Local invariants attached to Weierstrass points

Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation d of the hyperelliptic discriminant of X/S, and the valuation δ of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable the invariants d and δ are known to satisfy certain inequalities. We prove an exact formula relating d and δ with intersection theoretic data determined by the distribution of Weierstrass points over the special fiber, in the semistable case. We also prove an exact formula for the stable Faltings height of an arbitrary curve over a number field, involving local contributions associated to its Weierstrass points.     

One half log discriminant and division polynomials

Szpiro and Tucker recently proved that, under mild conditions, the valuation of the minimal discriminant of an elliptic curve with semistable reduction over a discrete valuation ring can be expressed in terms of intersections between n-torsion and 2-torsion, where n tends to infinity. The argument of Szpiro and Tucker is geometric in nature. We give a proof based on the arithmetic of division polynomials, and generalize the result to the case of hyperelliptic curves.     

False Hyperelliptic Surfaces with Section

Let X be a nonsingular relatively minimal projective surface over an algebraically closed field of characteristic p > 0. We call X a false hyperelliptic surface if X satisfies the following conditions: (1) c₂(X) = 0, c₁(X)² = 0, dim Alb (X) = 1, and (2) All fibres of the Albanese mapping of X are rational curves with only one cusp of type xp^v + yⁿ = 0. In this article, we consider a false hyperelliptic surface whose Albanese mapping has a cross-section. We prove that every false hyperellyptic surface with section arises from an elliptic ruled surface and that every false hyperelliptic surface has an elliptic fibration with multiple fibre. Moreover, we construct an example of false hyperelliptic surface with section, whose elliptic fibration has a multiple fibre of supersingular elliptic curve of multiplicity p^v (v > 1).     

On the indices of curves over local fields

Fix a non-negative integer g and a positive integer I dividing 2g − 2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we construct a curve C _/K of genus g and index I. This is obtained via a systematic analysis of local points on arithmetic surfaces with semistable reduction. Applications are discussed to the corresponding problem over number fields.     

Torsion Packets on Curves

Let X be a curve of genus g2 over a field of characteristic zero. Then X has at most finitely many torsion packets of size greater than 2. Moreover, X has infinitely many torsion packets of size 2 if and only if either g=2, or g=3 and X is both hyperelliptic and bielliptic.     

Numerical Semigroup Rings and Almost Prüfer v-Multiplication Domains

Let D be an integral domain with quotient field K, X be an indeterminate over D, Γ be a numerical semigroup with Γ ? ?₀, D[Γ] be the semigroup ring of Γ over D (and hence D ? D[Γ] ? D[X]), and D + X ⁿ K[X] = {a + X ⁿ g∣a ∈ D and g ∈ K[X]}. We show that there exists an order-preserving bijection between Spec(D[X]) and Spec(D[Γ]), which also preserves t-ideals. We also prove that D[Γ] is an APvMD (resp., AGCD-domain) if and only if D[X] is an APvMD (resp., AGCD-domain) and char(D) ≠ 0. We show that if n ≥ 2, then D is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain) and char(D) ≠ 0 if and only if D + X ⁿ K[X] is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain). Finally, we give some examples of APvMDs which are not AGCD-domains by using the constructions D[Γ] and D + X ⁿ K[X].     

Automorphisms of curves fixing the order two points of the Jacobian

Let X be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic different from two. If X admits a nontrivial automorphism σ that fixes pointwise all the order two points of Pic⁰(X), then we prove that X is hyperelliptic with σ being the unique hyperelliptic involution. As a corollary, if a nontrivial automorphisms of X fixes pointwise all the theta characteristics on X, then X is hyperelliptic with being its hyperelliptic involution.      

Actions of compact groups on coherent sheaves

We consider actions of compact real Lie GroupsK on complex spacesX such that the associated reducedK-space admits a semistable quotient, e.g.X is a Stein space. We show that there is a complex spaceX ^c endowed with a holomorphic action of the universal complexificationG ofK that containsX as an openK-stable subset. As our main result, we prove that every coherentK-sheaf onX extends uniquely to a holomorphicG-sheaf onX ^c .Supported by a Heisenberg Stipendium of the Deutsche Forschungsgemeinschaft.     

Existence of <Emphasis Type="Italic">g</Emphasis><Superscript>1</Superscript><Subscript><Emphasis Type="Italic">g</Emphasis>–2</Subscript>’s on a double covering of a hyperelliptic curve

Abstract Let X be a non–hyperelliptic curve of genus g which is a double covering of a hyperelliptic curve C of genus h. In this paper, we prove that, if h≥ 3 and g≥ 4h+5, then X admits a complete, base point free g¹_g–2. Moreover, if h=3, this result holds under the mild condition g≥ 4h+3=15. Keywords: Double covering of hyperelliptic curves, Pencil of degree g–2 Mathematics Subject Classification (2000:) 14H30, 14H45     

The stringy E‐function of the moduli space of Higgs bundles with trivial determinant

Let M be the moduli space of semistable rank 2 Higgs pairs (V, ϕ) with trivial determinant over a smooth projective curve X of genus g ≥ 2. We provide an explicit formula for the stringy E‐function of M . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sur certains sous-groupes de torsion de jacobiennes de courbes hyperelliptiques de genreg ≥ 1

We construct a family of hyperelliptic curves of genusg defined over Q whose Jacobians have a rational point of order 2g(2g+1). Forl = 2g ² + 5g + 5, we construct a family of genusg hyperelliptic curves defined over Q, such that their Jacobians have a rational point of orderl orl / 2 orl / 4. We also construct a hyperelliptic curve of genusg defined over Q, which does not belong to the previous family, and whose Jacobian has a rational point of orderl.      

