- Morita equivalences of Zhu’s associative algebra and mode transition algebras

Abstract: The mode transition algebra 𝔄, and d-th mode transition subalgebras 𝔄_d⊂𝔄 are associative algebras attached to vertex operator algebras. Here, under natural assumptions, we establish Morita equivalences involving Zhu’s associative algebra 𝖠 and 𝔄_d. As an application, we obtain explicit expressions for higher level Zhu algebras as products of matrices, generalizing the result for the 1 dimensional Heisenberg vertex operator algebra from our previous work. This theory applies for instance, to certain VOAs with commutative, and connected Zhu algebra, and to rational vertex operator algebras. Examples are given.

(Arxiv)

- Conformal blocks on smoothings via mode transition algebras (with Chiara Damiolini and Daniel Krashen)

ABSTRACT: Here we define a series of associative algebras attached to a vertex operator algebra V, called mode transition algebras, showing they reflect both algebraic properties of V and geometric constructions on moduli of curves. One can define sheaves of coinvariants on pointed coordinatized curves from V-modules. We show that if the mode transition algebras admit multiplicative identities with certain properties, these sheaves deform as wanted on families of curves with nodes (so V satisfies *smoothing*). Consequently, coherent sheaves of coinvariants defined by vertex operator algebras that satisfy smoothing form vector bundles. We also show that mode transition algebras give information about higher level Zhu algebras and generalized Verma modules. As an application, we completely describe higher level Zhu algebras of the Heisenberg vertex algebra for all levels, proving a conjecture of Addabbo–Barron.

(arXiv)

- Oberwolfach Report : Algebraic structures on moduli of curves from vertex operator algebras

ABSTRACT: This is an extended abstract for a talk given at the workshop “Recent Trends in Algebraic Geometry”, held at Oberwolfach during 18-23 June 2023. The presentation was about a series of associative algebras attached to a vertex operator algebra V, called mode transition algebras, defined in a recent paper with Damiolini and Krashen. Mode transition algebras reflect both the algebraic structure of V and the geometry of sheaves of coinvariants on the moduli space of curves derived from representations of V. On the geometric side, we show coherent sheaves of coinvariants are locally free when the mode transition algebras admit unities that act as identities on modules (we call these strong unities). Some questions are suggested.

- Factorization Presentations (with Chiara Damiolini and Daniel Krashen)

(arxiv)

ABSTRACT. Modules over a vertex operator algebra V give rise to sheaves of coinvariants on moduli of stable pointed curves. If V satisfies finiteness and semi-simplicity conditions, these sheaves are vector bundles. This relies on factorization, an isomorphism of spaces of coinvariants at a nodal curve with a finite sum of analogous spaces on the normalization of the curve. Here we introduce the notion of a factorization presentation, and using this, we show that finiteness conditions on V imply the sheaves of coinvariants are coherent on moduli spaces of pointed stable curves without any assumption of semisimplicity.

- On global generation of vector bundles on the moduli space of curves from representations of vertex algebras (with C. Damiolini). In Algebraic Geometry.

ABSTRACT. We prove sheaves of coinvariants on the moduli space of stable pointed rational curves defined by simple modules over a vertex operator algebra V, are globally generated when V is of CFT-type, and generated in degree 1. Examples where global generation fails, and evidence of positivity are given for more general V.

- On an equivalence of divisors on $\overline{M}_{0,n}$ from Gromov-Witten theory and Conformal Blocks (with L. Chen, L. Heller, E. Kalashnikov, H. Larson, and W. Xu), in Transformation Groups.

ABSTRACT. We consider a conjecture that identifies two types of base point free divisors on $\overline{M}_{0,n}$. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated to simple Lie algebras in type A. Here we reduce this conjecture on $\overline{M}_{0,n}$ to the same statement for n = 4. A reinterpretation leads to a proof of the conjecture on $\overline{M}_{0,n}$ for a large class, and we give sufficient conditions for the non-vanishing of these divisors.

