Torelli Theorem for the Moduli Spaces of Rank 2 Quadratic Pairs

Let X be a smooth projective complex curve. We prove that a Torelli type theorem holds, under certain conditions, for the moduli space of α-polystable quadratic pairs on X of rank 2.     

Moduli spaces of convex projective structures on surfaces

We introduce explicit parametrisations of the moduli space of convex projective structures on surfaces, and show that the latter moduli space is identified with the higher Teichmüller space for SL₃(R) defined in [V.V. Fock, A.B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, math.AG/0311149]. We investigate the cluster structure of this moduli space, and define its quantum version.     

The Moduli Space of 3-Dimensional Associative Algebras

In this article, we give a classification of the 3-dimensional associative algebras over the complex numbers, including a construction of the moduli space, using versal deformations to determine how the space is glued together.     

Moduli of Polarized Calabi–Yau Pairs

We prove that the irreducible components of the moduli space of polarized Calabi–Yau pairs are projective.     

Beauville Surfaces,Moduli Spaces and Finite Groups

In this article we give the asymptotic growth of the number of connected components of the moduli space of surfaces of general type corresponding to certain families of Beauville surfaces with group either PSL(2, p), or an alternating group, or a symmetric group or an abelian group. We moreover extend these results to regular surfaces isogenous to a higher product of curves.     

On families of quadratic surfaces having fixed intersections with two hyperplanes

We investigate families of quadrics all of which have the same intersection with two given hyperplanes. The cases when the two hyperplanes are parallel and when they are nonparallel are discussed. We show that these families can be described with only one parameter and describe how the quadrics are transformed as the parameter changes. This research was motivated by an application in mixed integer conic optimization. In that application, we aimed to characterize the convex hull of the union of the intersections of an ellipsoid with two half-spaces arising from the imposition of a linear disjunction.     

Left and right centers in quasi-Poisson geometry of moduli spaces

We introduce left central and right central functions and left and right leaves in quasi-Poisson geometry, generalizing central (or Casimir) functions and symplectic leaves from Poisson geometry. They lead to a new type of (quasi-)Poisson reduction, which is both simpler and more general than known quasi-Hamiltonian reductions. We study these notions in detail for moduli spaces of flat connections on surfaces, where the quasi-Poisson structure is given by an intersection pairing on homology.     

Tautological rings of stable map spaces

We find a set of generators and relations for the system of extended tautological rings associated to the moduli spaces of stable maps in genus zero, admitting a simple geometrical interpretation. In particular, when the target is Pn, these give a complete presentation for the cohomology and Chow rings in the cases with/without marked points.     

On Cremona transformations and quadratic complexes

We study the relation between Cremona transformations in space and quadratic line complexes. We show that it is possible to associate a space Cremona transformation to each smooth quadratic line complex once we choose two distinct lines contained in the complex. Such Cremona transformations are cubo-cubic and we classify them in terms of the relative position of the lines chosen. It turns out that the base locus of such a transformation contains a smooth genus two quintic curve. Conversely, we show that given a smooth quintic curve C of genus 2 in ℙ³ every Cremona transformation containing C in its base locus factorizes through a smooth quadratic line complex as before. We consider also some cases where the curve C is singular, and we give examples both when the quadratic line complex is smooth and singular.     

Disjointly strictly singular inclusions of symmetric spaces

In this paper, the disjoint strict singularity of inclusions of symmetric spaces of functions on an interval is considered. A condition for the presence of a “gap” between spaces sufficient for the inclusion of one of these spaces into the other to be disjointly strictly singular is found. The condition is stated in terms of fundamental functions of spaces and is exact in a certain sense. In parallel, necessary and sufficient conditions for an inclusion of Lorentz spaces to be disjointly strictly singular (and similar conditions for Marcinkiewicz spaces) are obtained and certain other assertions are proved. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 3–14, January, 1999.     

Bilinear pseudodifferential operators with forbidden symbols on Lipschitz and Besov spaces

We show that bilinear pseudodifferential operators with symbols in the forbidden class are bounded on products of Lipschitz and Besov spaces.     

Stability of the quadratic functional equation in Lipschitz spaces

Let G be an Abelian group with a metric d and E a normed space. For any f:G→E we define the quadratic difference of the function f by the formula

Qf(x,y):=2f(x)+2f(y)−f(x+y)−f(x−y)     

Harmonic forms and cohomology of singular stratified spaces

We construct a potential theory for differential forms on compact stratified spaces, and we show that the space of harmonic forms of degree k is isomorphic to the singular cohomology of degree k. This theory applies to any triangulable topological space.     

Moduli Spaces of Curves with Quasi-Symmetric Weierstrass GAp Seqnences

In this paper we give an explicit construction of the moduli space of the pointed complete Gorenstein curves of arithmetic genus g with a given quasi-symmetric Weierstrass semigroup, that is, a Weierstrass semigroup whose last gap is equal to 2g – 2. We identify such a curve with its image under the canonical embedding in the (g – 1)-dimensional projective space. By normalizing the coefficients of the quadratic relations and by constructing Gröbner bases of the canonical ideal, we obtain the equations of the moduli space in terms of Buchberger's criterion. Moreover, by analyzing syzygies of the canonical ideal we establish criteria that make the computations less expensive.     

Siegel coordinates and moduli spaces for morphisms of Abelian varieties

We describe the moduli spaces of morphisms between polarized complex abelian varieties. The discrete invariants, derived from a Poincaré decomposition of morphisms, are the types of polarizations and of lattice homomorphisms occurring in the decomposition. For a given type of morphisms the moduli variety is irreducible, and is obtained from a product of Siegel spaces modulo the action of a discrete group. Mathematics Subject Classification (2000) 14K20     

Square-tiled surfaces and rigid curves on moduli spaces

We study the algebro-geometric aspects of Teichmüller curves parameterizing square-tiled surfaces with two applications.(a) There exist infinitely many rigid curves on the moduli space of hyperelliptic curves. They span the same extremal ray of the cone of moving curves. Their union is a Zariski dense subset. Hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n-pointed rational curves for even n.(b) The limit of slopes of Teichmüller curves and the sum of Lyapunov exponents for the Teichmüller geodesic flow determine each other, which yields information about the cone of effective divisors on the moduli space of curves.     

Moduli of sheaves and the Chow group of K3 surfaces

Let X be a projective complex K  3 surface. Beauville and Voisin singled out a 0-cycle _cX$c_X$ on X of degree 1 and Huybrechts proved that the second Chern class of a rigid simple vector-bundle on X   is a multiple of _cX$c_X$ if certain hypotheses hold. We believe that the following generalization of Huybrechts? result holds. Let M be a moduli space of stable pure sheaves on X with fixed cohomological Chern character: the set whose elements are second Chern classes of sheaves parametrized by the closure of M (in the corresponding moduli spaces of semistable sheaves) depends only on the dimension of M. We will prove that the above statement holds under some additional assumptions on the Chern character.