Moduli Space Of Stable Curves Research Articles

The geometry of the moduli space of odd spin curves

The spin moduli spaceSg is the parameter space of theta characteristics (spin structures) on stable curves of genus g. It has two connected components, S g andS + g , depending on the parity of the spin structure. We establish a complete birational classication by Kodaira dimension of the odd componentS g of the spin moduli space. We show thatS g is uniruled for g < 12 and even unirational for g 8. In this range, introducing the concept of cluster for the Mukai variety whose one-dimensional linear sections are general canonical curves of genus g, we construct new birational models ofS g . These we then use to explicitly describe the birational structure of S g . For instance, S 8 is birational to a locally trivial P 7 -bundle over the moduli space of elliptic curves with seven pairs of marked points. For g 12, we prove thatS g is a variety of general type. In genus 12, this requires the construction of a counterexample to the Slope Conjecture on eective divisors on the moduli space of stable curves of genus 12.

The Class of the Bielliptic Locus in Genus 3

Let the bielliptic locus be the closure in the moduli space of stable curves of the locus of smooth curves that are double covers of genus 1 curves. In this paper, we compute the class of the bielliptic locus in the moduli space \overline{M}_3 of stable curves of genus three in terms of a standard basis of the rational Chow group of codimension-2 classes in the moduli space. Our method is to test the class on the hyperelliptic locus: this gives the desired result up to two free parameters, which are then determined by intersecting the locus with two surfaces in \overline{M}_3 .

Canonical systems and their limits on stable curves

We propose an object called ‘sepcanonical system’ on a stable curve X0 which is to serve as limiting object – distinct from other such limits introduced previously – for the canonical system, as a smooth curve degenerates to X0. First, for curves which cannot be separated by 2 or fewer nodes (the so-called ‘2-inseparable’ curves), the sepcanonical system consists of the sections of the dualizing sheaf, and fails to be very ample iff X0 is a limit of smooth hyperelliptic curves (such X0 are called 2-inseparable hyperelliptics). For general, 2-separable curves X0, this assertion is false, leading us to introduce the sepcanonical system, which is a collection of linear systems on the ‘2-inseparable parts’ of X0, each associated to a different twisted limit of the canonical system, where the entire collection varies smoothly with X0. To define sepcanonical system, we must endow the curve with extra structure called an ‘azimuthal structure’. We show (Theorem 6.5) that the sepcanonical system is ‘essentially very ample’ unless the curve is a tree-like arrangement of 2-inseparable hyperelliptics. In a subsequent paper [11] we will show that the latter property is equivalent to the curve being a limit of smooth hyperelliptics, and will essentially give defining equation for the closure of the locus of smooth hyperelliptic curves in the moduli space of stable curves.

Brill–Noether loci in codimension two

AbstractLet 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.

Log minimal model program for the moduli space of stable curves: the first flip

We give a geometric invariant theory (GIT) construction of the log canonical model M ¯ g (a) of the pairs (M ¯ g ,ad) for a?(7/10�?,7/10] for small ??Q + . We show that M ¯ g (7/10) is isomorphic to the GIT quotient of the Chow variety of bicanonical curves; M ¯ g (7/10-?) is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions; e.g., they do not contain elliptic chains. We show that there is a small contraction ?:M ¯ g (7/10+?)?M ¯ g (7/10) that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction ? + :M ¯ g (7/10-?)?M ¯ g (7/10) that is the Mori flip of ? .

The structure of the tautological ring in genus one

We prove Getzler’s claims about the cohomology of the moduli space of stable curves of genus one, that is, that the even cohomology ring is spanned by the strata classes and that all relations between these classes follow from the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) relation and Getzler’s relation. In particular, the even cohomology ring is isomorphic to the tautological ring.

