Higher-Level Conformal Blocks Divisors on

2014 ◽  
Vol 57 (1) ◽  
pp. 7-30 ◽  
Author(s):  
Valery Alexeev ◽  
Angela Gibney ◽  
David Swinarski
Keyword(s):  
Moduli Space ◽  
Picard Group ◽  
Conformal Blocks ◽  
Nef Cone ◽  
Ample Divisors

AbstractWe study a family of semi-ample divisors on the moduli space of n-pointed genus 0 curves given by higher-level conformal blocks. We derive formulae for their intersections with a basis of 1-cycles, show that they form a basis for the Sn-invariant Picard group, and generate a full-dimensional subcone of the Sn-invariant nef cone. We find their position in the nef cone and study their associated morphisms.

scholarly journals Numerical criteria for divisors on to be ample

2009 ◽  
Vol 145 (5) ◽  
pp. 1227-1248 ◽  
Author(s):  
Angela Gibney
Keyword(s):  
Moduli Space ◽  
General Type ◽  
Topological Type ◽  
Rational Curves ◽  
Canonical Divisor ◽  
Content Type ◽  
One Dimensional ◽  
Stable Curves ◽  
The One ◽  
Ample Divisors

AbstractThe moduli space $\M _{g,n}$ of n-pointed stable curves of genus g is stratified by the topological type of the curves being parameterized: the closure of the locus of curves with k nodes has codimension k. The one-dimensional components of this stratification are smooth rational curves called F-curves. These are believed to determine all ample divisors. F-conjecture A divisor on $\M _{g,n}$ is ample if and only if it positively intersects theF-curves. In this paper, proving the F-conjecture on $\M _{g,n}$ is reduced to showing that certain divisors on $\M _{0,N}$ for N⩽g+n are equivalent to the sum of the canonical divisor plus an effective divisor supported on the boundary. Numerical criteria and an algorithm are given to check whether a divisor is ample. By using a computer program called the Nef Wizard, written by Daniel Krashen, one can verify the conjecture for low genus. This is done on $\M _g$ for g⩽24, more than doubling the number of cases for which the conjecture is known to hold and showing that it is true for the first genera such that $\M _g$ is known to be of general type.

scholarly journals THE STRONG FRANCHETTA CONJECTURE IN ARBITRARY CHARACTERISTICS

2003 ◽  
Vol 14 (04) ◽  
pp. 371-396
Author(s):  
STEFAN SCHRÖER
Keyword(s):  
Moduli Space ◽  
Rational Points ◽  
Picard Group ◽  
Moduli Space Of Curves ◽  
Canonical Class ◽  
Generic Curve ◽  
Picard Scheme

Using Moriwaki's calculation of the ℚ-Picard group for the moduli space of curves, I prove the strong Franchetta Conjecture in all characteristics. That is, the canonical class generates the group of rational points on the Picard scheme for the generic curve of genus g ≥ 3. Similar results hold for generic pointed curves. Moreover, I show that Hilbert's Irreducibility Theorem implies that there are many other nonclosed points in the moduli space of curves with such properties.

Basepoint Free Cycles on $\overline{\operatorname{\textbf{M}}}_{\boldsymbol{0,n}}$ from Gromov–Witten Theory

2019 ◽  
Author(s):  
P Belkale ◽  
A Gibney
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Homogeneous Spaces ◽  
Type A ◽  
Rational Curves ◽  
Conformal Blocks ◽  
Witten Theory ◽  
Level 1

Abstract Basepoint free cycles on the moduli space $\overline{\operatorname{M}}_{0,n}$ of stable $n$-pointed rational curves, defined using Gromov–Witten invariants of smooth projective homogeneous spaces are introduced and studied. Intersection formulas to find classes are given. Gromov–Witten divisors for projective space are shown equivalent to conformal blocks divisors for type A at level 1.

scholarly journals Finite Generation of the Algebra of Type A Conformal Blocks via Birational Geometry

2018 ◽  
Author(s):  
Han-Bom Moon ◽  
Sang-Bum Yoo
Keyword(s):  
Moduli Space ◽  
Type A ◽  
Projective Line ◽  
Birational Geometry ◽  
Finite Generation ◽  
Conformal Blocks ◽  
Parabolic Bundles ◽  
Mori Dream Space ◽  
Birational Model ◽  
Effective Cone

Abstract We study the birational geometry of the moduli space of parabolic bundles over a projective line, in the framework of Mori’s program. We show that the moduli space is a Mori dream space. As a consequence, we obtain the finite generation of the algebra of type A conformal blocks. Furthermore, we compute the H-representation of the effective cone that was previously obtained by Belkale. For each big divisor, the associated birational model is described in terms of moduli space of parabolic bundles.

scholarly journals Picard Group and Fundamental Group of the Moduli of Higgs Bundles on Curves

2018 ◽  
Vol 5 (1) ◽  
pp. 146-149
Author(s):  
Sujoy Chakraborty ◽  
Arjun Paul
Keyword(s):  
Fundamental Group ◽  
Algebraic Group ◽  
Moduli Space ◽  
Topological Type ◽  
Picard Group ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve

Abstract Let X be an irreducible smooth projective curve of genus g ≥ 2 over ℂ. Let MG, Higgsδbe a connected reductive affine algebraic group over ℂ. Let Higgs be the moduli space of semistable principal G-Higgs bundles on X of topological type δ∈π1(G). In this article,we compute the fundamental group and Picard group of

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