scholarly journals Transitive Factorizations in the Hyperoctahedral Group

2008 ◽  
Vol 60 (2) ◽  
pp. 297-312
Author(s):  
G. Bini ◽  
I. P. Goulden ◽  
D. M. Jackson
Keyword(s):  
Symmetric Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Point Of View ◽  
Moduli Space Of Curves ◽  
Reflection Groups ◽  
Hurwitz Numbers ◽  
Hyperoctahedral Group ◽  
Geometric Point ◽  
Enumeration Problem

AbstractThe classical Hurwitz enumeration problem has a presentation in terms of transitive factorizations in the symmetric group. This presentation suggests a generalization from type A to other finite reflection groups and, in particular, to type B. We study this generalization both from a combinatorial and a geometric point of view, with the prospect of providing a means of understanding more of the structure of the moduli spaces of maps with an S2-symmetry. The type A case has been well studied and connects Hurwitz numbers to the moduli space of curves. We conjecture an analogous setting for the type B case that is studied here.

scholarly journals Geometric Perspective on Piecewise Polynomiality of Double Hurwitz Numbers

2014 ◽  
Vol 57 (4) ◽  
pp. 749-764 ◽  
Author(s):  
Renzo Cavalieri ◽  
Steffen Marcus
Keyword(s):  
Moduli Space ◽  
Moduli Space Of Curves ◽  
Intersection Numbers ◽  
Hurwitz Numbers ◽  
Suggestive Evidence ◽  
Wall Crossing ◽  
Double Ramification Cycle ◽  
Correction Terms

AbstractWe describe doubleHurwitz numbers as intersection numbers on the moduli space of curves Using a result on the polynomiality of intersection numbers of psi classes with the Double Ramification Cycle, our formula explains the polynomiality in chambers of double Hurwitz numbers and the wall-crossing phenomenon in terms of a variation of correction terms to the ψ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle (which is only known in genera 0 and 1).

scholarly journals Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields

2014 ◽  
Vol 17 (A) ◽  
pp. 128-147 ◽  
Author(s):  
Reynald Lercier ◽  
Christophe Ritzenthaler ◽  
Florent Rovetta ◽  
Jeroen Sijsling
Keyword(s):  
Finite Fields ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Moduli Space Of Curves ◽  
Isomorphism Classes ◽  
Families Of Curves ◽  
Plane Quartics ◽  
Smooth Plane

AbstractWe study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.

TWO REMARKS ON SUBVARIETIES OF MODULI SPACES

2008 ◽  
Vol 19 (02) ◽  
pp. 237-243 ◽  
Author(s):  
KIRTI JOSHI
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Abelian Varieties ◽  
Moduli Space Of Curves ◽  
Rank Zero ◽  
Zero Locus

We study two natural questions about subvarieties of moduli spaces. In the first section, we study the locus of curves equipped with F-nilpotent bundles and its relationship to the p-rank zero locus of the moduli space of curves of genus g. In the second section, we study subvarieties of moduli spaces of vector bundles on curves. We prove an analogue of a result of F. Oort about proper subvarieties of moduli of abelian varieties.

scholarly journals MODULI SPACES OF TOPOLOGICAL CALIBRATIONS, CALABI–YAU, HYPERKÄHLER, G2 and SPIN(7) STRUCTURES

2004 ◽  
Vol 15 (03) ◽  
pp. 211-257 ◽  
Author(s):  
RYUSHI GOTO
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Finite Dimension ◽  
Type Theorem ◽  
Differential Forms ◽  
Direct Proof ◽  
Stability Theorem ◽  
Point Of View ◽  
Geometric Structures ◽  
New Approach

This paper focuses on a geometric structure defined by a system of closed exterior differential forms and develops a new approach to deformation problems of geometric structures. We obtain a criterion for unobstructed deformations from a cohomological point of view (Theorem 1.7). Further we show that under a cohomological condition, the moduli space of the geometric structures becomes a smooth manifold of finite dimension (Theorem 1.8). We apply our approach to the geometric structures such as Calabi–Yau, HyperKähler, G2 and Spin(7) structures and then obtain a unified construction of smooth moduli spaces of these four geometric structures. We generalize the Moser's stability theorem to provide a direct proof of the local Torelli type theorem in these four geometric structures (Theorem 1.10).

