Moduli Space Of Curves Research Articles

Integrability, Quantization and Moduli Spaces of Curves

This paper has the purpose of presenting in an organic way a new approach to integrable (1+1)-dimensional field systems and their systematic quantization emerging from intersection theory of the moduli space of stable algebraic curves and, in particular, cohomological field theories, Hodge classes and double ramification cycles. This methods are alternative to the traditional Witten-Kontsevich framework and its generalizations by Dubrovin and Zhang and, among other advantages, have the merit of encompassing quantum integrable systems. Most of this material originates from an ongoing collaboration with A. Buryak, B. Dubrovin and J. Gu\'er\'e.

Stationary Gromov–Witten invariants of projective spaces

We represent stationary descendant Gromov–Witten invariants of projective space, up to explicit combinatorial factors, by polynomials. One application gives the asymptotic behaviour of the large degree behaviour of stationary descendant Gromov–Witten invariants in terms of intersection numbers over the moduli space of curves. We also show that primary Gromov–Witten invariants are “virtual” stationary descendants and hence the string and divisor equations can be understood purely in terms of stationary invariants.

Double ramification cycles on the moduli spaces of curves

Curves of genus $g$ which admit a map to $\mathbf {P}^{1}$ with specified ramification profile $\mu$ over $0\in \mathbf {P}^{1}$ and $\nu$ over $\infty\in \mathbf {P}^{1}$ define a double ramification cycle $\mathsf{DR}_{g}(\mu,\nu)$ on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves. The cycle $\mathsf{DR}_{g}(\mu,\nu)$ for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for $\mathsf{DR}_{g}(\mu,\nu)$ in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case. When $\mu=\nu=\emptyset$ , the formula for double ramification cycles expresses the top Chern class $\lambda_{g}$ of the Hodge bundle of $\overline {\mathcal{M}}_{g}$ as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.

Counting curves on surfaces

We consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology: for a compact surface [Formula: see text], with a finite set of points [Formula: see text] fixed on its boundary, how many configurations of disjoint arcs are there on [Formula: see text] whose boundary is [Formula: see text]? We find that this enumerative problem, counting curves on surfaces, has a rich structure. We show that such curve counts obey an effective recursion, in the general spirit of topological recursion, and exhibit quasi-polynomial behavior. This “elementary curve-counting” is in fact related to a more advanced notion of “curve-counting” from algebraic geometry or symplectic geometry. The asymptotics of this enumerative problem are closely related to the asymptotics of volumes of moduli spaces of curves, and the quasi-polynomials governing the enumerative problem encode intersection numbers on moduli spaces. Among several other results, we show that generating functions and differential forms for these curve counts exhibit structure that is reminiscent of the mathematical physics of free energies, partition functions and quantum curves.

Singularities of Moduli of Curves with a Universal Root

In a series of recent papers, Chiodo, Farkas and Ludwig carry out a deep analysis of the singular locus of the moduli space of stable (twisted) curves with an \ell -torsion line bundle. They show that for \ell\leq 6 and \ell\ne 5 pluricanonical forms extend over any desingularization. This opens the way to a computation of the Kodaira dimension without desingularizing, as done by Farkas and Ludwig for \ell=2 , and by Chiodo, Eisenbud, Farkas and Schreyer for \ell=3 . Here we treat roots of line bundles on the universal curve systematically: we consider the moduli space of curves C with a line bundle L such that L^{\otimes\ell}\cong \omega_C^{\otimes k} . New loci of canonical and non-canonical singularities appear for any k\not\in\ell\Bbb Z and \ell>2 , we provide a set of combinatorial tools allowing us to completely describe the singular locus in terms of dual graphs. We characterize the locus of non-canonical singularities, and for small values of \ell we give an explicit description.

Chiodo formulas for the r-th roots and topological recursion

We analyze Chiodo’s formulas for the Chern classes related to the r-th roots of the suitably twisted integer powers of the canonical class on the moduli space of curves. The intersection numbers of these classes with psi -classes are reproduced via the Chekhov–Eynard–Orantin topological recursion. As an application, we prove that the Johnson-Pandharipande-Tseng formula for the orbifold Hurwitz numbers is equivalent to the topological recursion for the orbifold Hurwitz numbers. In particular, this gives a new proof of the topological recursion for the orbifold Hurwitz numbers.

