Normal Rational Curves Over Prime Fields

It is shown that in the projective spaces PG(n,p),p prime, 2 n p-2, the normal rational curves are the only (p+1)-arcs fixed by a projective group G isomorphic to PSL(2,p).     

Curves in Projective Spaces and Almost MDS Codes

A k-track in PG(n,q) is a set of k points such that every n of them are in general position. Here, we construct a class of n-tracks arising from algebraic curves.     

Projective Varieties Swept Out by Rational Normal Curves

Let X ? ? ⁿ be a complex nondegenerate projective variety of dimension m ≥ 2. For t ≤ n ? m and a general q ∈ ? ⁿ , the linear space L _q spanned by q and t general points of X meets X in a finite set of points. We classify those X ? ? ⁿ for which there exists a point q ∈ ? ⁿ such that L _q meets X in a positive dimensional variety. If this occurs, there exists d ≤ n ? m such that a degree d rational normal curve through d general points of X is contained in X. Examples of this situation are provided. An infinitesimal generalization of part of the main result is also stated.     

Computing the Rational Torsion of an Elliptic Curve Using Tate Normal Form

It is a classical result (apparently due to Tate) that all elliptic curves with a torsion point of order n(4?n?10, or n=12) lie in a one-parameter family. However, this fact does not appear to have been used ever for computing the torsion of an elliptic curve. We present here an extremely down-to-earth algorithm using the existence of such a family.     

On Rational Cuspidal Projective Plane Curves

In 2002, L. Nicolaescu and the fourth author formulated a verygeneral conjecture which relates the geometric genus of a Gorensteinsurface singularity with rational homology sphere link withthe Seiberg--Witten invariant (or one of its candidates) ofthe link. Recently, the last three authors found some counterexamplesusing superisolated singularities. The theory of superisolatedhypersurface singularities with rational homology sphere linkis equivalent with the theory of rational cuspidal projectiveplane curves. In the case when the corresponding curve has onlyone singular point one knows no counterexample. In fact, inthis case the above Seiberg--Witten conjecture led us to a veryinteresting and deep set of ‘compatibility properties’of these curves (generalising the Seiberg--Witten invariantconjecture, but sitting deeply in algebraic geometry) whichseems to generalise some other famous conjectures and propertiesas well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yauinequalities). Namely, we provide a set of ‘compatibilityconditions’ which conjecturally is satisfied by a localembedded topological type of a germ of plane curve singularityand an integer d if and only if the germ can be realized asthe unique singular point of a rational unicuspidal projectiveplane curve of degree d. The conjectured compatibility propertieshave a weaker version too, valid for any rational cuspidal curvewith more than one singular point. The goal of the present articleis to formulate these conjectured properties, and to verifythem in all the situations when the logarithmic Kodaira dimensionof the complement of the corresponding plane curves is strictlyless than 2. 2000 Mathematics Subject Classification 14B05,14J17, 32S25, 57M27, 57R57 (primary), 14E15, 32S45, 57M25 (secondary).     

Cyclic Arcs in PG(2, q)

B.C. Kestenband [9], J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas [3], E. Boros, and T. Szönyi [1] constructed complete (q ² ? q + l)-arcs in PG(2, q ²), q ≥ 3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q ² ? q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.     

Explicit Rational Functions on Fermat Curves and a Theorem of Greenberg

This paper is concerned with the arithmetic of curves of the form v^p=u^s(1-u), where p is a prime with p 5 and s is an integer such that 1 s p-2. The Jacobians of these curves admit complex multiplication by a primitive p-th root of unity . We find explicit rational functions on these curves whose divisors are p-multiples of divisors representing (1-)² - and (1-)³-division points on the corresponding Jacobians. This also gives an effective version of a theorem of Greenberg.     

Enumerating Rational Curves: The Rational Fibration Method

A new method to approach enumerative questions about rational curves on algebraic varieties is described. The idea is to reduce the counting problems to computations on the Néron-Severi group of a ruled surface. Applications include a short proof of Kontsevich's formula for plane curves and the solution of the analogous problem for the Hirzebruch surface F₃.     

有限域上代数曲线有理点个数

设Ｘ是有限域上光滑、绝对不可约的射影曲线，表示其有理点集，ｇ（Ｘ）表示Ｘ的亏格，令证明了本文利用构造无Ｈｉｌｂｅｒｔ类域塔方法证明了     

Extrema Constrained by a Family of Curves and Local Extrema

This paper considers the connections between the local extrema of a function f:DR and the local extrema of the restrictions of f to specific subsets of D. In particular, such subsets may be parametrized curves, integral manifolds of a Pfaff system, Pfaff inequations. The paper shows the existence of C ¹ or C ²-curves containing a given sequence of points. Such curves are then exploited to establish the connections between the local extrema of f and the local extrema of f constrained by the family of C ¹ or C ²-curves. Surprisingly, what is true for C ¹-curves fails to be true in part for C ²-curves. Sufficient conditions are given for a point to be a global minimum point of a convex function with respect to a family of curves.     

有理曲线和曲面的分片多项式逼近

本文利用摄动的思想,以摄动有理曲线（曲面）的系数的无穷模作为优化目标,给出了用多项式曲线（曲面）逼近有理曲线（曲面）的一种新方法．同以前的各种方法相比,该方法不仅收敛而且具有更快的收敛速度,并且可以与细分技术相结合,得到有理曲线与曲面的整体光滑、分片多项式的逼近．     

Rational Surfaces with Many Nodes

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain n b ₂ –2 disjoint smooth rational curves with self-intersection –2, where b ₂ is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with p _g = 0 and K² = 8, 9 which admit an automorphism of order 2.     

On Normal Embedding of Subgroups

We survey conditions for a group to allow an embedding as a normal subgroup of another finite group, with an additional condition of containment in some given characteristic subgroup. The main objects will be finite p-groups.     

On the Group of Automorphisms of Shimura Curves and Applications

Let V _D be the Shimura curve over attached to the indefinite rational quaternion algebra of discriminant D. In this note we investigate the group of automorphisms of V _D and prove that, in many cases, it is the Atkin–Lehner group. Moreover, we determine the family of bielliptic Shimura curves over and over and we use it to study the set of rational points on V _D over quadratic fields. Finally, we obtain explicit equations of elliptic Atkin–Lehner quotients of V _D .     

On Linear Systems of Curves on Rational Scrolls

In this paper we prove a conjecture on the dimension of linear systems, with base points of multiplicity 2 and 3, on an Hirzebruck surface.     

Rigid Analytic Uniformization of Curves and the Study of Isogenies

Complex uniformization of curves is a popular tool in Number Theory. There are, however, some arithmetic and computational advantages in the use of p-adic uniformization. This paper compares the two theories and discusses how they can be used to study isogenies, with explicit examples of p-adic uniformization of hyperelliptic curves.      

Disconnected Equivariant Rational Homotopy Theory

We present a variant of the disconnected equivariant rational homotopy theory to complete the result shown in [8]. For a finite group G let O(G) be the category of its canonical orbits. We prove that the category O(G)-DGA _Q of O(G)S-complete differential graded algebras over the rationals is a closed model category, where S runs over all O(G)-sets. Then, by means of the equivariant KS-minimal models, we show that the homotopy category of O(G)-DGA _Q is equivalent to the rational homotopy category of G-nilpotent disconnected simplicial sets provided G is a finite Hamiltonian group.     

On k-primitivity of two classes of digraphs

In this paper, we study two classes of primitive digraphs, and show that there are k-colorings that are k-primitive.