Flag Hilbert schemes and moduli spaces of torsion plane sheaves

We find certain relations between flag Hilbert schemes of points on plane curves and moduli spaces of one-dimensional plane sheaves. We show that some of these moduli spaces are stably rational.     

On Moduli Spaces,Equidistribution, Estimates,and Rational Points of Algebraic Curves

We consider the moduli spaces of hyperelliptic curves, Artin–Schreier coverings, and some other families of curves of this type over fields of characteristic p. By using the Postnikov method, we obtain expressions for the Kloosterman sums. The distribution of angles of the Kloosterman sums was investigated on a computer. For small prime p, we study rational points on curves y ² = f(x). We consider the problem of the accuracy of estimates of the number of rational points of hyperelliptic curves and the existence of rational points of curves of the indicated type on the moduli spaces of these curves over a prime finite field.     

Hitchin System on Singular Curves

We study the Hitchin system on singular curves. We consider curves obtainable from the projective line by matching at several points or by inserting cusp singularities. It appears that on such singular curves, all basic ingredients of Hitchin integrable systems (moduli space of vector bundles, dualizing sheaf, Higgs field, etc.) can be explicitly described, which can be interesting in itself. Our main result is explicit formulas for the Hitchin Hamiltonians. We also show how to obtain the Hitchin integrable system on such curves by Hamiltonian reduction from a much simpler system on a finite-dimensional space. We pay special attention to a degenerate curve of genus two for which we find an analogue of the Narasimhan–Ramanan parameterization of the moduli space of SL(2) bundles as well as the explicit expressions for the symplectic structure and Hitchin-system Hamiltonians in these coordinates. We demonstrate the efficiency of our approach by rederiving the rational and trigonometric Calogero–Moser systems, which are obtained from Hitchin systems on curves with a marked point and with the respective cusp and node.     

The Néron model over the Igusa curves

We analyze the geometry of rational p-division points in degenerating families of elliptic curves in characteristic p. We classify the possible Kodaira symbols and determine for the Igusa moduli problem the reduction type of the universal curve. Special attention is paid to characteristic 2 and 3, where wild ramification and stacky phenomena show up.     

Moduli Stacks of Permutation Classes of Pointed Stable Curves

The notion of m/Γ-pointed stable curves is introduced. It should be viewed as a generalization of the notion of m-pointed stable curves of a given genus, where the labels of the marked points are only determined up to the action of a group of permutations Γ. The classical moduli spaces and moduli stacks are generalized to this wider setting. Finally, an explicit construction of the new moduli stack of m/Γ-pointed stable curves as a quotient stack is given. Received: February 2008     

The action of S n on the cohomology of \overline{M}_{0,n}(\mathbb{R})

In recent work by Etingof, Henriques, Kamnitzer, and the author, a presentation and explicit basis was given for the rational cohomology of the real locus of the moduli space of stable genus 0 curves with n marked points. We determine the graded character of the action of S_n on this space (induced by permutations of the marked points), both in the form of a plethystic formula for the cycle index, and as an explicit product formula for the value of the character on a given cycle type.      

Elliptic and hyperelliptic curves over supersimple fields in characteristic 2

In this paper, we extend a previous result of A. Pillay and the author regarding existence of rational points over elliptic and hyperelliptic curves with generic moduli defined over supersimple fields to the even characteristic case. We give a detailed exposition of the affine models of these families of curves in characteristic 2 and the transformations between members in the same rational isomorphism class.     

Polynomial Hurwitz Numbers and Intersections on \overline M _{0,k}

We express Hurwitz numbers of polynomials of arbitrary topological type in terms of intersection numbers on the moduli space of curves of genus zero with marked points.     

The rational cohomology of $${\overline{\mathcal{M}_4}}$$

We present two approaches to the study of the cohomology of moduli spaces of curves. Together, they allow us to compute the rational cohomology of the moduli space of stable complex curves of genus 4, with its Hodge structure.     

