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.

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 Birational Geometry of Blow-ups of Projective Spaces Along Points and Lines

2020 ◽  
Author(s):  
Zhuang He ◽  
Lei Yang
Keyword(s):  
General Position ◽  
Infinite Order ◽  
Blow Up ◽  
General Point ◽  
Birational Geometry ◽  
Picard Number ◽  
Kummer Surface ◽  
Mori Dream Space ◽  
Effective Divisors ◽  
Effective Cone

Abstract Consider the blow-up $X$ of ${\mathbb{P}}^3$ at $6$ points in very general position and the $15$ lines through the $6$ points. We construct an infinite-order pseudo-automorphism $\phi _X$ on $X$. The effective cone of $X$ has infinitely many extremal rays and, hence, $X$ is not a Mori Dream Space. The threefold $X$ has a unique anticanonical section, which is a Jacobian K3 Kummer surface $S$ of Picard number 17. The restriction of $\phi _X$ on $S$ realizes one of Keum’s 192 infinite-order automorphisms. We show the blow-up of ${\mathbb{P}}^n$ ($n\geq 3$) at $(n+3)$ very general points and certain $9$ lines through them is not a Mori Dream Space. As an application, for $n\geq 7$, the blow-up of $\overline{M}_{0,n}$ at a very general point has infinitely many extremal effective divisors.

scholarly journals Goldman Systems and Bending Systems

2015 ◽  
Vol 67 (5) ◽  
pp. 1109-1143
Author(s):  
Yuichi Nohara ◽  
Kazushi Ueda
Keyword(s):  
Moduli Space ◽  
Integrable Systems ◽  
Projective Line ◽  
Symplectic Manifolds ◽  
Complex Manifolds ◽  
Completely Integrable Systems ◽  
Stability Parameters ◽  
Completely Integrable ◽  
Parabolic Bundles ◽  
The Stability

AbstractWe show that the moduli space of parabolic bundles on the projective line and the polygon space are isomorphic, both as complex manifolds and as symplectic manifolds equipped with structures of completely integrable systems, if the stability parameters are small.

scholarly journals THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

2020 ◽  
pp. 1-23
Author(s):  
MICHELE BOLOGNESI ◽  
NÉSTOR FERNÁNDEZ VARGAS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Hyperelliptic Curve ◽  
Geometric Description ◽  
Birational Model ◽  
Rank 2 ◽  
Semistable Vector Bundles ◽  
Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

The Effective Cone of the Kontsevich Moduli Space

2008 ◽  
Vol 51 (4) ◽  
pp. 519-534 ◽  
Author(s):  
Izzet Coskun ◽  
Joe Harris ◽  
Jason Starr
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Complete Characterization ◽  
Stable Maps ◽  
Linear Combinations ◽  
Effective Divisors ◽  
Effective Cone

AbstractIn this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, , stabilize when r ≥ d. We give a complete characterization of the effective divisors on . They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.

scholarly journals Principal co-Higgs bundles on ℙ1

2020 ◽  
Vol 63 (2) ◽  
pp. 512-530 ◽  
Author(s):  
Indranil Biswas ◽  
Oscar García-Prada ◽  
Jacques Hurtubise ◽  
Steven Rayan
Keyword(s):  
Moduli Space ◽  
Borel Subgroup ◽  
Algebraic Groups ◽  
Projective Line ◽  
Higgs Bundles ◽  
Complex Projective Line ◽  
Theoretic Characterization ◽  
Complex Projective

AbstractFor complex connected, reductive, affine, algebraic groups G, we give a Lie-theoretic characterization of the semistability of principal G-co-Higgs bundles on the complex projective line ℙ1 in terms of the simple roots of a Borel subgroup of G. We describe a stratification of the moduli space in terms of the Harder–Narasimhan type of the underlying bundle.

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