- Vertex algebras of CohFT-type (with Chiara Damiolini and Nicola Tarasca),

*Vol. I, *164–189, London Math. Soc. Lecture Note Ser., 472, *Cambridge Univ. Press, Cambridge,* 2022. volume in honor of William Fulton on the occasion of his 80th birthday

ABSTRACT. Representations of certain vertex algebras, here called of CohFT-type, can be used to construct vector bundles of coinvariants and conformal blocks on moduli spaces of stable curves [DGT2]. We show that such bundles define semisimple cohomological field theories. As an application, we give an expression for their total Chern character in terms of the fusion rules, following the approach and computation in [MOPPZ] for bundles given by integrable modules over affine Lie algebras. It follows that the Chern classes are tautological. Examples and open problems are discussed.

- On Factorization and vector bundles of conformal blocks from vertex algebras (
*with Chiara Damiolini and Nicola Tarasca), to appear in Annales scientifiques de l’École normale supérieure .**(arXiv)*

ABSTRACT. Modules over conformal vertex algebras give rise to sheaves of coinvariants and conformal blocks on moduli of stable pointed curves. Here we prove the factorization conjecture for these sheaves. Our results apply in arbitrary genus and for a large class of vertex algebras. As an application, sheaves defined by finitely generated admissible modules over vertex algebras satisfying natural hypotheses are shown to be vector bundles. Factorization is essential to a recursive formulation of invariants, like ranks and Chern classes, and to produce new constructions of rational conformal field theories and cohomological field theories.

- Conformal blocks from vertex algebras and their connections on \overline{
*M}_g,n (with Chiara Damiolini and Nicola Tarasca), Geometry and Topology (journal),**(arxiv)*

ABSTRACT. We show that coinvariants of modules over conformal vertex algebras give rise to quasi-coherent sheaves on moduli of stable pointed curves. These generalize Verlinde bundles or vector bundles of conformal blocks defined using affine Lie algebras studied first by Tsuchiya-Kanie, Tsuchiya-Ueno-Yamada, and extend work of a number of researchers. The sheaves carry a twisted logarithmic D-module structure, and hence support a projectively flat connection. We identify the logarithmic Atiyah algebra acting on them, generalizing work of Tsuchimoto for affine Lie algebras.

- Basepoint free cycles on \overline{
*M}*from Gromov-Witten theory (with Prakash Belkale) IMRN 2019 (arxiv, journal)_{0,n }

ABSTRACT. Basepoint free cycles on the moduli space of stable n-pointed rational curves, defined using Gromov-Witten invariants of smooth projective homogeneous spaces X are studied. Intersection formulas to find classes are given, with explicit examples for X a projective space, and X a smooth projective quadric hypersurface. When X is projective space, divisors are shown equivalent to conformal blocks divisors for type A at level one, giving maps from $\overline{M}_{0,n}$ to birational models constructed as GIT quotients, parametrizing configurations of weighted points supported on (generalized) Veronese curves.

- On finite generation of the section ring of the determinant of cohomology line bundle (with Prakash Belkale), Transactions of the AMS, 2018 (arxiv , journal)

ABSTRACT. For C a stable curve of arithmetic genus g ≥ 2, and D the determinant of cohomology line bundle on Bun_{SL(r)}(C), we show the section ring for the pair (BunSL(r)(C), D) is finitely generated. Three applications are given.

- On higher Chern classes of vector bundles of conformal blocks (with Swarnava Mukhopadhyay) (arxiv)

ABSTRACT. Here we consider higher Chern classes of vector bundles of conformal blocks on the moduli space of stable pointed curves of genus zero, giving explicit formulas for them, and extending various results that hold for first Chern classes to them. We use these classes to form a full dimensional subcone of the Pliant cone on $\overline{M}_{0,n}$.

- Scaling of conformal blocks and generalized theta functions over \overline{M}
_{g,n }(with Prakash Belkale and Anna Kazanova), Mathematische Zeitschrift, 2016, (arxiv, journal)

ABSTRACT. By way of intersection theory on the moduli space of curves, we show that geometric interpretations for conformal blocks, as sections of ample line bundles over projective varieties, do not have to hold at points on the boundary. We show such a translation would imply certain recursion relations for first Chern classes of these bundles. While recursions can fail, geometric interpretations are shown to hold under certain conditions.

