Weighted Projective Lines as Fine Moduli Spaces of Quiver Representations

We describe weighted projective lines in the sense of Geigle and Lenzing by a moduli problem on the canonical algebra of Ringel. We then go on to study generators of the derived categories of coherent sheaves on the total spaces of their canonical bundles, and show that they are rarely tilting. We also give a moduli construction for these total spaces for weighted projective lines with three orbifold points.     

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.     

An equilateral four point property which characterizes generalized euclidean spaces

Various characterizations of generalized euclidean spaces among complete metric spaces which contain a metric line joining each two points make use of euclidean four point properties, which require that every quadruple from a suitably chosen class of quadruples of points of the metric space be isometric with a quadruple of euclidean points. The present paper shows if every quadruple which contains an equilateral triple and a linear triple is embeddable, the space is generalized euclidean.     

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.     

Moduli spaces of local systems and higher Teichmüller theory

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.     

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.     

Matrix Models: Geometry of Moduli Spaces and Exact Solutions

We study the connection between characteristics of moduli spaces of Riemann surfaces with marked points and matrix models. The Kontsevich matrix model describes intersection indices on continuous moduli spaces, and the Kontsevich–Penner matrix model describes intersection indices on discretized moduli spaces. Analyzing the constraint algebras satisfied by various generalized Kontsevich matrix models, we derive time transformations that establish exact relations between different models appearing in mathematical physics. We solve the Hermitian one-matrix model using the moment technique in the genus expansion and construct a recursive procedure for solving this model in the double scaling limit.     

Fano varieties of cubic fourfolds containing a plane

We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3 surface. The moduli spaces of complexes and of sheaves are related by wall-crossing in the derived category of twisted sheaves on the corresponding K3 surface.     

Equivariant cohomology of certain moduli of weighted pointed rational curves

We determine the action of the product of symmetric groups on the cohomology of certain moduli of weighted pointed rational curves. The moduli spaces that we study are of stable rational curves with m + n marked points where the first m marked points are distinct from all the others where as the last n may coincide among themselves. We give a recipe for calculating the equivariant Poincaré polynomials and list them for small m and n.     

A new class of four-point properties which characterize euclidean spaces

Characterizations of generalized euclidean spaces by means of euclidean four-point properties.state that every metric space which is complete, and which contains a metric line joining each two of its points is a generalized euclidean space if and only if each quadruple from a certain class of quadruples of the space is congruent with a quadruple of points in a euclidean space. It is known that it suffices to consider only quadruples containing a linear triple, or quadruples in which one of the linear points is a metric midpoint of the other two. Another class of four-point properties involves quadruples which contain a linear triple and a point equidistant from two of the linear points. The present paper presents three characterizations of euclidean spaces based on four-point properties in which the embedded quadruples contain a linear triple and some three of the distances determined by the four points are equal.     

The index of Floer moduli problems for parametrized action functionals

We define an index for the critical points of parametrized Hamiltonian action functionals. The expected dimension of moduli spaces of parametrized Floer trajectories equals the difference of indices of the asymptotes.     

The Voronoi Implicit Interface Method and Computational Challenges in Multiphase Physics

We review some of the motivation and development of the Voronoi Implicit Interfaces Method (VIIM), first introduced in [10], for tracking multiple interacting and evolving regions, whose motion is determined by complex physics that include hydrodynamic, elastic, and geometric forces. The method automatically handles multiple junctions, triple points and quadruple points in two dimensions, as well as triple lines, etc. in higher dimensions, and topological changes in the system occur naturally, with no surgery required.     

Moduli dimensions of lattice polygons

Journal of Algebraic Combinatorics - Given a lattice polygon P with g interior lattice points, we can associate to $$P$$ two moduli spaces: the moduli space of algebraic curves that are...     

Four-point characterizations of real inner product spaces

Generalized euclidean spaces have been characterized among metric spaces by the requirement that each member of certain classes of quadruples of points of the metric space be congruent to a quadruple of points of a euclidean space. The present paper strengthens earlier characterizations which only require the embedding of certain classes of quadruples which contain a linear triple and in which some three of the six distances between pairs of points are equal. These results generalize some similar characterizations of euclidean spaces among normed linear spaces. Received 4 January 1999; revised 12 August 2002.     

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.     

On the moduli spaces of multipolygonal linkages in the plane

One can form a polygonal linkage by identifying initial and terminal points of two free linkages. Likewise, one can form a multipolygonal linkage by identifying initial and terminal points of three free linkages. The geometric and topological properties of moduli spaces of multipolygonal linkages in the plane are studied. These spaces are compact algebraic varieties. Some conditions under which these spaces are smooth manifolds, cross products or disjoint unions of moduli spaces of polygonal linkages, or connected, are determined. Dimensions in smooth manifold cases and some Euler characteristics are computed. A classification of generic multiquadrilateral linkages is also made.     

Counting bitangents with stable maps

This paper is an elementary introduction to the theory of moduli spaces of curves and maps. As an application to enumerative geometry, we show how to count the number of bitangent lines to a projective plane curve of degree d by doing intersection theory on moduli spaces.     

Calabi–Yau Threefolds and Moduli of Abelian Surfaces I

We describe birational models and decide the rationality/unirationality of moduli spaces A _d (and A _d ^lev ) of (1, d)-polarized Abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of Abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi–Yau threefolds with Euler characteristic zero.     

Moduli spaces of metric graphs of genus 1 with marks on vertices

In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces , which parametrize the isometry classes of metric graphs of genus 1 with n marks on vertices are homotopy equivalent to the spaces TM1,n, which are the moduli spaces of tropical curves of genus 1 with n marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the so-called scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus.     

On moduli spaces of sheaves on K3 or abelian surfaces

We investigate the moduli spaces of one- and two-dimensional sheaves on projective K3 and abelian surfaces that are semistable with respect to a nongeneral ample divisor with regard to the symplectic resolvability. We can exclude the existence of new examples of projective irreducible symplectic manifolds lying birationally over components of the moduli spaces of one-dimensional semistable sheaves on K3 surfaces, and over components of many of the moduli spaces of two-dimensional sheaves on K3 surfaces, in particular, of those for rank two sheaves.