Normalization of the 1-stratum of the moduli space of stable curves

Let \mathcal{M}_g(\Gamma) be the stack of stable curves of genus g with a given dual graph \Gamma and let \overline{\mathcal{M}}_g(\Gamma) be its closure in \overline{\mathcal{M}}_g . We consider the normalization of \overline{\mathcal{M}}_g(\Gamma) in order to classify the residual orbifolds of the normalizations of the irreducible components of the locus in \overline{\mathcal{M}}_g corresponding to curves with at least 3g - 4 nodes.

Towards a classification of modular compactifications of $\mathcal {M}_{g,n}$

The moduli space of smooth curves admits a beautiful compactification $\mathcal{M}_{g,n} \subset \overline{\mathcal{M}}_{g,n}$ by the moduli space of stable curves. In this paper, we undertake a systematic classification of alternate modular compactifications of $\mathcal{M}_{g,n}$ . Let $\mathcal{U}_{g,n}$ be the (non-separated) moduli stack of all n-pointed reduced, connected, complete, one-dimensional schemes of arithmetic genus g. When g=0, $\mathcal{U}_{0,n}$ is irreducible and we classify all open proper substacks of $\mathcal{U}_{0,n}$ . When g≥1, $\mathcal{U}_{g,n}$ may not be irreducible, but there is a unique irreducible component $\mathcal{V}_{g,n} \subset\mathcal{U}_{g,n}$ containing $\mathcal{M}_{g,n}$ . We classify open proper substacks of $\mathcal {V}_{g,n}$ satisfying a certain stability condition.

Extended Torelli Map to the Igusa Blowup in Genus 6, 7, and 8

It was conjectured in [Namikawa 73] that the Torelli map associating to a curve its Jacobian extends to a regular map from the Deligne–Mumford moduli space of stable curves to the (normalization of the) Igusa blowup . A counterexample in genus g=9 was found in [Alexeev and Brunyate 11]. Here, we prove that the extended map is regular for all g⩽8, thus completely solving the problem in every genus.

The Final Log Canonical Model of the Moduli Space of Stable Curves of Genus 4

The Final Log Canonical Model of the Moduli Space of Stable Curves of Genus 4

Hodge integrals with one λ-class

Hodge integrals over moduli space of stable curves play an important roles in understanding the topological properties of moduli space. ELSV formula connects the Hodge integrals with Hurwitz numbers, and the generating function of Hurwitz numbers satisfies the cut-and-join equation. Therefore, it is natural to consider how to use the cut-and-join equation for Hurwitz numbers to compute Hodge integrals which appear in ELSV formula. In this paper, at first, we will review the method introduced in Goulden et al.’s paper to get the λg conjecture for Hodge integral. Through some variables transformation, the generating function of Hurwitz number becomes a symmetric polynomial which satisfies a symmetrized cut-and-join equation. By comparing the coefficients of the lowest degree term of both sides in this equation, we can get the λg conjecture. Then, in a similar way, we obtain our main result in this paper: a recursive formula for Hodge integral of type contains only one λg−1-class. We also point out that our results are closely related to the degree 0 Virasoro conjecture for a curve.

Torelli theorem for stable curves

We study the Torelli morphism from the moduli space of stable curves to the moduli space of principally polarized stable semi-abelic pairs. We give two characterizations of its fibers, describe its injectivity locus, and give a sharp upper bound on the cardinality of finite fibers. We also bound the dimension of infinite fibers.

A Geometric Invariant Theory Construction of Moduli Spaces of Stable Maps

We construct the moduli spaces of stable maps, \bar M_g,n(P^r,d), via geometric invariant theory (GIT). This construction is only valid over Spec C, but a special case is a GIT presentation of the moduli space of stable curves of genus g with n marked points, \bar M_g,n; this is valid over Spec Z. Our method follows that used in the case n=0 by Gieseker to construct \bar M_g, though our proof that the semistable set is nonempty is entirely different.