- Nonvanishing of conformal blocks divisors on \overline{
*M}*(with Prakash Belkale and Swarnava Mukhopadhyay), Transformation Groups, 2016 (arXiv, journal)_{0,n}

ABSTRACT. We introduce and study the problem of finding necessary and sufficient conditions under which a conformal blocks divisor on the moduli space of curves is nonzero. We give necessary conditions in type A, which are sufficient when theta and critical levels coincide. We show that divisors are subject to additive identities, dependent on ranks of the underlying bundle. These identities amplify vanishing and nonvanishing results and have other applications.

- Vanishing and identities of conformal blocks divisors, (with Prakash Belkale and Swarnava Mukhopadhyay), Algebraic Geometry, 2 (1) 2015 (arxiv, journal)

ABSTRACT. Conformal block divisors in type A on the moduli space of stable pointed rational curves are shown to satisfy new symmetries when levels and ranks are interchanged in non-standard ways. A connection with the quantum cohomology of Grassmannians reveals that these divisors vanish above the critical level.

- Higher level sl_2 conformal blocks divisors on \overline{M}
_{0,n}, (with Valery Alexeev, and David Swinarski), Proceedings of the Edinburgh Mathematical Society, 2014, (arxiv, journal)

ABSTRACT. We study a family of semi-ample divisors on the moduli space of stable pointed rational curves defined using conformal blocks and analyze their associated morphisms.

- Veronese quotient models of \overline{M}
_{0,n}and conformal blocks, (with Dave Jensen, Han-Bom Moon, and David Swinarski), the Michigan Mathematical Journal, 2013, (arxiv, journal)

ABSTRACT. The moduli space 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 associated to these maps and show that they arise as first Chern classes of vector bundles of conformal blocks.

- Conformal blocks divisors and the birational geometry of \overline{M}
_{g,n}, Mathematisches Forschungsinstitut Oberwolfach, Moduli Spaces in Algebraic Geometry 2013, (Abstract)

ABSTRACT. This is an abstract of a talk given at Oberwolfach.

- The cone of type A, level one conformal blocks divisors, (with Noah Giansiracusa), Advances in Mathematics, 2012, Pages 798-814, (arxiv, journal)

ABSTRACT. We prove that the type A, level one, conformal blocks divisors on the moduli space of stable pointed rational curves span a finitely generated, full-dimensional subcone of the nef cone. Each such divisor induces a morphism from the moduli space, and we identify its image as a GIT quotient parameterizing configurations of points supported on a flat limit of Veronese curves. We show how scaling GIT linearizations gives geometric meaning to certain identities among conformal blocks divisor classes. This also gives modular interpretations, in the form of GIT constructions, to the images of the hyperelliptic and cyclic trigonal loci under an extended Torelli map.

- On extensions of the Torelli Map, EMS Series of Congress Reports, Geometry and Arithmetic, October 2012, (arxiv, journal)

ABSTRACT. The divisors on the moduli space of stable curves of genus g that arise as the pullbacks of ample divisors along any extension of the Torelli map to any toroidal compactification of $A_g$ form a 2-dimensional extremal face of the nef cone of $\overline{M}_g$, which is explicitly described.

- sl
_{n}level 1 conformal blocks divisors on \overline{M}_{0,n}, (with M. Arap, J. Stankewicz and D. Swinarski), International Maths. Research Notices, 2011, (arxiv, journal)

ABSTRACT. We study a family of semi-ample divisors on the moduli space of stable pointed rational curves that come from the theory of conformal blocks for the Lie algebra sl_n and level 1. The divisors we study are invariant under the action of the symmetric group. We compute their classes and prove that they generate extremal rays in the cone of symmetric nef divisors on $\overline{M}_{0,n}$. In particular, these divisors define birational contractions, which we show factor through reduction morphisms to moduli spaces of weighted pointed curves defined by Hassett.

