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Han-Bom Moon

Publications and source records attributed to Han-Bom Moon.

36 records · Page 2Linked to original sources
arXiv2017

Finite generation of the algebra of type A conformal blocks via birational geometry

We study 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 which was previously obtained by Belkale. For each big divisor, the associated birational model is described in terms of moduli space of parabolic bundles.

math.AG↗
arXiv2017

On the $S_n$-invariant F-conjecture

By using classical invariant theory, we reduce the $S_{n}$-invariant F-conjecture to a feasibility problem in polyhedral geometry. We show by computer that for $n \le 19$, every integral $S_{n}$-invariant F-nef divisor on the moduli space of genus zero stable pointed curves is semi-ample, over arbitrary characteristic. Furthermore, for $n \le 16$, we show that for every integral $S_{n}$-invariant nef (resp. ample) divisor $D$ on the moduli space, $2D$ is base-point-free (resp. very ample). As applications, we obtain the nef cone of the moduli space of stable curves without marked points, and the semi-ample cone that of the moduli space of genus 0 stable maps to Grassmannian for small numerical values.

math.AG↗
arXiv2016

Mori's program for the moduli space of conics in Grassmannian

We complete Mori's program for Kontsevich's moduli space of degree 2 stable maps to Grassmannian of lines. We describe all birational models in terms of moduli spaces (of curves and sheaves), incidence varieties, and Kirwan's partial desingularization.

math.AG↗
arXiv2016

GIT Compactifications of M_{0,n} and Flips

We use geometric invariant theory (GIT) to construct a large class of compactifications of the moduli space M_{0,n}. These compactifications include many previously known examples, as well as many new ones. As a consequence of our GIT approach, we exhibit explicit flips and divisorial contractions between these spaces.

math.AG↗
arXiv2015

Moduli of sheaves, Fourier-Mukai transform, and partial desingularization

We study birational maps among 1) the moduli space of semistable torsion sheaves of Hilbert polynomial $4m+2$ on a smooth quadric surface, 2) the moduli space of semistable torsion sheaves of Hilbert polynomial $m^{2}+3m+2$ on $\mathbb{P}^{3}$, 3) Kontsevich's moduli space of genus zero stable maps of degree 2 to Grassmannian $Gr(2, 4)$. A regular birational morphism from 1) to 2) is described in terms of Fourier-Mukai transform. The map from 3) to 2) is Kirwan's partial desingularization. Also we investigate several geometric properties of 1) by using the variation of moduli spaces of stable pairs.

math.AG↗
arXiv2015

Birational contractions of $\overline{\mathrm{M}}_{0,n}$ and combinatorics of extremal assignments

From Smyth's classification, modular compactifications of pointed smooth rational curves are indexed by combinatorial data, so-called extremal assignments. We explore their combinatorial structures and show that any extremal assignment is a finite union of atomic extremal assignments. We discuss a connection with the birational geometry of the moduli space of stable pointed curves. As applications, we study three special classes of extremal assignments: smooth, toric, and invariant with respect to the symmetric group action. We identify them with three combinatorial objects: simple intersecting families, complete multipartite graphs, and special families of integer partitions, respectively.

math.AG↗
arXiv2015

Chow ring of the moduli space of stable sheaves supported on quartic curves

Motivated by the computation of the BPS-invariants on a local Calabi-Yau threefold suggested by S. Katz, we compute the Chow ring and the cohomology ring of the moduli space of stable sheaves of Hilbert polynomial $4m+1$ on the projective plane. As a byproduct, we obtain the total Chern class and Euler characteristics of all line bundles, which provide a numerical data for the strange duality on the plane.

math.AG↗
arXiv2014

Effective curves on $\overline{M}_{0,n}$ from group actions

We study new effective curve classes on the moduli space of stable pointed rational curves given by the fixed loci of subgroups of the permutation group action. We compute their numerical classes and provide a strategy for writing them as effective linear combinations of F-curves, using Losev-Manin spaces and toric degeneration of curve classes.

math.AG↗
arXiv2014

Mori's program for $\bar{M}_{0,6}$ with symmetric divisors

We complete Mori's program with symmetric divisors for the moduli space of stable six pointed rational curves. As an application, we give an alternative proof of the complete Mori's program of the moduli space of genus two stable curves, done by Hassett.

math.AG↗
arXiv2014

Mori's program for $\bar{M}_{0,7}$ with symmetric divisors

We complete Mori's program with symmetric divisors for the moduli space of stable seven pointed rational curves. We describe all birational models in terms of explicit blow-ups and blow-downs. We also give a moduli theoretic description of the first flip, which have not appeared in literature.

math.AG↗
arXiv2012

Veronese quotient models of $\bar{M}_{0,n}$ and conformal blocks

The moduli space $\bar{M}_{0,n}$ of Deligne-Mumford stable n-pointed rational curves admits morphisms to spaces recently constructed by Giansiracusa, Jensen, and Moon that we call Veronese quotients. We study divisors on $\bar{M}_{0,n}$ associated to these maps and show that these divisors arise as first Chern classes of vector bundles of conformal blocks.

math.AG↗
arXiv2011

Log canonical models for $\bar{M}_{g,n}$

We prove a formula of log canonical models for moduli space $\bar{M}_{g,n}$ of pointed stable curves which describes all Hassett's moduli spaces of weighted pointed stable curves in a single equation. This is a generalization of the preceding result for genus zero to all genera.

math.AG↗
arXiv2011

Log canonical models for the moduli space of pointed stable rational curves

We run Mori's program for the moduli space of pointed stable rational curves with divisor $K +\sum a_{i}ψ_{i}$. We prove that, without assuming the F-conjecture, the birational model for the pair is the Hassett's moduli space of weighted pointed stable rational curves, without any modification of weight coefficients.

math.AG↗
arXiv2010

Mori's program for the moduli space of pointed stable rational curves

We prove that, assuming the F-conjecture, the log canonical model of the pair $(\bar{M}_{0,n}, \sum a_i ψ_i)$ is the Hassett's moduli space of weighted pointed stable rational curves without any modification of weight coefficients. For the boundary weight cases, we prove that the birational model is the GIT quotient of the product of the projective lines. This is a generalization of Simpson's theorem for symmetric weight cases.

math.AG↗
arXiv2010

Moduli spaces of weighted pointed stable rational curves via GIT

We construct the Mumford-Knudsen space of n pointed stable rational curves by a sequence of explicit blow-ups from the GIT quotient (P^1)^n//SL(2) with respect to the symmetric linearization O(1,...,1). The intermediate blown-up spaces turn out to be the moduli spaces of weighted pointed stable curves for suitable ranges of weights. As an application, we provide a new unconditional proof of M. Simpson's Theorem about the log canonical models of the Mumford-Knudsen space. We also give a basis of the Picard group of the moduli spaces of weighted pointed stable curves.

math.AG↗
arXiv2008

Moduli space of stable maps to projective space via GIT

We compare the Kontsevich moduli space of genus 0 stable maps to projective space with the quasi-map space when $d=3$. More precisely, we prove that when $d=3$, the obvious birational map from the quasi-map space to the moduli space of stable maps is the composition of three blow-ups followed by two blow-downs. Furthermore, we identify the blow-up/down centers explicitly in terms of the moduli spaces for lower degrees. Using this, we calculate the Betti numbers, the integral Picard group, and the rational cohomology ring. The degree two case is worked out as a warm-up.

math.AG↗