scholarly journals REPRESENTATION VARIETIES OF ALGEBRAS WITH NODES

2021 ◽  
pp. 1-31
Author(s):  
Ryan Kinser ◽  
András C. Lőrincz
Keyword(s):  
Moduli Spaces ◽  
Rational Singularities ◽  
Orbit Closures ◽  
Node Splitting ◽  
Irreducible Components ◽  
Representation Varieties

Abstract We study the behaviour of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed subvarieties of representation varieties for different algebras, which preserves properties like normality or having rational singularities. Furthermore, we describe how the defining equations of such closed subvarieties change under the correspondence. By working in the ‘relative setting’ (splitting one node at a time), we demonstrate that there are many nonhereditary algebras whose irreducible components of representation varieties are all normal with rational singularities. We also obtain explicit generators of the prime defining ideals of these irreducible components. This class contains all radical square zero algebras, but also many others, as illustrated by examples throughout the paper. We also show that this is true when irreducible components are replaced by orbit closures, for a more restrictive class of algebras. Lastly, we provide applications to decompositions of moduli spaces of semistable representations of certain algebras.

scholarly journals Homology TQFT's and the Alexander–Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory

2003 ◽  
Vol 55 (4) ◽  
pp. 766-821 ◽  
Author(s):  
Thomas Kerler
Keyword(s):  
Moduli Spaces ◽  
Hopf Algebras ◽  
Large Family ◽  
Quantum Invariants ◽  
Topological Quantum ◽  
Irreducible Components ◽  
Skein Theory ◽  
Higher Rank ◽  
The One ◽  
Representation Varieties

AbstractWe develop an explicit skein-theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the homology ofU(1)-representation varieties on the one side and the combinatorially constructed Hennings TQFT based on the quasitriangular Hopf algebra= ℤ/2 n ⋊ Λ* ℝ2on the other side. We find that both TQFT's are SL(2; ℝ)-equivariant functors and, as such, are isomorphic. The SL(2; ℝ)-action in the Hennings construction comes from the natural action onand in the case of the Frohman–Nicas theory from the Hard–Lefschetz decomposition of theU(1)-moduli spaces given that they are naturally Kähler. The irreducible components of this TQFT, corresponding to simple representations of SL(2; ℤ) and Sp(2g; ℤ), thus yield a large family of homological TQFT's by taking sums and products. We give several examples of TQFT's and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion, Seiberg–Witten theories, Casson type theories for homology circlesà laDonaldson, higher rank gauge theories following Frohman and Nicas, and the ℤ=pℤ reductions of Reshetikhin.Turaev theories over the cyclotomic integers ℤ[ζp]. We also conjecture that the Hennings TQFT for quantum-sl2is the product of the Reshetikhin–Turaev TQFT and such a homological TQFT.

scholarly journals A modular compactification of ℳ1,n from A∞-structures

2019 ◽  
Vol 2019 (755) ◽  
pp. 151-189
Author(s):  
Yankı Lekili ◽  
Alexander Polishchuk
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Rational Singularities ◽  
Picard Groups

AbstractWe show that a certain moduli space of minimal A_{\infty}-structures coincides with the modular compactification {\overline{\mathcal{M}}}_{1,n}(n-1) of \mathcal{M}_{1,n} constructed by Smyth in [26]. In addition, we describe these moduli spaces and the universal curves over them by explicit equations, prove that they are normal and Gorenstein, show that their Picard groups have no torsion and that they have rational singularities if and only if n\leq 11.

scholarly journals Non-abelian cohomology jump loci from an analytic viewpoint

2014 ◽  
Vol 16 (04) ◽  
pp. 1350025 ◽  
Author(s):  
Alexandru Dimca ◽  
Ştefan Papadima
Keyword(s):  
Minimal Model ◽  
Nilpotent Groups ◽  
Exponential Map ◽  
Flat Connections ◽  
Differential Graded Algebra ◽  
Irreducible Components ◽  
Rank One ◽  
Positive Weights ◽  
Projective Manifolds ◽  
Representation Varieties

For a space, we investigate its CJL (cohomology jump loci), sitting inside varieties of representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its CJL, sitting inside varieties of flat connections. The analytic germs at the origin 1 of representation varieties are shown to be determined by the Sullivan 1-minimal model of the space. Up to a degree q, the two types of CJL have the same analytic germs at the origins, when the space and the algebra have the same q-minimal model. We apply this general approach to formal spaces (obtaining the degeneration of the Farber–Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic CJL: all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic CJL into their topological counterpart.

scholarly journals GEOMETRY OF MODULI SPACES OF FLAT BUNDLES ON PUNCTURED SURFACES

1998 ◽  
Vol 09 (01) ◽  
pp. 63-73 ◽  
Author(s):  
PHILIP A. FOTH
Keyword(s):  
Riemann Surface ◽  
Conjugacy Class ◽  
Moduli Spaces ◽  
Flat Connections ◽  
Rational Singularities ◽  
Take All

For a Riemann surface with one puncture we consider moduli spaces of flat connections such that the monodromy transformation around the puncture belongs to a given conjugacy class with the property that a product of its distinct eigenvalues is not equal to 1 unless we take all of them. We prove that these moduli spaces are smooth and their natural closures are normal with rational singularities.

scholarly journals Degenerate Affine Flag Varieties and Quiver Grassmannians

2020 ◽  
Author(s):  
Alexander Pütz
Keyword(s):  
Flag Varieties ◽  
Rational Singularities ◽  
Quiver Grassmannians ◽  
Finite Dimensional ◽  
Irreducible Components ◽  
Affine Flag

AbstractWe study finite dimensional approximations to degenerate versions of affine flag varieties using quiver Grassmannians for cyclic quivers. We prove that they admit cellular decompositions parametrized by affine Dellac configurations, and that their irreducible components are normal Cohen-Macaulay varieties with rational singularities.

