Deligne-Mumford Compactification Research Articles

Tropical invariants for binary quintics and reduction types of Picard curves

Abstract In this paper, we express the reduction types of Picard curves in terms of tropical invariants associated with binary quintics. We also give a general framework for tropical invariants associated with group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification $\overline{M}_{0,n}$ .

A compactification of the moduli space of multiple-spin curves

We construct a smooth Deligne–Mumford compactification for the moduli space of curves with an m-tuple of spin structures using line bundles on quasi-stable curves as limiting objects, as opposed to line bundles on stacky curves. For all m, we give a combinatorial description of the local structure of the corresponding coarse moduli spaces. We also classify all irreducible and connected components of the resulting moduli spaces of multiple-spin curves.

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.

Limit of Weierstrass measure on stable curves

The goal of the paper is to study the limiting behavior of the Weierstrass measures on a smooth curve of genus $g \geqslant 2$ as the curve approaches a certain nodal stable curve represented by a point in the Deligne–Mumford compactification $\overline{\mathcal{M}}_g$ of the moduli $\mathcal{M}_g$, including irreducible ones or those of compact type. As a consequence, the Weierstrass measures on a stable rational curve at the boundary of $\mathcal{M}_g$ are completely determined. In the process, the asymptotic behavior of the Bergman measure is also studied.

The Divisors of Prym Semicanonical Pencils

Abstract In the moduli space ${{\mathcal {R}}}_g$ of double étale covers of curves of a fixed genus $g$, the locus of covers of curves with a semicanonical pencil decomposes as the union of two divisors—${{\mathcal {T}}}^e_g$ and ${{\mathcal {T}}}^o_g$. Adapting arguments of Teixidor for the divisor of curves having a semicanonical pencil, we prove that both divisors are irreducible and compute their divisor classes in the Deligne–Mumford compactification ${\overline {{\mathcal {R}}}}_g$.

Stability in the homology of Deligne–Mumford compactifications

AbstractUsing the theory of ${\mathbf {FS}} {^\mathrm {op}}$ modules, we study the asymptotic behavior of the homology of ${\overline {\mathcal {M}}_{g,n}}$, the Deligne–Mumford compactification of the moduli space of curves, for $n\gg 0$. An ${\mathbf {FS}} {^\mathrm {op}}$ module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via copies that glue on marked projective lines, we give the homology of ${\overline {\mathcal {M}}_{g,n}}$ the structure of an ${\mathbf {FS}} {^\mathrm {op}}$ module and bound its degree of generation. As a consequence, we prove that the generating function $\sum _{n} \dim (H_i({\overline {\mathcal {M}}_{g,n}})) t^n$ is rational, and its denominator has roots in the set $\{1, 1/2, \ldots, 1/p(g,i)\},$ where $p(g,i)$ is a polynomial of order $O(g^2 i^2)$. We also obtain restrictions on the decomposition of the homology of ${\overline {\mathcal {M}}_{g,n}}$ into irreducible $\mathbf {S}_n$ representations.

Coordinates on the augmented moduli space of convex RP2 structures

Let S be an orientable, finite type surface with negative Euler characteristic. The augmented moduli space of convex real projective structures on S was first defined and topologized by the first author. In this article, we give an explicit description of this topology using explicit coordinates. More precisely, given every point in this augmented moduli space, we find explicit continuous coordinates on the quotient of a suitable open neighborhood about this point by a suitable subgroup of the mapping class group of S. Using this, we give a simpler proof of the fact that the augmented moduli space of convex real projective structures on S is homeomorphic to the orbifold vector bundle of regular cubic differentials over the Deligne-Mumford compactification of the moduli space of Riemann surfaces homeomorphic to S.

Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: The sl2 case

In this paper, we show that it is always possible to deform a differential equation ∂xΨ(x) = L(x)Ψ(x) with L(x)∈sl2(C)(x) by introducing a small formal parameter ℏ in such a way that it satisfies the topological type properties of Bergère, Borot, and Eynard [Annales Henri Poincaré 16(12), 2713–2782 (2015)]. This is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce ℏ. The topological recursion is then proved to provide a formal series expansion of the corresponding tau-function whose coefficients can thus be expressed in terms of intersections of tautological classes in the Deligne–Mumford compactification of the moduli space of surfaces. We present a few examples including any Fuchsian system of sl2(C)(x) as well as some elements of Painlevé hierarchies.

Rational Picard Group of Moduli of Pointed Hyperelliptic Curves

Abstract We determine the rational divisor class group of the moduli spaces of smooth pointed hyperelliptic curves and of their Deligne–Mumford compactification, over the field of complex numbers.

