On the Euler characteristics of certain moduli spaces of $1$-dimensional closed subschemes

2021 ◽  
Vol 17 (1) ◽  
pp. 349-384
Author(s):  
Mazen M. Alhwaimel
Keyword(s):  
Moduli Spaces ◽  
Euler Characteristics

scholarly journals On the Euler characteristics of certain moduli spaces of 1-dimensional subschemes

2017 ◽  
Author(s):  
◽  
Mazen M. Alhwaimel
Keyword(s):  
Generating Function ◽  
Moduli Spaces ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Euler Characteristics ◽  
Torus Actions ◽  
3 Dimensional ◽  
Hodge Polynomials

Generalizing the ideas in [LQ] and using virtual Hodge polynomials as well as torus actions, we compute the Euler characteristics of some moduli spaces of 1-dimensional closed subschemes when the ambient smooth projective variety admits a Zariskilocally trivial fibration to a codimension-1 base. As a consequence, we partially verify a conjecture of W.-P. Li and Qin [LQ]. We also calculate the generating function for the number of certain punctual 3-dimensional partitions, which is used to compute the above Euler characteristics.

scholarly journals Instantons, Topological Strings, and Enumerative Geometry

2010 ◽  
Vol 2010 ◽  
pp. 1-70 ◽  
Author(s):  
Richard J. Szabo
Keyword(s):  
Moduli Spaces ◽  
Gauge Theories ◽  
Topological String ◽  
Two Dimensions ◽  
Enumerative Geometry ◽  
Topological String Theory ◽  
Euler Characteristics ◽  
Wall Crossing ◽  
Supersymmetric Gauge ◽  
Torsion Free

We review and elaborate on certain aspects of the connections between instanton counting in maximally supersymmetric gauge theories and the computation of enumerative invariants of smooth varieties. We study in detail three instances of gauge theories in six, four, and two dimensions which naturally arise in the context of topological string theory on certain noncompact threefolds. We describe how the instanton counting in these gauge theories is related to the computation of the entropy of supersymmetric black holes and how these results are related to wall-crossing properties of enumerative invariants such as Donaldson-Thomas and Gromov-Witten invariants. Some features of moduli spaces of torsion-free sheaves and the computation of their Euler characteristics are also elucidated.

scholarly journals Moduli Spaces of Point Configurations and Plane Curve Counts

2019 ◽  
Author(s):  
Markus Reineke ◽  
Thorsten Weist
Keyword(s):  
Recurrence Relation ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Plane Curve ◽  
Projective Spaces ◽  
Rational Curves ◽  
Euler Characteristics ◽  
Point Configurations

Abstract We identify certain Gromov–Witten invariants counting rational curves with given incidence and tangency conditions with the Euler characteristics of moduli spaces of point configurations in projective spaces. On the Gromov–Witten side, S. Fomin and G. Mikhalkin established a recurrence relation via tropicalization, which is realized on the moduli space side using Donaldson–Thomas invariants of subspace quivers.

scholarly journals Cohomology of quiver moduli, functional equations, and integrality of Donaldson–Thomas type invariants

2011 ◽  
Vol 147 (3) ◽  
pp. 943-964 ◽  
Author(s):  
Markus Reineke
Keyword(s):  
Moduli Spaces ◽  
Hilbert Scheme ◽  
Functional Equations ◽  
Euler Characteristics ◽  
System Of Functional Equations ◽  
Representations Of Quivers ◽  
Wall Crossing

AbstractA system of functional equations relating the Euler characteristics of moduli spaces of stable representations of quivers and the Euler characteristics of (Hilbert-scheme-type) framed versions of quiver moduli is derived. This is applied to wall-crossing formulas for the Donaldson–Thomas type invariants of M. Kontsevich and Y. Soibelman, in particular confirming their integrality.

scholarly journals Moduli spaces for Lamé functions and Abelian differentials of the second kind

2021 ◽  
pp. 2150028
Author(s):  
Alexandre Eremenko ◽  
Andrei Gabrielov ◽  
Gabriele Mondello ◽  
Dmitri Panov
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Positive Curvature ◽  
Connected Components ◽  
Geometric Description ◽  
Euler Characteristics ◽  
Conic Singularity ◽  
Constant Positive Curvature ◽  
Analytic Curves

The topology of the moduli space for Lamé functions of degree [Formula: see text] is determined: this is a Riemann surface which consists of two connected components when [Formula: see text]; we find the Euler characteristics and genera of these components. As a corollary we prove a conjecture of Maier on degrees of Cohn’s polynomials. These results are obtained with the help of a geometric description of these Riemann surfaces, as quotients of the moduli spaces for certain singular flat triangles. An application is given to the study of metrics of constant positive curvature with one conic singularity with the angle [Formula: see text] on a torus. We show that the degeneration locus of such metrics is a union of smooth analytic curves and we enumerate these curves.

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