Topological Recursion and TQFTs

The topological recursion is an ubiquitous structure in enumerative geometry of surfaces and topological quantum field theories. Since its invention in the context of matrix models, it has been found or conjectured to compute intersection numbers in the moduli space of curves, topological string amplitudes, asymptotics of knot invariants, and more generally semiclassical expansion in topological quantum field theories. This workshop brought together mathematicians and theoretical physicists with various background to understand better the underlying geometry, learn about recent advances (notably on quantisation of spectral curves, topological strings and quantum gauge theories, and geometry of moduli spaces) and discuss the hot topics in the area.

Algebraic points, non-anticanonical heights and the Severi problem on toric varieties

In this article, we apply counting formulas for the number of morphisms from a curve to a toric variety to three different though related contexts (the first two are to be understood over global function fields): Manin's problem for rational points of bounded non-anticanonical height, asymptotics for algebraic points of bounded height and irreducibility of certain moduli spaces of curves, with application to the Severi problem for toric surfaces.

An Outline of the Log Minimal Model Program for the Moduli Space of Curves

ABSTRACTWe consider the log minimal model program for the moduli space of stable curves. It is widely believed now that there is a descending sequence of critical $α$ values where the log canonical model for the moduli space of stable curves with respect to $αδ$ changes, where $δ$ denotes the divisor of singular curves. We derive a conjectural formula for the critical values in two different ways: By working out the intersection theory of the moduli space of hyperelliptic curves and by computing the GIT stability of certain curves with tails and bridges. The results give a rough outline of how the log minimal model program would proceed, predicting when the log canonical model changes and which curves are to be discarded and acquired at the critical steps.

Moriwaki divisors and the augmented base loci of divisors on the moduli space of curves

We study the cone of Moriwaki divisors on \bar{M}_g by means of augmented base loci. Using a result of Moriwaki, we prove that an R-divisor D satisfies the strict Moriwaki inequalities if and only if the augmented base locus of D is contained in the boundary of \bar{M}_g. Then we draw some interesting consequences on the Zariski decomposition of divisors on \bar{M}_g, on the minimal model program of \bar{M}_g and on the log canonical models \bar{M}_g(\alpha).

Recursions and asymptotics of intersection numbers

We establish the asymptotic expansion of certain integrals of [Formula: see text] classes on moduli spaces of curves [Formula: see text], when either the [Formula: see text] or [Formula: see text] goes to infinity. Our main tools are cut-join type recursion formulae from the Witten–Kontsevich theorem, as well as asymptotics of solutions to the first Painlevé equation. We also raise a conjecture on large genus asymptotics for [Formula: see text]-point functions of [Formula: see text] classes and partially verify the positivity of coefficients in generalized Mirzakhani’s formula of higher Weil–Petersson volumes.

Hurwitz theory and the double ramification cycle

This survey grew out of notes accompanying a cycle of lectures at the workshop Modern Trends in Gromov–Witten Theory, in Hannover. The lectures are devoted to interactions between Hurwitz theory and Gromov–Witten theory, with a particular eye to the contributions made to the understanding of the Double Ramification Cycle, a cycle in the moduli space of curves that compactifies the double Hurwitz locus. We explore the algebro-combinatorial properties of single and double Hurwitz numbers, and the connections with intersection theoretic problems on appropriate moduli spaces. We survey several results by many groups of people on the subject, but, perhaps more importantly, collect a number of conjectures and problems which are still open.