Limits of Special Weierstrass Points

Let C be the union of two general connected, smooth, nonrational curves X and Y inter- secting transversally at a point P. Assume that P is a general point of X or of Y. Our main result describes all limits on C of special Weierstrass points along smooth curves degenerating to C . As an application, we recover in a unified and conceptually simpler way the computations made by Diaz and Cukierman of divisor classes of curves with special Weierstrass points in the moduli space of stable curves. In our approach there are no multiplicity issues, an usual nuisance of the method of test curves

LOCAL SIGNATURE OF FIBERED COMPLEX SURFACES VIA MODULI AND MONODROMY

The signature of fibered complex surfaces is sometimes localized at finite fiber germs. We analyze this phenomenon by using datum of local monodromy in the mapping class group and a certain rational divisor on the moduli space of stable curves.

The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points

We compute the Poincare polynomial and the cohomology algebra with rational coefficients of the manifold M_n of real points of the moduli space of algebraic curves of genus 0 with n labeled points. This cohomology is a quadratic algebra, and we conjecture that it is Koszul. We also compute the 2-local torsion in the cohomology of M_n. As was shown by the fourth author, the cohomology of M_n does not have odd torsion, so that the above determines the additive structure of the integral homology and cohomology. Further, we prove that the rational homology operad of M_n is the operad of 2-Gerstenhaber algebras, which is closely related to the Hanlon-Wachs operad of 2-Lie algebras (generated by a ternary bracket). Finally, using Drinfeld’s theory of quantization of coboundary Lie quasibialgebras, we show that a large series of representations of the quadratic dual Lie algebra L_n of H^*(M_n,Q) (associated to such quasibialgebras) factors through the the natural projection of L_n to the associated graded Lie algebra of the prounipotent completion of the fundamental group of M_n. This leads us to conjecture that the said projection is an isomorphism, which would imply a formula for lower central series ranks of the fundamental group. On the other hand, we show that the spaces M_n are not formal starting from n = 6.

Log minimal model program for the moduli space of stable curves of genus three

We completely carry out the log minimal model program for the moduli space of stable curves of genus three with respect to the total boundary $\delta$: We give a modular interpretation of the log canonical model for the pair $(\bar{\mathcal M}_3, \a\d)$ for each $\a \in [0,7/10)$ and of the birational maps between them. By using the modular description, we are able to identify all but one log canonical models $\Mg(\a)$, $\a \in [0,1]$, with existing compactifications of $M_3$, some new and others classical, while the exception gives a new modular compactification of $M_3$.

Local signature of fibered complex surfaces via moduli and monodromy

AbstractThe signature of fibered complex surfaces is sometimes localized at finite fiber germs. We analyze this phenomenon by using datum of local monodromy in the mapping class group and a certain rational divisor on the moduli space of stable curves.

Covers of elliptic curves and the moduli space of stable curves

Consider genus g curves that admit degree d covers of an elliptic curve. Varying a branch point, we get a one-parameter family W of simply branched covers. Varying the target elliptic curve, we get another one-parameter family Y of covers that have a unique branch point. We investigate the geometry of W and Y by using admissible covers to study their slopes, genera and components. The results can be applied to study slopes of effective divisors on the moduli space of stable genus g curves.

Rational maps between moduli spaces of curves and Gieseker-Petri divisors

We study contractions of the moduli space of stable curves beyond the minimal model ofM¯g′\overline {\mathcal {M}}_{g’}by giving a complete enumerative description of the rational map between two moduli spaces of curvesM¯g⇢M¯g′\overline {\mathcal {M}}_g \dashrightarrow \overline {\mathcal {M}}_{g’}which associates to a curveCCof genusggits Brill–Noether locus of special divisors in the case this locus is a curve. As an application we construct many examples of moving effective divisors onM¯g\overline {\mathcal {M}}_gof small slope, which in turn can be used to show that various moduli space of curves with level structure are of general type. For lowg′g’our calculation can be used to study the intersection theory of the moduli space of Prym varieties of dimension55.