Crystallographic Groups Arising from Teichmüller Spaces

The purpose of this paper is to show that there is a natural connection between Teichmüller spaces and crystallographic groups. This connection was first discovered by the author when discussing the Deligne-Mumford compactification of moduli spaces (see \lq\lq {\it The Deligne-Mumford compactification and crystallographic groups}\rq\rq, preprint, 2018). In this paper, we will discuss the same connection from a slightly different viewpoint having in mind an analogy of the maximally degenerate frontier points of the augmented Teichmüller spaces and the cusp points of hyperbolic spaces.

Hurwitz correspondences on compactifications of [formula omitted

Hurwitz correspondences are certain multi-valued self-maps of the moduli space M0,N. They arise in the study of Thurston's topological characterization of rational functions. We compare the dynamics of Hurwitz correspondence H on two different compactifications of M0,N: the Deligne-Mumford compactification M‾0,N, as well as a Hassett space of weighted stable curves. We use this comparison to show that the k-th dynamical degree of H is the absolute value of the dominant eigenvalue of the pushforward induced by H on a natural quotient of H2k(M‾0,N).

Boundary Behaviour of Weil–Petersson and Fibre Metrics for Riemann Moduli Spaces

Abstract The Weil–Petersson and Takhtajan–Zograf metrics on the Riemann moduli spaces of complex structures for an $n$-fold punctured oriented surface of genus $g,$ in the stable range $g+2n>2,$ are shown here to have complete asymptotic expansions in terms of Fenchel–Nielsen coordinates at the exceptional divisors of the Knudsen–Deligne–Mumford compactification. This is accomplished by finding a full expansion for the hyperbolic metrics on the fibres of the universal curve as they approach the complete metrics on the nodal curves above the exceptional divisors and then using a push-forward theorem for conormal densities. This refines a two-term expansion due to Obitsu–Wolpert for the conformal factor relative to the model plumbing metric which in turn refined the bound obtained by Masur. A similar expansion for the Ricci metric is also obtained.

Limits of equisymmetric $1$-complex dimensional families of Riemann surfaces

We describe the limit surfaces of some equisymmetric $1$-complex dimensional families of Riemann surfaces in the boundary of the Deligne-Mumford compactification of moduli space. We provide a description of such nodal Riemann surfaces in terms of the group of automorphisms defining the family. We apply our method to some known examples.

On the automorphisms of Hassett’s moduli spaces

Let $\overline{\mathcal{M}}_{g,A[n]}$ be the moduli stack parametrizing weighted stable curves, and let $\overline{M}_{g,A[n]}$ be its coarse moduli space. These spaces have been introduced by B. Hassett, as compactifications of $\mathcal{M}_{g,n}$ and $M_{g,n}$ respectively, by assigning rational weights $A = (a_{1},...,a_{n})$, $0< a_{i} \leq 1$ to the markings. In particular, the classical Deligne-Mumford compactification arises for $a_1 = ... = a_n = 1$. In genus zero some of these spaces appear as intermediate steps of the blow-up construction of $\overline{M}_{0,n}$ developed by M. Kapranov, while in higher genus they may be related to the LMMP on $\overline{M}_{g,n}$. We compute the automorphism groups of most of the Hassett's spaces appearing in the Kapranov's blow-up construction. Furthermore, if $g\geq 1$ we compute the automorphism groups of all Hassett's spaces. In particular, we prove that if $g\geq 1$ and $2g-2+n\geq 3$ then the automorphism groups of both $\overline{\mathcal{M}}_{g,A[n]}$ and $\overline{M}_{g,A[n]}$ are isomorphic to a subgroup of $S_{n}$ whose elements are permutations preserving the weight data in a suitable sense.

On Riemann surfaces of genus g with 4g automorphisms

We determine, for all genus g≥2 the Riemann surfaces of genus g with exactly 4g automorphisms. For g≠ 3,6,12,15 or 30, these surfaces form a real Riemann surface Fg in the moduli space Mg: the Riemann sphere with three punctures. We obtain the automorphism groups and extended automorphism groups of the surfaces in the family. Furthermore we determine the topological types of the real forms of real Riemann surfaces in Fg. The set of real Riemann surfaces in Fg consists of three intervals its closure in the Deligne–Mumford compactification of Mg is a closed Jordan curve. We describe the nodal surfaces that are limits of real Riemann surfaces in Fg.

