Papers
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and zbMATH.

Higher rank series and root puzzles for plumbed 3manifolds
with Allison Moore
Submitted
For a triple consisting of a weakly negative definite plumbed 3manifold, a
root lattice, and a generalized Spin^cstructure, 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 GukovPeiPutrovVafa,
GukovManolescu, 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.

Coinvariants of metaplectic representations on moduli of abelian varieties
Submitted
We construct spaces of coinvariants at principally polarized abelian varieties with respect to the action of an infinitedimensional Lie algebra. We show how these spaces globalize to twisted Dmodules 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.

A pointed PrymPetri Theorem
Transactions of the American Mathematical Society, 376:4 (2023), pp. 2641–2656
We construct pointed PrymBrillNoether 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 PrymPetri map and prove a pointed version of the PrymPetri theorem implying that the expected dimension holds in the general case. These results build on work of Welters and De ConciniPragacz on the unpointed case.
Finally, we show that Prym varieties are PrymTyurin varieties for PrymBrillNoether curves of exponent enumerating standard shifted tableaux, extending to the Prym setting work of Ortega.

kcanonical divisors through BrillNoether special points
with Iulia Gheorghita
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, to appear
Inside the projectivized kth Hodge bundle, we construct a collection of divisors obtained by imposing vanishing at a BrillNoether 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 2canonical divisor passing through the marked point.

Incidence varieties in the projectivized kth Hodge bundle over curves with rational tails
with Iulia Gheorghita
Communications in Contemporary Mathematics, 26:5 (2024)
Over the moduli space of pointed smooth algebraic curves, the projectivized kth Hodge bundle is the space of kcanonical divisors. The incidence loci are defined by requiring the kcanonical divisors to have prescribed multiplicities at the marked points. We compute the classes of the closure of the incidence loci in the projectivized kth 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.

Motivic classes of degeneracy loci and pointed BrillNoether varieties
with Dave Anderson and Linda Chen
Journal of the London Mathematical Society, 105:3 (2022), pp. 17871822
Motivic Chern and Hirzebruch classes are polynomials with Ktheory and homology classes as coefficients, which specialize to ChernSchwartzMacPherson classes, Ktheory classes, and CappellShaneson Lclasses.
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 χ_{y}genus of classical and onepointed BrillNoether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.

Vertex Algebras of CohFTtype
with Chiara Damiolini and Angela Gibney
Facets of Algebraic Geometry: A Collection in Honor of William Fulton’s 80th Birthday
London Mathematical Society Lecture Note Series (2022), pp. 164–189. Cambridge University Press
Representations of certain vertex algebras, here called of CohFTtype, can be used to construct vector bundles of coinvariants and conformal blocks on moduli spaces of stable curves [DamioliniGibneyTarasca].
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 [MarianOpreaPandharipandePixtonZvonkine] 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 Angela Gibney
Annales Scientifiques de l'École Normale Supérieure, to appear
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.

Conformal Blocks from Vertex Algebras and their Connections on M_{g,n}
with Chiara Damiolini and Angela Gibney
Geometry & Topology, 25:5 (2021), pp. 2235–2286
We show that coinvariants of modules over conformal vertex algebras give rise to quasicoherent 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 TsuchiyaKanie,
TsuchiyaUenoYamada, and extend work of a number of researchers.
The sheaves carry a twisted logarithmic Dmodule 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.

Kclasses of BrillNoether Loci and a Determinantal Formula
with Dave Anderson and Linda Chen
International Mathematics Research Notices, 16 (2022), pp. 12653–12698
We compute the Euler characteristic of the structure sheaf of the BrillNoether locus of linear series with special vanishing at up to two marked points.
When the BrillNoether 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 EisenbudHarris, Pirola, and ChanLópezPfluegerTeixidor for the arithmetic genus of a BrillNoether curve of special divisors.
These computations are obtained as applications of a new determinantal formula for the Ktheory class of certain degeneracy loci.
Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321avoiding permutations,
and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of BilleyJockuschStanley
expressing Schubert polynomials for 321avoiding permutations as generating functions for flagged skew tableaux.

Classes of Weierstrass Points on Genus 2 Curves
with Renzo Cavalieri
Transactions of the American Mathematical Society, 372:4 (2019), pp. 2467–2492
We study the codimension n locus of curves of genus 2 with n distinct marked Weierstrass points inside the moduli space of genus 2, npointed 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.

Du Val Curves and the Pointed BrillNoether Theorem
with Gavril Farkas
Selecta Mathematica, 23:3 (2017), pp. 22432259
We show that a general curve in an explicit class of what we call Du Val pointed curves satisfies the BrillNoether Theorem for pointed curves.
Furthermore, we prove that a generic pencil of Du Val pointed curves is disjoint from all BrillNoether divisors on the universal curve.
This provides explicit examples of smooth pointed curves of arbitrary genus defined over Q which are BrillNoether general.
A similar result is proved for 2pointed curves as well using explicit curves on elliptic ruled surfaces.

Extremality of Loci of Hyperelliptic Curves with Marked Weierstrass Points
with Dawei Chen
Algebra & Number Theory, 10:9 (2016), pp. 1935–1948
The locus of genustwo curves with n marked Weierstrass points has codimension n inside the moduli space of genustwo 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 genustwo 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.

Loci of Curves with Subcanonical Points in Low Genus
with Dawei Chen
Mathematische Zeitschrift, 284:3 (2016), pp. 683–714
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 nonhyperelliptic 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.

Divisors of Secant Planes to Curves
Journal of Algebra, 454 (2016), pp. 113
Inside the symmetric product of a very general curve, we consider the codimensionone 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.

Pointed Castelnuovo Numbers
with Gavril Farkas
Mathematical Research Letters, 23:2 (2016), pp. 389404
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 FultonPragacz 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.

Double Total Ramifications for Curves of Genus 2
International Mathematics Research Notices, 19 (2015), pp. 95699593
Inside the moduli space of curves of genus 2 with 2 marked points we consider the loci of curves admitting a map to P^{1} 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.

BrillNoether Loci in Codimension Two
Compositio Mathematica, 149:09 (2013), pp. 15351568
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 BrillNoether 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.
Extended abstract for Oberwolfach talk, in Oberwolfach Reports, 10:1 (2013), pp. 343–392

Double Points of Plane Models in M_{6,1}
Journal of Pure and Applied Algebra, 216:4 (2012), pp. 766774
The aim of this paper is to compute the class of the closure of the effective divisor
in M_{6,1} 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 M_{6,1} as shown by Jensen.
A general result on some families of linear series with adjusted BrillNoether number 0 or 1 is introduced to complete the computation.

Geometric Cycles in Moduli Spaces of Curves
PhD Thesis, Humboldt University in Berlin, 2012