Parabolic Subgroups with Abelian Unipotent Radical as a Testing Site for Invariant Theory

1999 ◽  
Vol 51 (3) ◽  
pp. 616-635 ◽  
Author(s):  
Dmitri I. Panyushev
Keyword(s):  
Polynomial Algebra ◽  
Free Module ◽  
Unipotent Radical ◽  
Parabolic Subgroups ◽  
Simple Algebraic Group ◽  
Orbit Closures ◽  
Determinantal Varieties ◽  
Commuting Variety ◽  
Irreducible Components ◽  
Testing Site

AbstractLet L be a simple algebraic group and P a parabolic subgroup with Abelian unipotent radical Pu. Many familiar varieties (determinantal varieties, their symmetric and skew-symmetric analogues) arise as closures of P-orbits in Pu. We give a unified invariant-theoretic treatment of various properties of these orbit closures. We also describe the closures of the conormal bundles of these orbits as the irreducible components of some commuting variety and show that the polynomial algebra k[Pu] is a free module over the algebra of covariants.

scholarly journals RATIONAL FAMILIES OF VECTOR BUNDLES ON CURVES

2004 ◽  
Vol 15 (01) ◽  
pp. 13-45 ◽  
Author(s):  
ANA-MARIA CASTRAVET
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Rational Curves ◽  
Complex Curve ◽  
Stable Vector Bundles ◽  
Irreducible Components ◽  
Rationally Connected ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

Let C be a smooth projective complex curve of genus g≥2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1. For any k≥1, we find all the irreducible components of the space of rational curves on M, of degree k. In particular, we find the maximal rationally connected fibrations of these components. We prove that there is a one-to-one correspondence between moduli spaces of rational curves on M and moduli spaces of rank 2 vector bundles on ℙ1×C.

scholarly journals GEOMETRY OF THE MODULI OF HIGHER SPIN CURVES

2000 ◽  
Vol 11 (05) ◽  
pp. 637-663 ◽  
Author(s):  
TYLER J. JARVIS
Keyword(s):  
Line Bundle ◽  
Moduli Spaces ◽  
Disjoint Union ◽  
Algebraic Curves ◽  
Quantum Cohomology ◽  
Smooth Curve ◽  
Natural Generalization ◽  
Intersection Theory ◽  
Universal Curve ◽  
Irreducible Components

This article treats various aspects of the geometry of the moduli [Formula: see text] of r-spin curves and its compactification [Formula: see text]. Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. In particular, these spaces are the subject of a remarkable conjecture of E. Witten relating their intersection theory to the Gelfand–Dikii (KdVr) heirarchy. There is also a W-algebra conjecture for these spaces [16] generalizing the Virasoro conjecture of quantum cohomology. For any line bundle [Formula: see text] on the universal curve over the stack of stable curves, there is a smooth stack [Formula: see text] of triples (X, ℒ, b) of a smooth curve X, a line bundle ℒ on X, and an isomorphism [Formula: see text]. In the special case that [Formula: see text] is the relative dualizing sheaf, then [Formula: see text] is the stack [Formula: see text] of r-spin curves. We construct a smooth compactification [Formula: see text] of the stack [Formula: see text], describe the geometric meaning of its points, and prove that its coarse moduli is projective. We also prove that when r is odd and g>1, the compactified stack of spin curves [Formula: see text] and its coarse moduli space [Formula: see text] are irreducible, and when r is even and [Formula: see text] is the disjoint union of two irreducible components. We give similar results for n-pointed spin curves, as required for Witten's conjecture, and also generalize to the n-pointed case the classical fact that when [Formula: see text] is the disjoint union of d(r) components, where d(r) is the number of positive divisors of r. These irreducibility properties are important in the study of the Picard group of [Formula: see text] [15], and also in the study of the cohomological field theory related to Witten's conjecture [16, 34].

MODULI SPACES OF STABLE BUNDLES ON K3 FIBERED CALABI–YAU THREEFOLDS

2003 ◽  
Vol 05 (01) ◽  
pp. 119-126 ◽  
Author(s):  
TOHRU NAKASHIMA
Keyword(s):  
Moduli Spaces ◽  
Projective Spaces ◽  
Stable Rank ◽  
Stable Bundles ◽  
Irreducible Components

In this paper we study stable rank two bundles on a Calabi–Yau threefold. For hypersurfaces in a ℙ3-bundle over ℙ1, we show that their moduli spaces have irreducible components which are birational to projective spaces.

scholarly journals Geometric Manin’s conjecture and rational curves

2019 ◽  
Vol 155 (5) ◽  
pp. 833-862 ◽  
Author(s):  
Brian Lehmann ◽  
Sho Tanimoto
Keyword(s):  
Growth Rate ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Counting Function ◽  
Rational Curves ◽  
Fano Variety ◽  
Complex Numbers ◽  
Number Of Components ◽  
Irreducible Components ◽  
Manin’S Conjecture

Let$X$be a smooth projective Fano variety over the complex numbers. We study the moduli space of rational curves on$X$using the perspective of Manin’s conjecture. In particular, we bound the dimension and number of components of spaces of rational curves on$X$. We propose a geometric Manin’s conjecture predicting the growth rate of a counting function associated to the irreducible components of these moduli spaces.

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