Models of discretized moduli spaces, cohomological field theories, and Gaussian means

We prove combinatorially the explicit relation between genus filtrated s -loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich–Penner matrix model (KPMM). The latter is the generating function for volumes of discretized (open) moduli spaces M g , s disc given by N g , s ( P 1 , … , P s ) for ( P 1 , … , P s ) ∈ Z + s . This generating function therefore enjoys the topological recursion, and we prove that it is simultaneously the generating function for ancestor invariants of a cohomological field theory thus enjoying the Givental decomposition. We use another Givental-type decomposition obtained for this model by the second authors in 1995 in terms of special times related to the discretization of moduli spaces thus representing its asymptotic expansion terms (and therefore those of the Gaussian means) as finite sums over graphs weighted by lower-order monomials in times thus giving another proof of (quasi)polynomiality of the discrete volumes. As an application, we find the coefficients in the first subleading order for M g , 1 in two ways: by using the refined Harer–Zagier recursion and by exploiting the above Givental-type transformation. We put forward the conjecture that the above graph expansions can be used for probing the reduction structure of the Deligne–Mumford compactification M ¯ g , s of moduli spaces of punctured Riemann surfaces.

Computing dynamical degrees of rational maps on moduli space

The dynamical degrees of a rational map$f:X{\dashrightarrow}X$are fundamental invariants describing the rate of growth of the action of iterates of$f$on the cohomology of$X$. When$f$has non-empty indeterminacy set, these quantities can be very difficult to determine. We study rational maps$f:X^{N}{\dashrightarrow}X^{N}$, where$X^{N}$is isomorphic to the Deligne–Mumford compactification$\overline{{\mathcal{M}}}_{0,N+3}$. We exploit the stratified structure of$X^{N}$to provide new examples of rational maps, in arbitrary dimension, for which the action on cohomology behaves functorially under iteration. From this, all dynamical degrees can be readily computed (given enough book-keeping and computing time). In this paper, we explicitly compute all of the dynamical degrees for all such maps$f:X^{N}{\dashrightarrow}X^{N}$, where$\text{dim}(X^{N})\leq 3$and the first dynamical degrees for the mappings where$\text{dim}(X^{N})\leq 5$. These examples naturally arise in the setting of Thurston’s topological characterization of rational maps.

Connecting $$p$$ p -gonal loci in the compactification of moduli space

Consider the moduli space \(\mathcal {M}_{g}\) of Riemann surfaces of genus \(g\ge 2\) and its Deligne–Mumford compactification \(\overline{\mathcal {M}_{g}}\). We are interested in the branch locus \({\mathcal {B}_{g}}\) for \(g>2\), i.e., the subset of \(\mathcal {M}_{g}\) consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in \(\mathcal {M}_{g}\) but the set of (cyclic) trigonal surfaces is not. By contrast, the set of (cyclic) trigonal surfaces is connected in \(\overline{\mathcal {M}_{g}}\). We exhibit an explicit nodal surface that lies in the completion of every equisymmetric set of \(3\)-gonal Riemann surfaces providing an alternative proof of a result of Achter and Pries (Math Ann 338:187–206, 2007). For \(p>3\) the connectivity of the \(p\)-gonal loci becomes more involved. We show that for \(p\ge 11\) prime and genus \(g=p-1\) there are one-dimensional strata of cyclic \(p\)-gonal surfaces that are completely isolated in the completion \(\overline{\mathcal {B}_{g}}\) of the branch locus in \(\overline{\mathcal {M}_{g}}\).

An analytic construction of the Deligne-Mumford compactification of the moduli space of curves

In 1969, P. Deligne and D. Mumford compactified the moduli space of curves. Their compactification is a projective algebraic variety, and as such, it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmueller space by the action of the mapping class group gives a compactification of the moduli space. We put an analytic structure on this compact quotient and prove that with respect to this structure, it is canonically isomorphic (as an analytic space) to the Deligne-Mumford compactification.

The double ramification cycle and the theta divisor

We compute the classes of universal theta divisors of degrees zero and g-1 over the Deligne-Mumford compactification of the moduli space of curves, with various integer weights on the points, in particular reproving a recent result of M\uller. We also obtain a formula for the class in the Chow ring of the moduli space of curves of compact type of the double ramification locus, given by the condition that a fixed linear combination of the marked points is a principal divisor, reproving a recent result of Hain. Our approach for computing the theta divisor is more direct, via test curves and the geometry of the theta divisor, and works easily over the entire Deligne-Mumford compactification. We use our extended result in another paper to study the partial compactification of the double ramification cycle.