scholarly journals Zero cycles on the moduli space of curves

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Rahul Pandharipande ◽  
Johannes Schmitt
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Rational Surface ◽  
K3 Surfaces ◽  
Moduli Of Curves ◽  
K3 Surface ◽  
Moduli Space Of Curves ◽  
Moduli Spaces Of Curves ◽  
Open Questions ◽  
Witten Theory

While the Chow groups of 0-dimensional cycles on the moduli spaces of Deligne-Mumford stable pointed curves can be very complicated, the span of the 0-dimensional tautological cycles is always of rank 1. The question of whether a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is subtle. Our main results address the question for curves on rational and K3 surfaces. If C is a nonsingular curve on a nonsingular rational surface of positive degree with respect to the anticanonical class, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the virtual dimension in Gromov-Witten theory of the moduli space of stable maps. If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the genus of C and every marking is a Beauville-Voisin point. The latter result provides a connection between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1 tautological 0-cycles on K3 surfaces. Several further results related to tautological 0-cycles on the moduli spaces of curves are proven. Many open questions concerning the moduli points of curves on other surfaces (Abelian, Enriques, general type) are discussed. Comment: Published version

scholarly journals Tropical covers of curves and their moduli spaces

2014 ◽  
Vol 17 (01) ◽  
pp. 1350045 ◽  
Author(s):  
Arne Buchholz ◽  
Hannah Markwig
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Stable Maps ◽  
Hurwitz Numbers ◽  
Polyhedral Complex ◽  
Local Duality ◽  
The One ◽  
Definition Of ◽  
Correspondence Theorem

We define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications. Our main result is the invariance of the degree of the branch map, which enables us to give a tropical intersection-theoretic definition of tropical triple Hurwitz numbers. We show that our intersection-theoretic definition coincides with the one given in [B. Bertrand, E. Brugallé and G. Mikhalkin, Tropical open Hurwitz numbers, Rend. Semin. Mat. Univ. Padova 125 (2011) 157–171] where a Correspondence Theorem for Hurwitz numbers is proved. Thus we provide a tropical intersection-theoretic justification for the multiplicities with which a tropical cover has to be counted. Our method of proof is to establish a local duality between our tropical moduli spaces and certain moduli spaces of relative stable maps to ℙ1.

scholarly journals Moduli spaces of Calabi-Yau d-folds as gravitational-chiral instantons

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Sergio Cecotti
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Equations Of Motion ◽  
Point Of View ◽  
Chiral Model ◽  
Gravitational Instanton ◽  
Complex Dimension ◽  
Effective Lagrangian ◽  
Finite Action ◽  
M Theory

Abstract Motivated by the swampland program, we show that the Weil-Petersson geometry of the moduli space of a Calabi-Yau manifold of complex dimension d ≤ 4 is a gravitational instanton (i.e. a finite-action solution of the Euclidean equations of motion of gravity with matter). More precisely, the moduli geometry of Calabi-Yau d-folds (d ≤ 4) describes instantons of (E)AdS Einstein gravity coupled to a standard chiral model.From the point of view of the low-energy physics of string/M-theory compactified on the Calabi-Yau X, the various fields propagating on its moduli space are the couplings appearing in the effective Lagrangian "Image missing".

scholarly journals A MODULI STACK OF TROPICAL CURVES

2020 ◽  
Vol 8 ◽  
Author(s):  
RENZO CAVALIERI ◽  
MELODY CHAN ◽  
MARTIN ULIRSCH ◽  
JONATHAN WISE
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Algebraic Curves ◽  
Tropical Geometry ◽  
Moduli Space Of Curves ◽  
Tropical Curves ◽  
Moduli Spaces Of Curves ◽  
Moduli Stack ◽  
Marked Points ◽  
Moduli Problems

We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves. Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.

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