A remark on very ample linear series

We give a criterium on the existence of (e - 1)-very ample linear series on a general k-gonal curve of genus $g (e \geq 1)$, and we add some general remarks on such series.     

Real line bundles on k-gonal real curves

For a generalk-gonal complex curve of genusg its variety of special line bundlesL with deg(L) =d andh ⁰(L) >r is known to contain an irreducible component of the expected dimension ρ_g (d, r) provided that the Brill-Noether number ρ_g (d, r) is non-negative andr ≤ k - 2. It is the purpose of this note to transfer this result of Brill-Noether type to the case ofk-gonal real curves, for real line bundles.     

Linear series on a general k-gonal curve

This paper is a contribution towards a Brill-Noether theory for the moduli space of smooth &-gonal curves of genusg. Specifically, we prove the existence of certain special divisors on a generalk-gonal curveC of genusg, and we detect an irreducible component of the “expected” dimension in the varietyW ^r _d (C), (r ≤k — 2) of special divisors ofC. The latter induces a new proof of the existence theorem for special divisors on a smooth curve.     

Koszul cohomology and singular curves

We investigate Koszul cohomology on irreducible nodal curves following the lines of [2]. In particular, we prove both Green and Green-Lazarsfeld conjectures for any k-gonal nodal curve which is general in the sense of [4].     

On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

Stronger linear programming relaxations of max-cut

We consider linear programming relaxations for the max cut problem in graphs, based on k-gonal inequalities. We show that the integrality ratio for random dense graphs is asymptotically 1+1/k and for random sparse graphs is at least 1+3/k. There are O(n^k)k-gonal inequalities. These results generalize work by Poljak and Tuza, who gave similar results for k=3.     

Representing a Functional Curve by Curves with Fewer Peaks

In this paper, we study the problems of (approximately) representing a functional curve in 2-D by a set of curves with fewer peaks. Representing a function (or its curve) by certain classes of structurally simpler functions (or their curves) is a basic mathematical problem. Problems of this kind also find applications in applied areas such as intensity-modulated radiation therapy (IMRT). Let f\bf f be an input piecewise linear functional curve of size n. We consider several variations of the problems. (1) Uphill–downhill pair representation (UDPR): Find two nonnegative piecewise linear curves, one nondecreasing (uphill) and one nonincreasing (downhill), such that their sum exactly or approximately represents f\bf f. (2) Unimodal representation (UR): Find a set of unimodal (single-peak) curves such that their sum exactly or approximately represents f\bf f. (3) Fewer-peak representation (FPR): Find a piecewise linear curve with at most k peaks that exactly or approximately represents f\bf f. Furthermore, for each problem, we consider two versions. For the UDPR problem, we study its feasibility version: Given ε>0, determine whether there is a feasible UDPR solution for f\bf f with an approximation error ε; its min-ε version: Compute the minimum approximation error ε ^∗ such that there is a feasible UDPR solution for f\bf f with error ε ^∗. For the UR problem, we study its min-k version: Given ε>0, find a feasible solution with the minimum number k ^∗ of unimodal curves for f\bf f with an error ε; its min-ε version: given k>0, compute the minimum error ε ^∗ such that there is a feasible solution with at most k unimodal curves for f\bf f with error ε ^∗. For the FPR problem, we study its min-k version: Given ε>0, find one feasible curve with the minimum number k ^∗ of peaks for f\bf f with an error ε; its min-ε version: given k≥0, compute the minimum error ε ^∗ such that there is a feasible curve with at most k peaks for f\bf f with error ε ^∗. Little work has been done previously on solving these functional curve representation problems. We solve all the problems (except the UR min-ε version) in optimal O(n) time, and the UR min-ε version in O(n+mlog m) time, where m<n is the number of peaks of f\bf f. Our algorithms are based on new geometric observations and interesting techniques.     

