Semistable Curves Research Articles

CHOW STABILITY OF CANONICAL GENUS 4 CURVES

In this paper, we give sufficient conditions on a canonical genus 4 curve for it to be Chow (semi)stable. A Deligne-Mumford stable curve is a complete connected curve C having ample dualising sheaf !C and admitting only nodes as singularities. An n- canonical curve C ⊂ P N is a Deligne-Mumford stable curve of arithmetic genus g embedded by the complete linear system |! ⊗n C | where N = (2n−1)(g−1)−1 if n ≥ 2, and N = g − 1 if n = 1. Let Chowg,n be the closure of the locus of the Chow forms of n-canonical curves of arithmetic genus g in the Chow variety of algebraic cycles of dimension 1 and degree 2g−2 in P N. The natural action of SLN+1 on P N induces an action on Chowg,n. Denote the corresponding GIT (Geometric Invariant Theory) quotient space by Chowg,n//SLN+1. To understand this quotient space as a parameter space of curves with some geometric properties, we need to find Chow stability conditions. Mumford showed that, for n ≥ 5 and g ≥ 2, the Chow stable curves are pre- cisely Deligne-Mumford stable curves and there is no strictly Chow semistable curve (cf. (14)). This implies that the quotient space is precisely the moduli space of Deligne-Mumford stable curves M4. The cases when n = 3 and g ≥ 3 were concerned by Schubert in (16). He proved that a 3-canonical curve of genus g ≥ 3 is Chow stable if and only if it is pseudo-stable and also showed that there is no strictly Chow semistable curve, and thus the quotient space is the moduli space of pseudo-stable curves M ps . A pseudo-stable curve is a complete connected curve C satisfying the following properties.

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 ? .

Linear Series on Semistable Curves

of degree parameterized by the compactified Picard scheme. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following cases: X has two components; X is any semistable curve, and d=0 or d=2g ! 2; X is stable, free from separating nodes, and d 4. These results are shown to be sharp. Applications to the Clifford index, to the combinatorial description of hyperelliptic curves, and to plane quintics are given.

BOUNDS FOR THE TORSION OF ELLIPTIC CURVES OVER EXTENSIONS WITH BOUNDED RAMIFICATION

Let E be a semi-stable elliptic curve defined over ℚ, and fix N ≥ 2. Let KN/ℚ be a maximal algebraic Galois extension of ℚ whose ramification indices are all at most N. We show that there exists a computable bound B(N), which depends only on N and not on the choice of E/ℚ, such that the size of E(KN) Tors is always at most B(N).

Note on families of semistable curves over ℙ1 with 4 singular fibers whose Jacobian are non-compact

Let f: S → ℙ1 be a semistable family of curves of genus g ⩾ 2. We prove that if f admits exactly 5 singular fibers and 4 of them have non-compact Jacobian, then g = 2.

Note on families of semistable curves over ℙ1 with 4 singular fibers whose Jacobian are non-compact

Note on families of semistable curves over ℙ1 with 4 singular fibers whose Jacobian are non-compact

On Abel maps of stable curves

We construct Abel maps for a stable curve X. Namely, for each oneparameter deformation of X to a smooth curve, having regular total space, and each d ≥ 1, we construct by specialization a map αd X : Ẋd → P d X , where Ẋ ⊆ X is the smooth locus, and P d X is the coarse moduli scheme for equivalence classes of degree-d “semibalanced” line bundles on semistable curves having X as a stable model. For d = 1, we show that α1 X extends to a map α1 X : X → P 1 X , and does not depend on the choice of the deformation. Finally, we give a precise description of when α1 X is injective. The theory of Abel maps for smooth curves goes back to the 19th century. In the modern language, let C be a smooth projective curve, and PicC its degree-d Picard variety parametrizing line bundles of degree d on C. For each d > 0 there exists a remarkable morphism, often called the d-th Abel map: Cd −→ PicC (p1, . . . , pd) 7→ OC( ∑ pi). This map has been extensively studied and used in the literature. For d = 1, after the choice of a “base” point on C, it gives the Abel–Jacobi embedding C ↪→ PicC ∼= PicC (unless C ∼= P1). For an interesting historic survey see [K04] or [K05]. What about Abel maps for singular curves? Abel maps were constructed for all integral curves in [AK], and further studied in [EGK00], [EGK02] and [EK05]. In [AK], it is shown that the first Abel map of an integral singular curve is an embedding into its compactified Picard scheme. However almost nothing is known for reducible curves, not even when they are stable. This lack of knowledge appears all the more regrettable because of the importance of stable curves in moduli theory. In the present paper we construct Abel maps for stable curves. As we see it, Abel maps should satisfy the following natural properties. First, they should have a geometric meaning. More explicitly, recall that for a smooth

