For a triple consisting of a weakly negative definite plumbed 3-manifold, a root lattice, and a generalized Spin^c-structure, we construct a family of invariants in the form of a Laurent series. Each series is an invariant of the triple up to orientation preserving homeomorphisms and the action of the Weyl group. We show that there are infinitely many such series for irreducible root lattices of rank at least 2, with each series depending on a solution to a combinatorial puzzle defined on the root lattice. Our series recover certain related series recently defined by Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park and Ri as special cases. Explicit computations are given for Brieskorn homology spheres, for which the series may be expressed as modified higher rank false theta functions.
@article{MooreTarasca2024,title={Higher Rank Series and Root Puzzles for Plumbed 3-Manifolds},author={Moore, Allison and Tarasca, Nicola},journal={Submitted},year={2024}}
Submitted
Coinvariants of Vertex Algebras on Moduli of Abelian Varieties
We construct spaces of coinvariants at principally polarized abelian varieties with respect to the action of an infinite-dimensional Lie algebra. We show how these spaces globalize to twisted D-modules on moduli of principally polarized abelian varieties, and we determine the Atiyah algebra of a line bundle acting on them. We prove analogous results on the universal abelian variety. An essential aspect of our arguments involves analyzing the Atiyah algebra of the Hodge and canonical line bundles on moduli of abelian varieties and the universal abelian variety.
@article{AVVA,author={Tarasca, Nicola},journal={Submitted},title={Coinvariants of Vertex Algebras on Moduli of Abelian Varieties},year={2024}}
Ann. SNS Pisa
k-canonical Divisors through Brill-Noether Special Points
Inside the projectivized k-th Hodge bundle, we construct a collection of divisors obtained by imposing vanishing at a Brill-Noether special point. We compute the classes of the closures of such divisors in two ways, using incidence geometry and restrictions to various families, including pencils of curves on K3 surfaces and pencils of Du Val curves. We also show the extremality and rigidity of the closure of the incidence divisor consisting of smooth pointed curves together with a canonical or 2-canonical divisor passing through the marked point.
@article{gt2,author={Gheorghita, Iulia and Tarasca, Nicola},journal={Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, to appear},title={k-canonical Divisors through Brill-Noether Special Points},year={2024}}
CCM
Incidence Varieties in the Projectivized k-th Hodge Bundle over Curves with Rational Tails
Over the moduli space of pointed smooth algebraic curves, the projectivized k-th Hodge bundle is the space of k-canonical divisors. The incidence loci are defined by requiring the k-canonical divisors to have prescribed multiplicities at the marked points. We compute the classes of the closure of the incidence loci in the projectivized k-th Hodge bundle over the moduli space of curves with rational tails. The classes are expressed as a linear combination of tautological classes indexed by decorated stable graphs with coefficients enumerating appropriate weightings. As a consequence, we obtain an explicit expression for some relations in tautological rings of moduli of curves with rational tails.
@article{gt,doi={https://www.worldscientific.com/doi/10.1142/S0219199723500153},author={Gheorghita, Iulia and Tarasca, Nicola},journal={Communications in Contemporary Mathematics},title={Incidence Varieties in the Projectivized k-th Hodge Bundle over Curves with Rational Tails},volume={26},number={5},year={2024}}
Ann. Sci. ÉNS
On Factorization and Vector Bundles of Conformal Blocks from Vertex Algebras
Representations of vertex operator algebras define sheaves of coinvariants and conformal blocks on moduli of stable pointed curves. Assuming certain finiteness and semisimplicity conditions, we prove that such sheaves satisfy the factorization conjecture and consequently are 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.
@article{dgt2,author={Damiolini, Chiara and Gibney, Angela and Tarasca, Nicola},journal={Annales Scientifiques de l'École Normale Supérieure},title={On Factorization and Vector Bundles of Conformal Blocks from Vertex Algebras},doi={https://smf.emath.fr/publications/sur-la-factorisation-et-les-fibres-vectoriels-de-blocs-conformes-des-algebres-vertex?language=en},volume={57},number={1},pages={241-292},year={2024}}
2023
Trans. AMS
A pointed Prym-Petri Theorem
Nicola Tarasca
Transactions of the American Mathematical Society, 2023
We construct pointed Prym-Brill-Noether varieties parametrizing line bundles assigned to an irreducible étale double covering of a curve with a prescribed minimal vanishing at a fixed point. We realize them as degeneracy loci in type D and deduce their classes in case of expected dimension. Thus, we determine a pointed Prym-Petri map and prove a pointed version of the Prym-Petri theorem implying that the expected dimension holds in the general case. These results build on work of Welters and De Concini-Pragacz on the unpointed case. Finally, we show that Prym varieties are Prym-Tyurin varieties for Prym-Brill-Noether curves of exponent enumerating standard shifted tableaux, extending to the Prym setting work of Ortega.
