Moduli Space Of Genus Research Articles

Computations of Gromov–Witten invariants of toric varieties

We present the Julia package ToricAtiyahBott.jl, providing an easy way to perform the Atiyah–Bott formula on the moduli space of genus 0 stable maps M‾0,m(X,β) where X is any smooth projective toric variety, and β is any effective 1-cycle. The list of the supported cohomological cycles contains the most common ones, and it is extensible. We provide a detailed explanation of the algorithm together with many examples and applications. The toric variety X, as well as the cohomology class β, must be defined using the package Oscar.jl.

A BALL QUOTIENT PARAMETRIZING TRIGONAL GENUS 4 CURVES

Abstract We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $\frac {1}{2}(3^{10}-1)$ cover of the nine-dimensional Deligne–Mostow ball quotient such that the natural divisors that live on that moduli space become totally geodesic (their normalizations are eight-dimensional ball quotients). This isomorphism differs from the one considered by S. Kondō, and its construction is perhaps more elementary, as it does not involve K3 surfaces and their Torelli theorem: the Deligne–Mostow ball quotient parametrizes certain cyclic covers of degree 6 of a projective line and we show how a level structure on such a cover produces a degree 3 cover of that line with the same discriminant, yielding a genus 4 curve endowed with a $g^1_3$ .

Polynomial Relations Among Kappa Classes on the Moduli Space of Curves

Abstract We construct an infinite collection of universal—independent of $(g,n)$—polynomials in the Miller–Morita–Mumford classes $\kappa _m\in H^{2m}( \overline{\mathcal{M}}_{g,n},{\mathbb{Q}})$, defined over the moduli space of genus $g$ stable curves with $n$ labeled points. We conjecture vanishing of these polynomials in a range depending on $g$ and $n$.

A transfer principle: from periods to isoperiodic foliations

We classify the possible closures of leaves of the isoperiodic foliation defined on the Hodge bundle over the moduli space of genus $$g\ge 2$$ curves and prove that the foliation is ergodic on those sets. The results derive from the connectedness properties of the fibers of the period map defined on the Torelli cover of the moduli space. Some consequences on the topology of Hurwitz spaces of primitive branched coverings over elliptic curves are also obtained. To prove the results we develop the theory of augmented Torelli space, the branched Torelli cover of the Deligne–Mumford compactification of the moduli space of curves.

On Polynomials Counting Essentially Irreducible Maps

We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice point in the moduli space of genus-$g$ curves with $n$ labeled points and is given by a symmetric polynomial $N_{g,n}(\ell_1,\ldots,\ell_n)$ in the face degrees $2\ell_1, \ldots, 2\ell_n$. We generalize this by restricting to genus-$g$ maps that are essentially $2b$-irreducible for $b\geq 0$, which loosely speaking means that they are not allowed to possess contractible cycles of length less than $2b$ and each such cycle of length $2b$ is required to bound a face of degree $2b$. The enumeration of such maps is shown to be again given by a symmetric polynomial $\hat{N}_{g,n}^{(b)}(\ell_1,\ldots,\ell_n)$ in the face degrees with a polynomial dependence on $b$. These polynomials satisfy (generalized) string and dilaton equations, which for $g\leq 1$ uniquely determine them. The proofs rely heavily on a substitution approach by Bouttier and Guitter and the enumeration of planar maps on genus-$g$ surfaces.

Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges

We show that the stringy Kähler moduli space of a generic genus one curve of degree N, for N ≤ 5, is the Γ1(N) modular curve X1(N). This implies a correspondence between the cusps of the modular curves and certain large volume limits in the stringy Kähler moduli spaces of genus one fibered Calabi-Yau manifolds with N-sections. Using Higgs transitions in M-theory and F-theory as well as modular properties of the topological string partition function, we identify these large volume limits with elements of the Tate-Shafarevich group of the genus one fibration. Singular elements appear in the form of non-commutative resolutions with a torsional B-field at the singularity. The topological string amplitudes that arise at the various large volume limits are related by modular transformations. In particular, we find that the topological string partition function of a smooth genus one fibered Calabi-Yau threefold is transformed into that of a non-commutative resolution of the Jacobian by a Fricke involution. In the case of Calabi-Yau threefolds, we propose an expansion of the partition functions of a singular fibration and its non-commutative resolutions in terms of Gopakumar-Vafa invariants that are associated to BPS states with discrete charges. For genus one fibrations with 5-sections, this provides an enumerative interpretation for the partition functions that arise at certain irrational points of maximally unipotent monodromy.