Curves with prescribed intersection with boundary divisors in moduli spaces of curves

Curves with prescribed intersection with boundary divisors in moduli spaces of curves

Blobbed topological recursion: properties and applications

AbstractWe study the set of solutions (ωg,n)g⩾0,n⩾1of abstract loop equations. We prove that ωg,nis determined by its purely holomorphic part: this results in a decomposition that we call “blobbed topological recursion”. This is a generalisation of the theory of the topological recursion, in which the initial data (ω0,1, ω0,2) is enriched by non-zero symmetric holomorphic forms innvariables (φg,n)2g−2+n>0. In particular, we establish for any solution of abstract loop equations: (1) a graphical representation of ωg,nin terms of φg,n; (2) a graphical representation of ωg,nin terms of intersection numbers on the moduli space of curves; (3) variational formulas under infinitesimal transformation of φg,n; (4) a definition for the free energies ωg,0=Fgrespecting the variational formulas. We discuss in detail the application to the multi-trace matrix model and enumeration of stuffed maps.

Pointed Castelnuovo numbers

Abstract. 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 numbers 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.

Rational self-maps of moduli spaces

We discuss some examples of geometrically meaningful rational self-maps of moduli space of curves of low genus and homogeneous forms.

Normal forms of hyperelliptic curves of genus 3

The main motivation for the paper is to understand which hyperelliptic curves of genus 3 defined over a field K of characteristic $$\ne $$Â?2 appear as the image of the Donagi---Livne---Smith construction. By results in Frey and Kani (Lecture Notes in Computer Science, vol. 7053, pp. 1---19. Springer, Heidelberg, 2012) this means that one has to determine the intersection W of a Hurwitz space defined by curves of genus 3 together with cover maps of degree 4 to $$\mathbb {P}^1_K$$PK1 and a certain ramification type with the hyperelliptic locus in the moduli space of curves of genus 3. To achieve this aim we first study hyperelliptic curves of genus g as smooth curves C in $$\mathbb {P}^1_K\times \mathbb {P}^1_K$$PK1A—PK1 and prove that, under mild conditions on K, the curve C can be given by a "$$(g+1,2)$$(g+1,2)-normal form", namely by an affine equation in two variables of partial degrees $$g+1$$g+1 and 2 and hence of total degree $$\le $$≤g + 3, which is smaller than the degree of Weierstraβ normal forms. Such curves are naturally parameterized by a Hurwitz space $$\overline{\mathcal{H}}_{g,g+1}$$H¯g,g+1. We then specialize to $$g=3$$g=3 and introduce Hurwitz spaces for 4-covers with special ramification types. The study of these spaces enables us to determine that W is irreducible of dimension 4. Moreover we find an explicitly given K-rational family of curves C in (4, 2)-normal form such that the isomorphism classes of its members are in W(K) and such that the image of the family in W is Zariski-dense. For these curves we describe the "inverse" of the Donagi---Livne---Smith construction.

Towards large genus asymptotics of intersection numbers on moduli spaces of curves

We explicitly compute the diverging factor in the large genus asymptotics of the Weil–Petersson volumes of the moduli spaces of n-pointed complex algebraic curves. Modulo a universal multiplicative constant we prove the existence of a complete asymptotic expansion of the Weil–Petersson volumes in the inverse powers of the genus with coefficients that are polynomials in n. This is done by analyzing various recursions for the more general intersection numbers of tautological classes, whose large genus asymptotic behavior is also extensively studied.

Extremal higher codimension cycles on moduli spaces of curves

We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli space M ¯ g , n of stable genus g curves with n ordered marked points. In particular, we prove that codimension 2 boundary strata are extremal and exhibit extremal boundary strata of higher codimension. We also show that the locus of hyperelliptic curves with a marked Weierstrass point in M ¯ 3 , 1 and the locus of hyperelliptic curves in M ¯ 4 are extremal cycles. In addition, we exhibit infinitely many extremal codimension 2 cycles in M ¯ 1 , n for n ⩾ 5 and in M ¯ 2 , n for n ⩾ 2 .

Weierstrass cycles and tautological rings in various moduli spaces of algebraic curves

We analyze Weierstrass cycles and tautological rings in moduli spaces of smooth algebraic curves and in moduli spaces of integral algebraic curves with embedded disks with special attention to moduli spaces of curves having genus less than or equal to 6. In particular, we show that our general formula gives a good estimate for the dimension of Weierstrass cycles for low genera.