Déformations formelles de revêtements: un principe local-global

LetC be a generically smooth, locally complete intersection curve defined over an algebraically closed fieldk of characteristicp≥0. LetG⊃ AutkC be a finite group of automorphisms ofC. We develop a theory ofG-equivariant deformations of the Galois coverC→C/G. We give a thorough study of the local obstructions, those localized at singular or widely ramified points, to deform equivariantly a cover. As an application, we discuss the case ofG-equivariant deformations of semistable curves.

Quelques résultats sur les déformations équivariantes des courbes stables

Let G be a finite group, let g≥2 and g ′ ≥ 0 be integers. We introduce the algebraic stack Open image in new window classifying the stable curves Open image in new window of genus g endowed with an action of G faithful in each geometric fiber and such that the quotient of each fiber is a semi-stable curve of genus g′. We study the completion of the local rings of this algebraic stack. They are closely related to universal equivariant deformation rings RC,G of stable curves Open image in new window endowed with a faithful action of G. A useful tool for this purpose is a local-global principle generalizing the one obtained by Bertin and Mezard in [BM00]. We then use the results we already proved in [Mau03b] and [Mau03a] to describe some properties of the space Open image in new window (purity, dimension).

Problem of formation of quotients and base change

Let X/S be a semistable curve equipped with the action of a finite group G and let H be a normal subgroup of G. The main result of this paper is the following. If the action of G is free on an open dense set on any geometric fiber, then for any base change T→S, (X/G)×ST is isomorphic to (X×ST)/G. As an application, this allows us to define induction and restriction morphisms between the G-equivariant deformation functor of X and the G/H-equivariant (resp. H-equivariant) deformation functor of X/H (resp. X).

Cornalba-Harris equality for semistable hyperelliptic curves in positive characteristic

Let Y be a nonsingular projective curve over an algebraically closed field k and f : X → Y a generically smooth semistable curve of genus g ≥ 2 with X nonsingular. Let ωX/Y denote the relative dualizing sheaf of f . Relation between deg ( f∗ωX/Y ) and discriminant divisors has been studied by many people. Here we consider the case of f hyperelliptic, i.e., the case where there exists a Y -automorphism ι inducing the hyperelliptic involution on the geometric generic fiber. Then for each node x of type 0 in a fiber, we can assign a non-negative integer, called the subtype, to x or the pair {x, ι(x)} (c.f. [CH] or §§ 1.2 for the definitions). Let δi(X/Y ) denote the number of the nodes of type i, ξ0(X/Y ) the number of nodes of subtype 0 and let ξj(X/Y ) denote the number of pairs of nodes {x, ι(x)} of subtype j > 0, in all the fibers. Cornalba and Harris proved in [CH] an equality

Families of $K3$ surfaces over curves reaching the Arakelov-Yau type upper bounds and Modularity