Subvarieties of quotients of bounded symmetric domains

We present a new criterion for the complex hyperbolicity of a non-compact quotient X of a bounded symmetric domain. For each \(p \ge 1\), this criterion gives a precise condition under which the subvarieties \(V \subset X\) with \(\dim V \ge p\) are of general type, and X is p-measure hyperbolic. Then, we give several applications related to ball quotients, or to the Siegel moduli space of principally polarized abelian varieties. In the case of smooth compactifications of ball quotients, we obtain a conditional upper bound on the dimension on the exceptional locus, assuming the Green–Griffiths–Lang conjecture holds true. As another example of application, we give effective lower bounds for levels l so that the moduli spaces of genus g curves with l-level structures are of general type.

Poisson equation for genus two string invariants: a conjecture

We consider some string invariants at genus two that appear in the analysis of the D8ℛ4 and D6ℛ5 interactions in type II string theory. We conjecture a Poisson equation involving them and the Kawazumi-Zhang invariant based on their asymptotic expansions around the non-separating node in the moduli space of genus two Riemann surfaces.

Integrating simple genus two string invariants over moduli space

We consider an Sp(4, ℤ) invariant expression involving two factors of the Kawazumi-Zhang (KZ) invariant each of which is a modular graph with one link, and four derivatives on the moduli space of genus two Riemann surfaces. Manipulating it, we show that the integral over moduli space of a linear combination of a modular graph with two links and the square of the KZ invariant reduces to a boundary integral. We also consider an Sp(4, ℤ) invariant expression involving three factors of the KZ invariant and six derivatives on moduli space, from which we deduce that the integral over moduli space of a modular graph with three links reduces to a boundary integral. In both cases, the boundary term is completely determined by the KZ invariant. We show that both the integrals vanish.

Equivariant cohomology of the moduli space of genus three curves with symplectic level two structure via point counts

We determine the cohomology groups of the quartic and hyperelliptic loci inside the moduli space of genus three curves with symplectic level two structure as representations of the symmetric group S_7 together with their mixed Hodge structures by means of making equivariant point counts over finite fields and via purity arguments. This determines the weighted Euler characteristic of the whole moduli space of genus three curves with level two structure.

Socle pairings on tautological rings

We study some aspects of the $\lambda_g$ pairing on the tautological ring of $M_g^c$, the moduli space of genus $g$ stable curves of compact type. We consider pairing kappa classes with pure boundary strata, all tautological classes supported on the boundary, or the full tautological ring. We prove that the rank of this restricted pairing is equal in the first two cases and has an explicit formula in terms of partitions, while in the last case the rank increases by precisely the rank of the $\lambda_g\lambda_{g - 1}$ pairing on the tautological ring of $M_g$.

Equivariant cohomology of moduli spaces of genus three curves with level two structure

We study the cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as representations of the symplectic group on a six dimensional vector space over the field of two elements. We also make the analogous computations for some related spaces such as moduli spaces of genus three curves with a marked point and strata of the moduli space of Abelian differentials of genus three.

Tropical geometry of genus two curves

We exploit three classical characterizations of smooth genus two curves to study their tropical and analytic counterparts. First, we provide a combinatorial rule to determine the dual graph of each algebraic curve and the metric structure on its minimal Berkovich skeleton. Our main tool is the description of genus two curves via hyperelliptic covers of the projective line with six branch points. Given the valuations of these six points and their differences, our algorithm provides an explicit harmonic 2-to-1 map to a metric tree on six leaves. Second, we use tropical modifications to produce a faithful tropicalization in dimension three starting from a planar hyperelliptic embedding.Finally, we consider the moduli space of abstract genus two tropical curves and translate the classical Igusa invariants characterizing isomorphism classes of genus two algebraic curves into the tropical realm. While these tropical Igusa functions do not yield coordinates in the tropical moduli space, we propose an alternative set of invariants that provides new length data.

Invariant theory of ∧3(9) and genus-2 curves

Previous work established a connection between the geometric invariant theory of the third exterior power of a 9-dimensional complex vector space and the moduli space of genus 2 curves with some additional data. We generalize this connection to arbitrary fields, and describe the arithmetic data needed to get a bijection between both sides of this story.

Symmetric powers of Severi–Brauer varieties

We classify products of symmetric powers of a Severi-Brauer variety, up-to stable birational equivalence. The description also includes Grassmannians and moduli spaces of genus 0 stable maps.