Weierstrass multiple points on algebraic curves and ramified coverings

Here we prove the following result on Weierstrass multiple points. Theorem:Fix integers k, g with k≥5 and g>4k. Then there exist a genus g, Riemann surface X and k points P ₁, …,P _k of X such that for all integers b ₁≥…≥b _k ≥0we have:

On q-n-gonal Klein Surfaces

We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism φ of prime order n with quotient space X/（φ） of algebraic genus q. These Klein surfaces axe called q-n-gonal surfaces and they are n-sheeted covers of surfaces of algebraic genus q. In this paper we extend the results of the already studied cases n ≤ 3 to this more general situation. Given p ≥ 2, we obtain, for each prime n, the （admissible） values q for which there exists a q-n-gonal surface of algebraic genus p. Furthermore, for each p and for each admissible q, it is possible to check all topological types of q-n-gonal surfaces with algebraic genus p. Several examples are given： q-pentagonal surfaces and q-n-gonal bordered surfaces with topological genus g = 0, 1.     

Multiple prime covers of the riemann sphere

A compact Riemann surface X of genus g≥2 which admits a cyclic group of automorphisms C _q of prime order q such that X/C _q has genus 0 is called a cyclic q-gonal surface. If a q-gonal surface X is also p-gonal for some prime p≠q, then X is called a multiple prime surface. In this paper, we classify all multiple prime surfaces. A consequence of this classification is a proof of the fact that a cyclic q-gonal surface can be cyclic p-gonal for at most one other prime p.     

Linear pencils on real algebraic curves

Our knowledge of linear series on real algebraic curves is still very incomplete. In this paper we restrict to pencils (complete linear series of dimension one). Let X denote a real curve of genus g with real points and let k(R) be the smallest degree of a pencil on X (the real gonality of X). Then we can find on X a base point free pencil of degree g+1 (resp. g if X is not hyperelliptic, i.e. if k(R)>2) with an assigned geometric behaviour w.r.t. the real components of X, and if we prove that which is the same bound as for the gonality of a complex curve of even genus g. Furthermore, if the complexification of X is a k-gonal curve (k≥2) one knows that k≤k(R)≤2k−2, and we show that for any two integers k≥2 and 0≤n≤k−2 there is a real curve with real points and k-gonal complexification such that its real gonality is k+n.     

On cyclic <Emphasis Type="Italic">p</Emphasis>-gonal Riemann surfaces with several <Emphasis Type="Italic">p</Emphasis>-gonal morphisms

Let p be a prime number, p > 2. A closed Riemann surface which can be realized as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. If the p-gonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic p-gonal Riemann surface. Accola showed that if the genus is greater than (p − 1)² the p-gonal morphism is unique. Using the characterization of p-gonality by means of Fuchsian groups we show that there exists a uniparametric family of cyclic p-gonal Riemann surfaces of genus (p − 1)² which admit two p-gonal morphisms. In this work we show that these uniparametric families are connected spaces and that each of them is the Riemann sphere without three points. We study the Hurwitz space of pairs (X, f), where X is a Riemann surface in one of the above families and f is a p-gonal morphism, and we obtain that each of these Hurwitz spaces is a Riemann sphere without four points.     

Equipartition of mass distributions by hyperplanes

We consider the problem of determining the smallest dimensiond=Δ(j, k) such that, for anyj mass distributions inR ^d , there arek hyperplanes so that each orthant contains a fraction 1/2 ^k of each of the masses. The case Δ(1,2)=2 is very well known. The casek=1 is answered by the ham-sandwich theorem with Δ(j, 1)=j. By using mass distributions on the moment curve the lower bound Δ(j, k)≥j(2 ^k −1)/k is obtained. We believe this is a tight bound. However, the only general upper bound that we know is Δ(j, k)≤j2 ^k−1. We are able to prove that Δ(j, k)=⌈j(^2k−1/k⌉ for a few pairs (j, k) ((j, 2) forj=3 andj=2 ⁿ withn≥0, and (2, 3)), and obtain some nontrivial bounds in other cases. As an intermediate result of independent interest we prove a Borsuk-Ulam-type theorem on a product of balls. The motivation for this work was to determine Δ(1, 4) (the only case forj=1 in which it is not known whether Δ(1,k)=k); unfortunately the approach fails to give an answer in this case (but we can show Δ(1, 4)≤5). This research was supported by the National Science Foundation under Grant CCR-9118874.     