- Lower and upper bounds on nef cones, (with Diane Maclagan), International Maths. Research Notices, 2011, (arxiv, journal)

ABSTRACT. The nef cone of a projective variety Y is an important and often elusive invariant. In this paper we construct two polyhedral lower bounds and one polyhedral upper bound for the nef cone of Y using an embedding of Y into a toric variety. The lower bounds generalize the combinatorial description of the nef cone of a Mori dream space, while the upper bound generalizes the F-conjecture for the nef cone of the moduli space $\overline{M}_{0,n}$ to a wide class of varieties.

- Equations for Chow and Hilbert Quotients, (with Diane Maclagan), Algebra and Number Theory, 2010, (arxiv, journal)

ABSTRACT. We give explicit equations for the Chow and Hilbert quotients of a projective scheme X by the action of an algebraic torus T in an auxiliary toric variety. As a consequence we provide GIT descriptions of these canonical quotients, and obtain other GIT quotients of X by variation of GIT quotient. We apply these results to find equations for the moduli space $\overline{M}_{0,n}$ of stable genus zero n-pointed curves as a subvariety of a smooth toric variety defined via tropical methods.

- Numerical criteria for divisors on \overline{M}
_{g}to be ample, Compositio Mathematica, 2009, (arxiv, journal)

ABSTRACT. The moduli space of n−pointed stable curves of genus g is stratified by the topological type of the curves being parametrized: The closure of the locus of curves with k nodes has codimension k. The one dimensional components of this stratification are smooth rational curves (whose numerical equivalence classes are) called F-curves. The F-conjecture asserts that a divisor on $\overline{M}_{g,n}$ is nef if and only if it nonnegatively intersects the F−curves. In this paper the F-conjecture on $\overline{M}_{g,n}$ is reduced to showing that certain divisors in $\overline{M}_{0,N}$ for $N \le g+n$ are equivalent to the sum of the canonical divisor plus an effective divisor supported on the boundary. As an application of the reduction, numerical criteria are given which if satisfied by a divisor D on $\overline{M}_g$, show that D is ample. Additionally, an algorithm is described to check that a given divisor is ample. Using a computer program called Nef Wizard, written by Daniel Krashen, one can use the criteria and the algorithm to verify the conjecture for low genus. This is done for $g\le 24$, more than doubling the known cases of the conjecture, and showing it is true for the first genus such that the moduli space is known to be of general type.

- Pointed trees of projective spaces, (with L. Chen and D. Krashen), Journal of Algebraic Geometry, 2008, (arxiv, journal)

ABSTRACT. We introduce a smooth projective variety $T_{d,n}$ which compactifies the space of configurations of n distinct points on affine d-space modulo translation and homothety. The points in the boundary correspond to n-pointed stable rooted trees of d-dimensional projective spaces, which for d=1, are (n+1)-pointed stable rational curves. In particular, $T_{1,n}$ is isomorphic to $\overline{M}_{0,n+1}$, the moduli space of such curves.

- The Mori cones of moduli spaces of pointed curves of small genus, (with G. Farkas), Trans. Amer. Math. Soc., 2003, (arxiv, journal)

ABSTRACT. We compute the Mori cone of curves of the moduli space of stable n-pointed curves of genus g in the case when g and n are relatively small. For instance, we show that for $g<14$ every curve in $\overline{M}_g$ is numerically equivalent to an effective sum of 1-strata (loci of curves with 3g-4 nodes). We also prove that the nef cone of $\overline{M}_{0,6}$ is composed of 11 natural subcones all contained in the convex hull of boundary classes. We apply this result to classify the fibrations of the moduli space of rational curves with $n<7$ marked points.

- Towards the ample cone of \overline{M}
_{g,n}_{, }(with S. Keel and I. Morrison), J. Amer. Math. Soc. 2002, (arxiv, journal)

ABSTRACT. In this paper we study the ample cone of the moduli space of stable n-pointed curves of genus g. Our motivating conjecture is that a divisor on $\overline{M}_{g,n}$ is ample iff it has positive intersection with all 1-dimensional strata (the components of the locus of curves with at least 3g+n−2 nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the 1-strata are rational, includes the conjecture that the Mori cone is polyhedral and generated by rational curves. Our main result is that the conjecture holds iff it holds for g=0.