharp de Rham realization

We introduce the sharp (universal) extension of a 1-motive (with additive factors and torsion) over a field of characteristic zero. We define the sharp de Rham realization T by passing to the Lie-algebra. Over the complex numbers we then show a (sharp de Rham) comparison theorem in the category of formal Hodge structures. For a free 1-motive along with its Cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations. Thus we show how to provide one-dimensional sharp de Rham cohomology of algebraic varieties.     

A characteristic-free criterion of birationality

One develops ab initio the theory of rational/birational maps over reduced, but not necessarily irreducible, projective varieties in arbitrary characteristic. A virtual numerical invariant of a rational map is introduced, called the Jacobian dual rank. It is proved that a rational map in this general setup is birational if and only if the Jacobian dual rank is well defined and attains its maximal possible value. Even in the “classical” case where the source variety is irreducible there is some gain for this invariant over the degree of the map because, on one hand, it relates naturally to constructs in commutative algebra and, on the other hand, is effectively computable. Applications are given to results only known so far in characteristic zero. One curious byproduct is an alternative approach to deal with the result of Dolgachev concerning the degree of a plane polar Cremona map.     

On Identity Bases of Epigroup Varieties

Let C _n and C _n be the varieties of all completely regular and of all completely simple semigroups, respectively, whose idempotent generated subsemigroups are periodic with period n. We use Ol'shanski 's theory of geometric group presentations to show that for large odd n these varieties (and similarly defined varieties of epigroups) do not have finitely axiomatizable equational theories.     

The relative pluricanonical stability for 3-folds of general type

The aim of this paper is to improve a theorem of János Kollár by a different method. For a given smooth complex projective threefold of general type, suppose the plurigenus . Kollár proved that the -canonical map is birational. Here we show that either the -canonical map or the -canonical map is birational and that the -canonical map is stably birational onto its image. Suppose . Then the -canonical map is birational for . In particular, is birational whenever and is birational whenever .

     

Automorphisms of a Generic Jacobian Kummer Surface

For a generic quartic surface F in ³ with sixteen nodes, we shall prove that every birational automorphism of F is induced by a Cremona transformation of ³.     

Supersingular <Emphasis Type="Italic">K</Emphasis>3 surfaces in odd characteristic and sextic double planes

We show that every supersingular K3 surface is birational to a double cover of a projective plane. Mathematics Subject Classification (2000):14J28, 14Q10, 11H55     

Surfaces of general type with <Emphasis Type="Italic">p</Emphasis><Subscript><Emphasis Type="Italic">g</Emphasis></Subscript>=0, <Emphasis Type="Italic">K</Emphasis><Superscript>2</Superscript>=6 and non birational bicanonical map

Let S be a minimal surface of general type with p_g=0 and K²=6, such that its bicanonical map is not birational. The map is a morphism of degree 4 onto a surface. The case of deg = 4 is completely classified in [Topology, 40 (5) (2001), 977–991] and the present paper completes the characterization of these surfaces. It is proven that the degree of cannot be equal to 3, and the geometry of surfaces with deg = 2 is analysed in detail. The last section contains three examples of such surfaces, two of which appear to be new.Mathematics Subject Classification (2000): 14J29     

Efficient Smooth Stratification of an Algebraic Variety in Zero Characteristic and Its Applications

Consider a projective algebraic variety V defined as the set of common zeros of a family of homogeneous polynomials of degree less than d in variables with coefficients from a field k of zero characteristic. We prove that V can be represented as a union (respectively, a disjoint union) of at most (respectively, ) smooth quasiprojective algebraic varieties such that the degrees of these varieties are bounded from above by , where depends only on n. We propose algorithms for constructing regular sequences and sequences of local parameters for irreducible components of V and for computing the dimension of a real variety. The complexity of these algorithms is polynomial in the size of the input and in . Bibliography: 15 titles.     

On the Codimension Growth of Finite-Dimensional Lie Algebras

We study the exponential growth of the codimensions c_n(L) of a finite-dimensional Lie algebra L over a field of characteristic zero. We show that if the solvable radical of L is nilpotent then exists and is an integer.     

Moduli of (1, 7)–Polarized Abelian Surfaces via Syzygies

We prove that the moduli space X(1,7) of (1,7)–polarized abelian surfaces with canonical level–structure is birational to the Fano 3–fold V₂₂ of polar hexagons of the Klein quartic (7). In particular X(1,7) is rational and the birational map to ℙ³ is defined over ℚ. As a byproduct we obtain explicitely the equations of the (1,7)–very–ample–polarized abelian surfaces embedded in ℙ⁶.     

Semigroup Varieties on Whose Free Objects Almost All Fully Invariant Congruences are Weakly Permutable

A semigroup variety is said to be of index 2 if all nil-semigroups of the variety are semigroups with zero multiplication. We describe all semigroup varieties V of index 2 on free objects of which every two fully invariant congruences contained in the least semilattice congruence are weakly permutable, and semigroup varieties of index 2 all of whose subvarieties share the above-mentioned property.     

Singular abelian covers of algebraic surfaces

In this paper we generalise F. Cataneses work on singular (/2)²-covers to arbitrary finite abelian covers of algebraic surfaces. We determine the contribution of singularities to the invariants , K ², q, p _g , P _n and the canonical sheaf. We use these computations to construct a surface of general type with birational canonical map ₁, p _g =4 and K ² =31. Mathematics Subject Classification (2000):14J29, 14J17, 14J25, 14E20     

Involutions of reductive Lie algebras in positive characteristic

Let G be a reductive group over a field k of characteristic ≠2, let , let θ be an involutive automorphism of G and let be the associated symmetric space decomposition. For the case of a ground field of characteristic zero, the action of the isotropy group G^θ on is well understood, since the well-known paper of Kostant and Rallis [B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971) 753–809]. Such a theory in positive characteristic has proved more difficult to develop. Here we use an approach based on some tools from geometric invariant theory to establish corresponding results in (good) positive characteristic.Among other results, we prove that the variety of nilpotent elements of has a dense open orbit, and that the same is true for every fibre of the quotient map . However, we show that the corresponding statement for G, conjectured by Richardson, is not true. We provide a new, (mostly) calculation-free proof of the number of irreducible components of , extending a result of Sekiguchi for . Finally, we apply a theorem of Skryabin to describe the infinitesimal invariants .     

Varieties of modules for

Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety , that is the set of pairs of (n×n)-matrices (A,B) such that A²=B²=[A,B]=0, is equidimensional. can be identified with the ‘variety of n-dimensional modules’ for , or equivalently, for k[X,Y]/(X²,Y²). On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic >2. We also prove that if e²=0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of as a direct sum of indecomposable components of varieties of -modules.