Some analogues of eberhard’s theorem on convex polytopes

Letp _k denote the number ofk-gonal faces of a simple 3-polytope. Euler’s relation leads to an equation between thep _k ’s which does not involvep ₆. Eberhard proved in 1891 that every sequence of non-negative integers (p ₃,p ₄,…) satisfying this equation corresponds to a polytope for suitable values ofp ₆. In the present paper it is established that ifp ₃=p ₄=0 then every valuep ₆≧8 is suitable. Research supported in part by the National Science Foundation under grant GP-7536     

Sur les espaces multisécants aux courbes algébriques

We give a general new algorithm to find formulas for the number of k-secant d-planes to a smooth curve in P^N, for any k, d, N.     

Hodge structures on abelian varieties of type IV

Let A be a general member of a PEL-family of abelian varieties with endomorphisms by an imaginary quadratic number field k, and let E be an elliptic curve with complex multiplications by k. We show that the usual Hodge conjecture for products of A with powers of E implies the general Hodge conjecture for all powers of A. We deduce the general Hodge conjecture for all powers of certain 5-dimensional abelian varieties. Mathematics Subject Classification (2000): Primary 14C30, 14K20.Research supported in part by a Research and Creative Activity Award for Summer 2001 from East Carolina University.     

On gonality of Riemann surfaces

A compact Riemann surface X is called a (p, n)-gonal surface if there exists a group of automorphisms C of X (called a (p, n)-gonal group) of prime order p such that the orbit space X/C has genus n. We derive some basic properties of (p, n)-gonal surfaces considered as generalizations of hyperelliptic surfaces and also examine certain properties which do not generalize. In particular, we find a condition which guarantees all (p, n)-gonal groups are conjugate in the full automorphism group of a (p, n)-gonal surface, and we find an upper bound for the size of the corresponding conjugacy class. Furthermore we give an upper bound for the number of conjugacy classes of (p, n)-gonal groups of a (p, n)-gonal surface in the general case. We finish by analyzing certain properties of quasiplatonic (p, n)-gonal surfaces. An open problem and two conjectures are formulated in the paper.     

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     

Scrollar invariants and resolutions of certaind-gonal curves

In this note we compute the scrollar invariants of certaind-gonal curves (e.g. Castelnuovo curves and bielliptic curves) by using appropriate plane models. Ford=4 andg(C)≥10, we show that those invariants discriminate bielliptic curves among tetragonal ones.

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Sunto In questa nota si determinano gli invarianti scrollari per alcuni tipi di curved-gonaliC (ad esempio curve di Castelnuovo, curve biellittiche) tramite appropriati modelli piani. Perd=4 eg(C)≥10, si mostra che tali invarianti individuano le curve biellittiche fra le tetragonali.

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An erratum to this article is available at .     

Disk-like Tiles Derived from Complex Bases

For each positive integer k, the radix representation of the complex numbers in the base –k+i gives rise to a lattice self-affine tile T _k in the plane, which consists of all the complex numbers that can be expressed in the form ∑ _j≥1 d _j (–k+i)^–j , where d _j ∈{0, 1, 2, ...,k ²}. We prove that T _k is homeomorphic to the closed unit disk {z∈C:∣z∣ ≤ 1} if and only if k ≠ 2. The first author is supported by Youth Project of Tianyuan Foundation (10226031) and Zhongshan University Promotion Foundation for Young Teachers (34100-1131206); the second author is supported by National Science Foundation (10041005) and Guangdong Province Science Foundation (011221)