@article{tarasca2023pointed,doi={https://www.ams.org/journals/tran/2023-376-04/S0002-9947-2023-08792-X/home.html},title={A pointed Prym-Petri Theorem},author={Tarasca, Nicola},journal={Transactions of the American Mathematical Society},volume={376},number={04},pages={2641--2656},year={2023}}
2022
J. London Math. Soc.
Motivic Classes of Degeneracy Loci and Pointed Brill-Noether Varieties
Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch χy-genus of classical and one-pointed Brill-Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.
@article{anderson2020motivic,doi={https://doi.org/10.1112/jlms.12547},author={Anderson, Dave and Chen, Linda and Tarasca, Nicola},journal={Journal of the London Mathematical Society},number={3},pages={1787-1822},title={Motivic Classes of Degeneracy Loci and Pointed Brill-Noether Varieties},volume={105},year={2022}}
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 [Damiolini-Gibney-Tarasca]. 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 [Marian-Oprea-Pandharipande-Pixton-Zvonkine] for bundles given by integrable modules over affine Lie algebras. It follows that the Chern classes are tautological. Examples and open problems are discussed.
@inproceedings{dgt3,doi={https://doi.org/10.1017/9781108877831.006},author={Damiolini, Chiara and Gibney, Angela and Tarasca, Nicola},booktitle={Facets of Algebraic Geometry: A Collection in Honor of William Fulton's 80th Birthday},journal={Facets of {Algebraic Geometry}, volume in honor of {William Fulton} on the occasion of his 80th birthday},pages={164--189},publisher={Cambridge University Press},series={London Mathematical Society Lecture Note Series},title={Vertex Algebras of CohFT-type},year={2022}}
IMRN
K-classes of Brill-Noether Loci and a Determinantal Formula
We compute the Euler characteristic of the structure sheaf of the Brill-Noether locus of linear series with special vanishing at up to two marked points. When the Brill-Noether number rho is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when rho=1, we recover the formulas of Eisenbud-Harris, Pirola, and Chan-L{\'o}pez-Pflueger-Teixidor for the arithmetic genus of a Brill-Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the K-theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations, and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey-Jockusch-Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for flagged skew tableaux.
@article{act,author={Anderson, Dave and Chen, Linda and Tarasca, Nicola},journal={International Mathematics Research Notices},pages={12653--12698},title={K-classes of Brill-Noether Loci and a Determinantal Formula},doi={https://academic.oup.com/imrn/article-abstract/2022/16/12653/6253768},volume={16},year={2022}}
2021
Geom. Topol.
Conformal Blocks from Vertex Algebras and their Connections on $\overline{\mathcal{M}}_{g,n}$
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.
@article{dgt,doi={https://msp.org/gt/2021/25-5/p03.xhtml},author={Damiolini, Chiara and Gibney, Angela and Tarasca, Nicola},journal={Geometry & Topology},number={5},pages={2235--2286},title={Conformal Blocks from Vertex Algebras and their Connections on $\overline{\mathcal{M}}_{g,n}$},volume={25},year={2021}}
We study the codimension n locus of curves of genus 2 with n distinct marked Weierstrass points inside the moduli space of genus 2, n-pointed curves, for n ≤ 6. We give a recursive description of the classes of the closure of these loci inside the moduli space of stable curves. For n ≤ 4, we express these classes using a generating function over stable graphs indexing the boundary strata of moduli spaces of pointed stable curves. Similarly, we express the closure of these classes inside the moduli space of curves of compact type for all n. This is a first step in the study of the structure of hyperelliptic classes in all genera.
@article{CaT,doi={https://www.ams.org/journals/tran/2019-372-04/S0002-9947-2019-07626-2/},author={Cavalieri, Renzo and Tarasca, Nicola},journal={Transactions of the American Mathematical Society},number={4},pages={2467--2492},title={Classes of Weierstrass Points on Genus 2 Curves},volume={372},year={2019}}
@article{tarasca2019embracing,title={Embracing the Job Search},author={Tarasca, Nicola},journal={Notices of the American Mathematical Society},volume={66},number={10},pages={1656--1657},year={2019},publisher={American Mathematical Society, AMS}}
2017
Selecta Math. (N.S.)
Du Val Curves and the Pointed Brill-Noether Theorem
We show that a general curve in an explicit class of what we call Du Val pointed curves satisfies the Brill-Noether Theorem for pointed curves. Furthermore, we prove that a generic pencil of Du Val pointed curves is disjoint from all Brill-Noether divisors on the universal curve. This provides explicit examples of smooth pointed curves of arbitrary genus defined over Q which are Brill-Noether general. A similar result is proved for 2-pointed curves as well using explicit curves on elliptic ruled surfaces.
