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Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra and the algebra of smooth functions on decorated Teichmuller space. In this paper, we consider surfaces with punctures which is not the 3-holed sphere and which have an ideal triangulation without self-folded edges or triangles. For those surfaces, we prove that Roger and Yang's Poisson algebra homomorphism is injective, and the skein algebra they defined have no zero divisors. A section about generalized corner coordinates for normal arcs may be of independent interest.

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and$U(2)\oplus D_4^{\oplus 2}$lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of$\mathbb {P}^{1}\times \mathbb {P}^{1}$branched along a specific$(4,\,4)$curve. We show that, up to a finite group action, this stable pairs compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient$(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$with the symmetric linearization.

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 ≤ 19 , every integral S n -invariant F-nef divisor on the moduli space of genus zero stable n-pointed curves is semi-ample, over arbitrary characteristic. Furthermore, for n ≤ 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 of the moduli space of genus 0 stable maps to Grassmannian for small numerical values.

In 2004 Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map with a weighted variant, we show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter--Speyer envisioned. Moreover, we provide a geometric interpretation in terms of configurations of points on rational normal curves and construct a finite tropical basis that yields an explicit characterization of weighted dissimilarity vectors., Final version. To appear in Proceedings of the National Academy of Sciences of the United States of America (PNAS)

We describe presentations of the Roger-Yang generalized skein algebras for punctured spheres with an arbitrary number of punctures. This skein algebra is a quantization of the decorated Teichmuller space and generalizes the construction of the Kauffman bracket skein algebra. In this paper, we also obtain a new interpretation of the homogeneous coordinate ring of the Grassmannian of planes in terms of skein theory., Comment: 19 pages, to appear on Algebraic & Geometric Topology

We study birational geometry of the moduli space of stable sheaves on a quadric surface with Hilbert polynomial 5 m + 1 and c 1 = ( 2 , 3 ) . We describe a birational map between the moduli space and a projective bundle over a Grassmannian as a composition of smooth blow-ups/downs.

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 has not appeared in the literature.

From Smyth’s classification, modular compactifications of the moduli space of pointed smooth rational curves are indexed by combinatorial data, the 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 rational 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.

We prove finite generation of the algebra of type A conformal blocks over arbitrary stable curves of any genus. As an application we construct a flat family of irreducible normal projective varieties over the moduli stack of stable pointed curves, whose fiber over a smooth curve is a moduli space of semistable parabolic bundles. This generalizes a construction of a degeneration of the moduli space of vector bundles presented in a recent work of Belkale and Gibney., Comment: 27 pages, Revised, Published version in Proceedings of the London Mathematical Society (2020)

We study birational maps among (1) the moduli space of semistable sheaves of Hilbert polynomial $$4m+2$$ on a smooth quadric surface, (2) the moduli space of semistable 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 the Grassmannian Gr(2, 4). A regular birational morphism from (1) to (2) is described in terms of Fourier–Mukai transforms. The map from (3) to (2) is Kirwan’s partial desingularization. We also investigate several geometric properties of 1) by using the variation of moduli spaces of stable pairs.

We prove a formula of log canonical models for moduli stack 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.

The parameter space of $n$ ordered points in projective $d$-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in $(\mathbb{P}^d)^n$. The resulting variety was used to study the birational geometry of the moduli space $\overline{\mathrm{M}}_{0,n}$ of $n$-tuples of points in $\mathbb{P}^1$. In this paper we turn to a more classical question, first asked independently by both Speyer and Sturmfels: what are the defining equations? For conics, namely $d=2$, we find scheme-theoretic equations revealing a determinantal structure and use this to prove some geometric properties; moreover, determining which subsets of these equations suffice set-theoretically is equivalent to a well-studied combinatorial problem. For twisted cubics, $d=3$, we use the Gale transform to produce equations defining the union of two irreducible components, the compactified configuration space we want and the locus of degenerate point configurations, and we explain the challenges involved in eliminating this extra component. For $d \ge 4$ we conjecture a similar situation and prove partial results in this direction., Comment: 28 pages. Minor correction. We removed the erroneous Lemma 4.7 in the previous version, but the remaining results are valid

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., Comment: 25 pages; comments welcome

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.

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, which was first done by Hassett.

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.

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., Comment: 25 pages

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., 22 pages. Introduction revised. Typos corrected

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., Comments are welcome

We investigate the birational geometry (in the sense of Mori's program) of the moduli space of rank 2 semistable parabolic vector bundles on a rational curve. We compute the effective cone of the moduli space and show that all birational models obtained by Mori's program are also moduli spaces of parabolic vector bundles with certain parabolic weights., Comment: 22 pages, Revised, Published version in International Mathematics Research Notices (2015)

The moduli space M0;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 M0;n associated to these maps and show that these divisors arise as rst Chern classes of vector bundles of conformal blocks. 0;n that receive morphisms from M0;n. From the perspective of Mori theory, this is tantamount to describing cer- tain semi-ample divisors on M0;n. This work is concerned with two recent constructions that each yield an abundance of such semi-ample divisors on M0;n, and the relationship between them. The rst comes from Geometric Invariant Theory (GIT), while the second from conformal eld theory. There are new natural birational models of M0;n obtained via GIT which are moduli spaces of pointed rational normal curves of a