Restriction of Stable Rank Two Vector Bundles in Arbitrary Characteristic

Let X be a smooth variety defined over an algebraically closed field of arbitrary characteristic and 𝒪 _X (H) be a very ample line bundle on X. We show that for a semistable X-bundle E of rank two, there exists an integer m depending only on Δ(E) · H ^dim(X)?2 and H ^dim(X) such that the restriction of E to a general divisor in |mH| is again semistable. As corollaries, we obtain boundedness results, and weak versions of Bogomolov's Theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.     

A new infinite class of S(3, {4, 5, 7}, v)

A t‐wise balanced design ( at BD) of order v and block sizes from K , denoted by S ( t , K , v ), is a pair ( X , ??), where X is a v ‐element set and ?? is a set of subsets of X , called blocks , with the property that | B |∈ K for any B ∈?? and every t ‐element subset of X is contained in a unique block. In this article, we shall show that there is an S ( 3 , { 4 , 5 , 7 }, v ) for any positive integer v ≡ 7 ( mod12 ) with v ≠ 19 . Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 20:68–80, 2012     

Über Bewertungen,welche unter einer reduktiven Gruppe invariant sind

Summary LetG be a reductive group defined over an algebraically closed fieldk and letX be aG-variety. In this paper we studyG-invariant valuationsv of the fieldK of rational functions onX. These objects are fundamental for the theory of equivariant completions ofX. LetB be a Borel subgroup andU the unipotent radical ofB. It is proved thatv is uniquely determined by its restriction toK ^U . Then we study the set of invariant valuations having some fixed restrictionv ₀, toK ^B . Ifv ₀ is geometric (i.e., induced by a prime divisor) then this set is a polyhedron in some vector space. In characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of finite reflection groupW ^X . Thus, the classification of invariant valuations is almost reduced to the classification of valuations ofK ^B .

Unterstützt durch den Schweizerischen Nationalfonds zur Förderung der wissenschaftlichen Forschung.     

On Invariants and Strict Systems of Irreducible Polynomials Over Henselian Valued Fields

Let g(x) be a monic irreducible defectless polynomial over a henselian valued field (K, v), i.e., K(θ) is a defectless extension of (K, v) for any root θ of g(x). It is known that a complete distinguished chain for θ with respect to (K, v) gives rise to several invariants associated with g(x). Recently Ron Brown studied certain invariants of defectless polynomials by introducing strict systems of polynomial extensions. In this article, the authors establish a one-to-one correspondence between strict systems of polynomial extensions and conjugacy classes of complete distinguished chains. This correspondence leads to a simple interpretation of various results proved for strict systems. The authors give new characterizations of an invariant γ _g introduced by Brown.     

Integer-Valued Polynomial Rings,t-Closure,and Associated Primes

Given an integral domain D with quotient field K, the ring of integer-valued polynomials on D is the subring {f(X) ∈ K[X]: f(D) ? D} of the polynomial ring K[X]. Using the tools of t-closure and associated primes, we generalize some known results on integer-valued polynomial rings over Krull domains, Prüfer v-multiplication domains, and Mori domains.     

Graph invariants and the positivity of the height of the Gross-Schoen cycle for some curves

Let X be a projective curve over a global field K. Gross and Schoen defined a modified diagonal cycle Δ on X ³, and showed that the height ${\langle \Delta, \Delta \rangle}$ is defined in general. Zhang recently proved a formula which describe ${\langle \Delta, \Delta \rangle}$ in terms of the self pairing of the admissible dualizing sheaf and the invariants arising from the reduction graphs. In this note, we calculate explicitly those graph invariants for the reduction graphs of curves of genus 3 and examine the positivity of ${\langle \Delta, \Delta \rangle}$ . We also calculate them for so-called hyperelliptic graphs. As an application, we find a characterization of hyperelliptic curves of genus 3 by the configuration of the reduction graphs and the property ${\langle \Delta, \Delta \rangle = 0}$ .     

A characterization of hyperelliptic Jacobians

In this paper we will prove a criterion for hyperelliptic Jacobians. LetD be a translation invariant vector field on an indecompssable principally polarized abelian variety (i.p.p.a.v.) (X, Θ), letDΘ be the divisor of the sectionDΘ∈H ⁰ (Θ,O(Θ)|Θ). We have that (X, Θ) is the Jacobian of an hyperelliptic curve iff (Theorem 1) all the component ofDΘ are non reduced and the singular locus of Θ has dimension less thang-2. We will prove our theorem by showing that under the above geometrical condition it is possible to construct a Kodomcev-Petviashvili (K.P.) equation which is satisfied by the theta function corresponding to the principal polarization onX.