@article{FT2,author={Farkas, Gavril and Tarasca, Nicola},journal={Selecta Mathematica},number={3},pages={2243--2259},title={Du Val Curves and the Pointed Brill-Noether Theorem},doi={http://link.springer.com/article/10.1007/s00029-017-0329-3?wt_mc=Internal.Event.1.SEM.ArticleAuthorOnlineFirst},volume={23},year={2017}}
2016
ANT
Extremality of Loci of Hyperelliptic Curves with Marked Weierstrass Points
The locus of genus-two curves with n marked Weierstrass points has codimension n inside the moduli space of genus-two curves with n marked points, for n ≤ 6. It is well known that the class of the closure of the divisor obtained for n=1 spans an extremal ray of the cone of effective divisor classes. We generalize this result for all n: we show that the class of the closure of the locus of genus-two curves with n marked Weierstrass points spans an extremal ray of the cone of effective classes of codimension n, for n≤6. A related construction produces extremal nef curve classes in moduli spaces of pointed elliptic curves.
@article{CT2,author={Chen, Dawei and Tarasca, Nicola},journal={Algebra & Number Theory},number={9},pages={1935--1948},title={Extremality of Loci of Hyperelliptic Curves with Marked Weierstrass Points},doi={http://dx.doi.org/10.2140/ant.2016.10.1935},volume={10},year={2016}}
Math. Z.
Loci of Curves with Subcanonical Points in Low Genus
Inside the moduli space of curves of genus three with one marked point, we consider the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-hyperelliptic curves with a marked hyperflex. These loci have codimension two. We compute the classes of their closures in the moduli space of stable curves of genus three with one marked point. Similarly, we compute the class of the closure of the locus of curves of genus four with an even theta characteristic vanishing with order three at a certain point. These loci naturally arise in the study of minimal strata of Abelian differentials.
@article{CT,author={Chen, Dawei and Tarasca, Nicola},journal={Mathematische Zeitschrift},number={3-4},pages={683--714},title={Loci of Curves with Subcanonical Points in Low Genus},doi={http://dx.doi.org/10.1007/s00209-016-1670-5},volume={284},year={2016}}
Inside the symmetric product of a very general curve, we consider the codimension-one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension. We compute the classes of such divisors, and thus obtain improved bounds for the slope of the cone of effective divisor classes on symmetric products of a very general curve. By letting the moduli of the curve vary, we study more generally the classes of the related divisors inside the moduli space of stable pointed curves.
@article{Secant,author={Tarasca, Nicola},journal={Journal of Algebra},pages={1--13},title={Divisors of Secant Planes to Curves},doi={http://dx.doi.org/10.1016/j.jalgebra.2016.01.022},volume={454},year={2016}}
The classical Castelnuovo numbers count linear series of minimal degree and fixed dimension on a general curve, in the case when this number is finite. For pencils, that is, linear series of dimension one, the Castelnuovo specialize to the better known Catalan numbers. Using the Fulton-Pragacz determinantal formula for flag bundles and combinatorial manipulations, we obtain a compact formula for the number of linear series on a general curve having prescribed ramification at an arbitrary point, in the case when the expected number of such linear series is finite. The formula is then used to solve some enumerative problems on moduli spaces of curves.
@article{FT,author={Farkas, Gavril and Tarasca, Nicola},journal={Mathematical Research Letters},number={2},pages={389--404},title={Pointed Castelnuovo Numbers},doi={http://intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0023/0002/a005/index.html},volume={23},year={2016}}
Inside the moduli space of curves of genus 2 with 2 marked points we consider the loci of curves admitting a map to P1 of degree d totally ramified over the two marked points, for d ≥ 2. Such loci have codimension two. We compute the class of their compactifications in the moduli space of stable curves. Several results will be deduced from this computation.
@article{DR,author={Tarasca, Nicola},journal={International Mathematics Research Notices},number={19},pages={9569--9593},title={Double Total Ramifications for Curves of Genus 2},doi={http://dx.doi.org/10.1093/imrn/rnu228},year={2015}}
Let us consider the locus in the moduli space of curves of genus 2k defined by curves with a pencil of degree k. Since the Brill-Noether number is equal to -2, such a locus has codimension two. Using the method of test surfaces, we compute the class of its closure in the moduli space of stable curves.
@article{MR3109733,oberwolfach={https://doi.org/10.14760/OWR-2013-6},author={Tarasca, Nicola},journal={Compositio Mathematica},number={9},pages={1535--1568},title={Brill-Noether Loci in Codimension Two},doi={http://dx.doi.org/10.1112/S0010437X13007215},volume={149},year={2013}}
2012
JPAA
Double Points of Plane Models in $\overline{M}_{6,1}$
The aim of this paper is to compute the class of the closure of the effective divisor in M6,1 given by pointed curves [C,p] with a sextic plane model mapping p to a double point. Such a divisor generates an extremal ray in the pseudoeffective cone of M6,1 as shown by Jensen. A general result on some families of linear series with adjusted Brill-Noether number 0 or -1 is introduced to complete the computation.
@article{MR2864851,author={Tarasca, Nicola},title={Double Points of Plane Models in $\overline{M}_{6,1}$},journal={Journal of Pure and Applied Algebra},number={4},pages={766--774},volume={216},year={2012},doi={https://doi.org/10.1016/j.jpaa.2011.10.035}}