Let f : X → C be a family of semi-stable curves of genus g over a smooth projective C of genus q, and S ⊂ C the degeneration locus of f. The so-called Arakelov inequality states that deg f∗ωX/C ≤ g2 deg Ω 1 C(log S) = g 2 (2q − 2 + #S). When g ≥ 2 and #S = 0, the Miyaoka-Yau inequality for surfaces implies a much stronger inequality deg f∗ωX/C ≤ g − 1 6 deg Ω 1 C . In general, Tan [28] proved that the Arakelov inequality for a family f : X → C of semi-stable curves of genus ≥ 2 holds strictly. If g = 1, then deg f∗ωX/C can reach the upper bound in the inequality. Beauville has classified such families over C = P with #S = 4. More precisely, there are exactly 6 non-isotrivial families of semi-stable elliptic curves over P with 4 singular fibres. All of them are modular families of elliptic curves [2]. In this paper, we will consider the similar question for families of higher dimensional varieties. The Arakelov inequality is a special case of some more general inequalities for Hodge bundles. To state them, let V denote a polarized real variation of Hodge structure on a smooth projective curve C S such that the local monodromies around S are all unipotent, let (⊕p+q=kE, θ) denote the corresponding Hodge bundles. In [8] the following Arakelov-Yau type inequality was proven (also see [18] for a similar inequality): If k = 2l + 1, then

Arakelov type inequalities for Hodge bundles over algebraic varieties, Part I: Hodge bundles over algebraic curves

We prove Arakelov inequalities for systems of Hodge bundles over algebraic varieties, generalizing the classical ones for families of semi-stable curves and abelian varieties. These inequalities are derived from the semi-stability of an associated Higgs bundle, a consequence of the existence of a Hermitian Yang-Mills metric.

Some Computations with Heeke Rings and Deformation Rings

lnthe proof by Wiles, completed by Taylor-Wiles, of the fact that all semistable elliptic curves over Q are modular, certain deformation rings play an important role. In this note, we explicitly compute these rings for the elliptic curve y 2 + XY = X 3 – X 2 – X – 3 of conductor 142.

Automorphismes d'ordre p des couronnes p-adiques ouvertes

Let k be an algebraically closed field of characteristic p > 0 and R a discrete valuation ring finite over W(k)[ P 2 √1]. We construct order p and p 2 automorphisms of open annuli over fr R, and we deduce a theorem concerning liftings of semi-stable curves (Y, G) over k, together with a finite group of automorphisms such that the inertia subgroups are cyclic of order p an, where aand (n,p) = 1. We also give a description of the metric geometry affixed points of order p automorphisms of open annuli with small conductors.

Height and arithmetic intersection for a family of semi-stable curves

In this paper, we consider an arithmetic Hodge index theorem for a family of semi-stable curves, generalizing Faltings-Hriljac's arithmetic Hodge index theorem for an arithmetic surface.

Moduli-stacks for bundles on semistable curves

Moduli-stacks for bundles on semistable curves

On the Weierstrass Class of Semistable Elliptic Curves Over Quadratic Fields

On the Weierstrass Class of Semistable Elliptic Curves Over Quadratic Fields

Moduli of Abelian Varieties

On extra components in the functorial compactification of Ag.- On Mumford's uniformization and Neron models of Jacobians of semistable curves over complete rings.- Torelli theorem via Fourier-Mukai transform.- On the Andre-Oort conjecture for Hilbert modular surfaces.- Toroidal resolutions for some matrix singularities.- Formal Brauer groups and moduli of abelian surfaces.- Isogeny classes of abelian varieties with no principal polarizations.- Igusa's modular form and the classification of Siegel modular threefolds.- Mirror symmetry and quantization of abelian varieties.- Group schemes with additional structures and Weyl group cosets.- Moduli space of elliptic curves with Heisenberg level structure.- Singularities of the height strata in the moduli of K3 surfaces.- A stratification of a moduli space of abelian varieties.- Newton polygon strata in the moduli space of abelian varieties.- The dimension of Oort strata of Shimura varieties of PEL-type.- Hyperelliptic Jacobians and modular representations.- Windows for displays of p-divisible groups.

Semi-stable curves on algebraic surfaces and logarithmic pluricanonical maps

Semi-stable curves on algebraic surfaces and logarithmic pluricanonical maps