Embedding pointed curves in K3 surfaces

Mathematische Zeitschrift - We analyze morphisms from pointed curves to K3 surfaces with a distinguished rational curve, such that the marked points are taken to the rational curve, perhaps with...     

Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry

We first build the moduli spaces of real rational pseudo-holomorphic curves in a given real symplectic 4-manifold. Then, following the approach of Gromov and Witten [3, 19, 11], we define invariants under deformation of real symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational J-holomorphic curves which realize a given homology class and pass through a given real configuration of points. Mathematics Subject Classification (2000) 14N10, 14P25, 53D05, 53D45     

The orbifold cohomology of moduli of genus 3 curves

In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of the covering curve is g. Then we work out the case of genus g =  3. Furthermore, we determine the part of the orbifold cohomology of the Deligne–Mumford compactification of the moduli space of genus 3 curves that comes from the Zariski closure of the inertia stack of ${\mathcal{M}3}$ .     

Representation stability for the cohomology of arrangements associated to root systems

From Smyth’s classification, modular compactifications of the moduli space of pointed smooth rational curves are indexed by combinatorial data, the so-called extremal assignments. We explore their combinatorial structures and show that any extremal assignment is a finite union of atomic extremal assignments. We discuss a connection with the birational geometry of the moduli space of stable pointed rational curves. As applications, we study three special classes of extremal assignments: smooth, toric, and invariant with respect to the symmetric group action. We identify them with three combinatorial objects: simple intersecting families, complete multipartite graphs, and special families of integer partitions, respectively.     

Quadratic algebras related to elliptic curves

We construct quadratic finite-dimensional Poisson algebras corresponding to a rank-N degree-one vector bundle over an elliptic curve with n marked points and also construct the quantum version of the algebras. The algebras are parameterized by the moduli of curves. For N = 2 and n = 1, they coincide with Sklyanin algebras. We prove that the Poisson structure is compatible with the Lie-Poisson structure defined on the direct sum of n copies of sl(N). The origin of the algebras is related to the Poisson reduction of canonical brackets on an affine space over the bundle cotangent to automorphism groups of vector bundles. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 163–183, August, 2008.     

The Chen–Ruan cohomology of moduli of curves of genus 2 with marked points

In this work we describe the Chen–Ruan cohomology of the moduli stack of smooth and stable genus 2 curves with marked points. In the first half of the paper we compute the additive structure of the Chen–Ruan cohomology ring for the moduli stack of stable n-pointed genus 2 curves, describing it as a rationally graded vector space. In the second part we give generators for the even Chen–Ruan cohomology ring as an algebra on the ordinary cohomology.     

Mori's program for with symmetric divisors

We complete Mori's program with symmetric divisors for the moduli space of stable six‐pointed rational curves. As an application, we give an alternative proof of the complete Mori's program of the moduli space of genus two stable curves, which was first done by Hassett.     

Moduli spaces of weighted pointed stable curves

A weighted pointed curve consists of a nodal curve and a sequence of marked smooth points, each assigned a number between zero and one. A subset of the marked points may coincide if the sum of the corresponding weights is no greater than one. We construct moduli spaces for these objects using methods of the log minimal model program, and describe the induced birational morphisms between moduli spaces as the weights are varied. In the genus zero case, we explain the connection to Geometric Invariant Theory quotients of points in the projective line, and to compactifications of moduli spaces studied by Kapranov, Keel, and Losev-Manin.     

Tropicalization of Classical Moduli Spaces

The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our primary example is the Burkhardt quartic, whose tropicalization is a 3-dimensional fan in 39-dimensional space. This effectuates a synthesis of concrete and abstract approaches to tropical moduli of genus 2 curves.     

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.     

2-Weierstrass points of genus 3 hyperelliptic curves with extra involutions

We consider families of curves with extra automorphisms in ?₃, the moduli space of smooth hyperelliptic curves of genus g = 3. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. For each family of positive dimension where |Aut (C)|>4, we determine the possible distributions of weights of 2-Weierstrass points.