Moduli Space of Semistable G-Bundles Over a Smooth Curve

2021 ◽  
pp. 277-327
Keyword(s):  
Moduli Space ◽  
Smooth Curve

scholarly journals HALF NIKULIN SURFACES AND MODULI OF PRYM CURVES

2019 ◽  
pp. 1-38
Author(s):  
Andreas Leopold Knutsen ◽  
Margherita Lelli-Chiesa ◽  
Alessandro Verra
Keyword(s):  
Irreducible Component ◽  
Moduli Space ◽  
Smooth Curve ◽  
Complete Picture ◽  
Boundary Points

Let ${\mathcal{F}}_{g}^{\mathbf{N}}$ be the moduli space of polarized Nikulin surfaces $(Y,H)$ of genus $g$ and let ${\mathcal{P}}_{g}^{\mathbf{N}}$ be the moduli of triples $(Y,H,C)$ , with $C\in |H|$ a smooth curve. We study the natural map $\unicode[STIX]{x1D712}_{g}:{\mathcal{P}}_{g}^{\mathbf{N}}\rightarrow {\mathcal{R}}_{g}$ , where ${\mathcal{R}}_{g}$ is the moduli space of Prym curves of genus $g$ . We prove that it is generically injective on every irreducible component, with a few exceptions in low genus. This gives a complete picture of the map $\unicode[STIX]{x1D712}_{g}$ and confirms some striking analogies between it and the Mukai map $m_{g}:{\mathcal{P}}_{g}\rightarrow {\mathcal{M}}_{g}$ for moduli of triples $(Y,H,C)$ , where $(Y,H)$ is any genus $g$ polarized $K3$ surface. The proof is by degeneration to boundary points of a partial compactification of ${\mathcal{F}}_{g}^{\mathbf{N}}$ . These represent the union of two surfaces with four even nodes and effective anticanonical class, which we call half Nikulin surfaces. The use of this degeneration is new with respect to previous techniques.

scholarly journals COMPACTIFICATION OF THE PRYM MAP FOR NON-CYCLIC TRIPLE COVERINGS

2013 ◽  
Vol 24 (03) ◽  
pp. 1350015 ◽  
Author(s):  
HERBERT LANGE ◽  
ANGELA ORTEGA
Keyword(s):  
Moduli Space ◽  
Smooth Curve ◽  
Jacobian Variety ◽  
Prym Variety ◽  
Prym Varieties ◽  
Jacobian Varieties ◽  
Genus 2 ◽  
Proper Map ◽  
Dimension 2 ◽  
Prym Map

According to [H. Lange and A. Ortega, Prym varieties of triple coverings, Int. Math. Res. Notices2011(22) (2011) 5045–5075], the Prym variety of any non-cyclic étale triple cover f : Y → X of a smooth curve X of genus 2 is a Jacobian variety of dimension 2. This gives a map from the moduli space of such covers to the moduli space of Jacobian varieties of dimension 2. We extend this map to a proper map Pr of a moduli space [Formula: see text] of admissible S3-covers of genus 7 to the moduli space [Formula: see text] of principally polarized abelian surfaces. The main result is that [Formula: see text] is finite surjective of degree 10.

scholarly journals On the theta divisor of SU(r; 1)

2002 ◽  
Vol 165 ◽  
pp. 179-193
Author(s):  
Sonia Brivio ◽  
Alessandro Verra
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Smooth Curve ◽  
Theta Divisor ◽  
Stable Vector Bundles

Let SU(r; 1) be the moduli space of stable vector bundles, on a smooth curve C of genus g ≥ 2, with rank r ≥ 3 and determinant OC(p), p ∈ C; let L be the generalized theta divisor on SU(r; 1). In this paper we prove that the map øL, defined by L, is a morphism and has degree 1.

scholarly journals MODULI SPACES OF PARABOLIC HIGGS BUNDLES AND PARABOLIC K(D) PAIRS OVER SMOOTH CURVES: I

1996 ◽  
Vol 07 (05) ◽  
pp. 573-598 ◽  
Author(s):  
HANS U. BODEN ◽  
KÔJI YOKOGAWA
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Morse Theory ◽  
Smooth Curve ◽  
Higgs Bundles ◽  
Poincaré Polynomial ◽  
Smooth Curves ◽  
Connected Manifold ◽  
Parabolic Bundles ◽  
Simply Connected

This paper concerns the moduli spaces of rank-two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to be a non-compact, connected, simply connected manifold, and a computation of its Poincaré polynomial is given.

scholarly journals Geometry of Vector Bundle Extensions and Applications to a Generalised Theta Divisor

2013 ◽  
Vol 112 (1) ◽  
pp. 61
Author(s):  
George H. Hitching
Keyword(s):  
Moduli Space ◽  
Tangent Cone ◽  
Vector Bundles ◽  
Smooth Curve ◽  
Line Bundles ◽  
Singularity Theorem ◽  
Theta Divisor ◽  
Stable Point ◽  
Roch Theorem ◽  
Complex Projective

Let $E$ and $F$ be vector bundles over a complex projective smooth curve $X$, and suppose that $0 \to E \to W \to F \to 0$ is a nontrivial extension. Let $G \subseteq F$ be a subbundle and $D$ an effective divisor on $X$. We give a criterion for the subsheaf $G(-D) \subset F$ to lift to $W$, in terms of the geometry of a scroll in the extension space ${\mathbf{P}} H^{1}(X, \mathrm{Hom}(F, E))$. We use this criterion to describe the tangent cone to the generalised theta divisor on the moduli space of semistable bundles of rank $r$ and slope $g-1$ over $X$, at a stable point. This gives a generalisation of a case of the Riemann-Kempf singularity theorem for line bundles over $X$. In the same vein, we generalise the geometric Riemann-Roch theorem to vector bundles of slope $g-1$ and arbitrary rank.

Topology of the quantum modified moduli space

2001 ◽  
Vol 15 (4) ◽  
pp. 279-289
Author(s):  
S. L. Dubovsky
Keyword(s):  
Moduli Space

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

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