The infinite topology of the hyperelliptic locus in Torelli space

Genus g Torelli space is the moduli space of genus g curves of compact type equipped with a homology framing. The hyperelliptic locus is a closed analytic subvariety consisting of finitely many mutually isomorphic components. We use properties of the hyperelliptic Torelli group to show that when \(g\ge 3\) these components do not have the homotopy type of a finite CW complex. Specifically, we show that the second rational homology of each component is infinite-dimensional. We give a more detailed description of the topological features of these components when \(g=3\) using properties of genus 3 theta functions.

Classification of higher rank orbit closures in ${\mathcal H^{\mathrm{odd}}(4)}$

The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component {\mathcal H^{\mathrm{odd}}(4)} the only {GL^+(2,\mathbb R)} orbit closures are closed orbits, the Prym locus {\tilde{\mathcal{Q}}(3,-1^3)} , and {\mathcal H^{\mathrm{odd}}(4)} . Together with work of Matheus–Wright, this implies that there are only finitely many non-arithmetic closed orbits (Teichmüller curves) in \mathcal H^{\mathrm{odd}}(4) outside of the Prym locus.

A virtual Kawasaki–Riemann–Roch formula

Kawasaki's formula is a tool to compute holomorphic Euler characteristics of vector bundles on a compact orbifold X. Let X be an orbispace with perfect obstruction theory which admits an embedding in a smooth orbifold. One can then construct the virtual structure sheaf and the virtual fundamental class of X. In this paper we prove that Kawasaki's formula behaves well " with working virtually" on X in the following sense: if we replace the structure sheaves, tangent and normal bundles in the formula by their virtual counterparts then Kawasaki's formula stays true. Our motivation comes from studying the quantum K-theory of a complex manifold X, with the formula applied to Kontsevich' moduli spaces of genus 0 stable maps to X.

Orbifold points on Teichmüller curves and Jacobians with complex multiplication

For each integer D≥5 with D≡0 or 1 mod4, the Weierstrass curve WD is an algebraic curve and a finite-volume hyperbolic orbifold which admits an algebraic and isometric immersion into the moduli space of genus two Riemann surfaces. The Weierstrass curves are the main examples of Teichmuller curves in genus two. The primary goal of this paper is to determine the number and type of orbifold points on each component of WD. Our enumeration of the orbifold points, together with Bainbridge [Geom. Topol. 11 (2007) 1887–2073] and McMullen [Math. Ann. 333 (2005) 87–130], completes the determination of the homeomorphism type of WD and gives a formula for the genus of its components. We use our formula to give bounds on the genus of WD and determine the Weierstrass curves of genus zero. We will also give several explicit descriptions of each surface labeled by an orbifold point on WD.

Modular invariant regularization of string determinants and the Serre GAGA principle

Since any string theory involves a path integration on the world-sheet metric, their partition functions are volume forms on the moduli space of genus $g$ Riemann surfaces ${\mathcal{M}}_{g}$, or on its super analog. It is well known that modular invariance fixes strong constraints that in some cases appear only at higher genus. Here we classify all the Weyl and modular invariant partition functions given by the path integral on the world-sheet metric, together with space-time coordinates, $b\text{\ensuremath{-}}c$ and/or $\ensuremath{\beta}\text{\ensuremath{-}}\ensuremath{\gamma}$ systems, that correspond to volume forms on ${\mathcal{M}}_{g}$. This was a long standing question, advocated by Belavin and Knizhnik, inspired by the Serre GAGA principle and based on the properties of the Mumford forms. The key observation is that the Bergman reproducing kernel provides a Weyl and modular invariant way to remove the point dependence that appears in the above string determinants, a property that should have its superanalog based on the super Bergman reproducing kernel. This is strictly related to the properties of the propagator associated to the space-time coordinates. Such partition functions $\mathcal{Z}[\mathcal{J}]$ have well-defined asymptotic behavior and can be considered as a basis to represent a wide class of string theories. In particular, since noncritical bosonic string partition functions ${\mathcal{Z}}_{D}$ are volume forms on ${\mathcal{M}}_{g}$, we suggest that there is a mapping, based on bosonization and degeneration techniques, from the Liouville sector to first order systems that may identify ${\mathcal{Z}}_{D}$ as a subclass of the $\mathcal{Z}[\mathcal{J}]$. The appearance of $b\text{\ensuremath{-}}c$ and $\ensuremath{\beta}\text{\ensuremath{-}}\ensuremath{\gamma}$ systems of any conformal weight shows that such theories are related to $W$ algebras. The fact that in a large $N$ 't Hooft-like limit two-dimensional ${W}_{N}$ minimal models conformal field theories are related to higher spin gravitational theories on ${\mathrm{AdS}}_{3}$, suggests that the string partition functions introduced here may lead to a formulation of higher spin theories in a string context.