Search results for "Moduli space"
showing 10 items of 45 documents
The unequal mass sunrise integral expressed through iterated integrals on M‾1,3
2020
Abstract We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter e. In order to do so, we transform the system of differential equations for the master integrals to an e-form. The sunrise integral with unequal masses depends on three kinematical variables. We perform a change of variables to standard coordinates on the moduli space M 1 , 3 of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on M ‾ 1 , 3 . On the hypersurface τ = const our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.
Moduli spaces of discrete gravity
2003
Spectral triples describe and generalize Riemannian spin geometries by converting the geometrical information into algebraic data, which consist of an algebra $A$, a Hilbert space $H$ carrying a representation of $A$ and the Dirac operator $D$ (a selfadjoint operator acting on $H$). The gravitational action is described by the trace of a suitable function of $D$. In this paper we examine the (path-integral-) quantization of such a system given by a finite dimensional commutative algebra. It is then (in concrete examples) possible to construct the moduli space of the theory, i.e. to divide the space of all Dirac operators by the action of the diffeomorphism group, and to construct an invaria…
On the Rational Cohomology of Moduli Spaces of Curves with Level Structures
2009
We investigate low degree rational cohomology groups of smooth compactifications of moduli spaces of curves with level structures. In particular, we determine $H^k(\sgbar, \Q)$ for $g \ge 2$ and $k \le 3$, where $\sgbar$ denotes the moduli space of spin curves of genus $g$.
On GIT quotients of Hilbert and Chow schemes of curves
2011
The aim of this note is to announce some results on the GIT problem for the Hilbert and Chow scheme of curves of degree d and genus g in P^{d-g}, whose full details will appear in a subsequent paper. In particular, we extend the previous results of L. Caporaso up to d>4(2g-2) and we observe that this is sharp. In the range 2(2g-2)<d<7/2(2g-2), we get a complete new description of the GIT quotient. As a corollary, we get a new compactification of the universal Jacobian over the moduli space of pseudo-stable curves.
La singularité de O’Grady
2006
Let M 2 v M_{2v} be the moduli space of semistable sheaves with Mukai vector 2 v 2v on an abelian or K 3 K3 surface where v v is primitive such that ⟨ v , v ⟩ = 2 \langle v,v \rangle =2 . We show that the blow-up of the reduced singular locus of M 2 v M_{2v} provides a symplectic resolution of singularities. This provides a direct description of O’Grady’s resolutions of M K 3 ( 2 , 0 , 4 ) M_{K3}(2,0,4) and M A b ( 2 , 0 , 2 ) M_{Ab}(2,0,2) . Résumé. Soit M 2 v M_{2v} l’espace de modules des faisceaux semi-stables de vecteur de Mukai 2 v 2v sur une surface K 3 K3 ou abélienne où v v est primitif tel que ⟨ v , v ⟩ = 2 \langle v,v \rangle =2 . Nous montrons que l’éclatement de M 2 v M_{2v} le…
Picard-Fuchs operators for octic arrangements, I: the case of orphans
2019
We report on $25$ families of projective Calabi-Yau threefolds that do not have a point of maximal unipotent monodromy in their moduli space. The construction is based on an analysis of certain pencils of octic arrangements that were found by C. Meyer. There are seven cases where the Picard-Fuchs operator is of order two and $18$ cases where it is of order four. The birational nature of the Picard-Fuchs operator can be used effectively to distinguish between families whose members have the same Hodge numbers.
The Lang–Vojta Conjectures on Projective Pseudo-Hyperbolic Varieties
2020
These notes grew out of a mini-course given from May 13th to May 17th at UQAM in Montreal during a workshop on Diophantine Approximation and Value Distribution Theory.
Stability conditions and related filtrations for $(G,h)$-constellations
2017
Given an infinite reductive algebraic group $G$, we consider $G$-equivariant coherent sheaves with prescribed multiplicities, called $(G,h)$-constellations, for which two stability notions arise. The first one is analogous to the $\theta$-stability defined for quiver representations by King and for $G$-constellations by Craw and Ishii, but depending on infinitely many parameters. The second one comes from Geometric Invariant Theory in the construction of a moduli space for $(G,h)$-constellations, and depends on some finite subset $D$ of the isomorphy classes of irreducible representations of $G$. We show that these two stability notions do not coincide, answering negatively a question raise…
On stability of logarithmic tangent sheaves. Symmetric and generic determinants
2021
We prove stability of logarithmic tangent sheaves of singular hypersurfaces D of the projective space with constraints on the dimension and degree of the singularities of D. As main application, we prove that determinants and symmetric determinants have stable logarithmic tangent sheaves and we describe an open dense piece of the associated moduli space.
A note on the unirationality of a moduli space of double covers
2010
In this note we look at the moduli space $\cR_{3,2}$ of double covers of genus three curves, branched along 4 distinct points. This space was studied by Bardelli, Ciliberto and Verra. It admits a dominating morphism $\cR_{3,2} \to {\mathcal A}_4$ to Siegel space. We show that there is a birational model of $\cR_{3,2}$ as a group quotient of a product of two Grassmannian varieties. This gives a proof of the unirationality of $\cR_{3,2}$ and hence a new proof for the unirationality of ${\mathcal A}_4$.