Factoriality of Certain Threefolds Complete Intersection in P 5 with Ordinary Double Points

Abstract

Let V ? P ⁵ be a reduced and irreducible threefold of degree s, complete intersection on a smooth hypersurface of degree t, with s > t ² ? t. In this paper, we prove that if the singular locus of V consists of δ < 3s/8t ordinary double points, then any projective surface contained in V is a complete intersection on V. In particular, V is Q-factorial.     

Curves on Generic Surfaces of High Degree Through a Complete Intersection in P 3

We consider general surfaces, S, of high degree containing a given complete intersection space curve, Y. We study integral curves in the subgroup of Pic(S) generated by Y and the plane section. We determine the cohomological invariants of these curves and classify the subcanonical ones. Then using these subcanonical curves we produce stable rank two vector bundles on P ³.     

Curves of Genus Three in Characteristic Two

For an algebraic nonsingular nonhyperelliptic complete curve of genus three defined over a field of characteristic two we show that the number of canonical Weierstrass points is one of the following: 24, 21, 20, 18, 17, 16, 15, 12, 11, 6 or 5. In most cases, we give also an explicit equation for the curve with the given number of Weierstrass points.     

Hyperbolic Hexagons and Algebraic Curves in Genus 3

One of the consequences of the uniformization theorem of Koebeand Poincaré is that any smooth complex algebraic curveC of genus g > 1 is conformally equivalent to H/G, whereG PSL₂(R) is a Fuchsian group and is naturally endowed witha hyperbolic metric. Conversely, any compact hyperbolic surfaceis isomorphic to an algebraic curve. Hence any curve of genusg > 1 may be described in two ways, either by an equationor by a Fuchsian group.     

The Moduli Spaces of Bielliptic Curves of Genus 4 with more Bielliptic Structures

Let C be an irreducible, smooth, projective curve of genus g 3 over the complex field C. The curve C is called biellipticif it admits a degree-two morphism : C E onto an ellipticcurve E such a morphism is called a bielliptic structure onC. If C is bielliptic and g6, then the bielliptic structureon C is unique, but if g=3,4,5, then this holds true only genericallyand there are curves carrying n>>1 bielliptic structures.The sharp bounds n 21,10,5 exist for g=3,4,5 respectively.Let M_g be the coarse moduli space of irreducible, smooth, projectivecurves of genus g=3,4,5. Denote by the locus of points in M_g $ representing curves carrying atleast n bielliptic structures. It is then natural to ask thefollowing questions. Clearly does hold? What are the irreducible components of ? Are the irreducible components of rational? How do the irreducible components of intersect each other? Let how many non-isomorphic elliptic quotients can it have? Completeanswers are given to the above questions in the case g=4.     

Quadrics of codimension 4 in ℂ7 and their automorphisms

Automorphisms of nondegenerate (4, 3)-quadrics are studied. We classify quadrics of this kind with nonlinear automorphisms and find the dimensions of their groups of automorphisms.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 803–811, December, 1998.     

分片代数曲线交点的结式求法

本文对确定分片代数曲线的二元样条函数的整体表达式中的截断引入参数表示,给出了分片代数曲线交点的结式求法．理论与实例表明,这种算法是有效的．     

Higher-Order Accurate Meshing of Nonsmooth Implicitly Defined Surfaces and Intersection Curves

Computational Mathematics and Mathematical Physics - A higher-order accurate meshing algorithm for nonsmooth surfaces defined via Boolean set operations from smooth surfaces is presented. Input...     

A Note on the Intersection of Dispersion Curves of a Fluid-loaded Elastic Plate

An investigation is made of the behaviour of a fluid-loadedelastic plate at the points of intersection of curves correspondingto symmetrical and antisymmetrical modes in the dispersion diagram.At these points one mode dominates the vibrational symmetryof the plate at the expense of the other. An approximate analysisis presented that provides a criterion, expressed in terms ofthe gradients of the dispersion curves, to indicate the dominantmode at the cross-over point.     

Correction: on the Intersection Matrices of Curves lying on Irregular Algebraic Surfaces

Professor W. F. Hammond has kindly drawn my attention to a blunderin 4 of the above paper. He referred to the ( – 2r) xß submatrix D of the skew-symmetric matrix displayednear the top of page 181, of which it is asserted that it issquare and non-singular, and pointed out that, from the factthat the matrix of which D forms part is regular, it may onlybe deduced that the columns of D are linearly independent; thatis, it only follows that – 2r ß. The validity of the equation – 2r = ß is essentialto the succeeding argument and, fortunately, may be establishedby alternative means. Using the nomenclature of the paper, wehave on F the set ₁^*, ..., _2r^*, ₁^*, ..., _ß^* of independent3-cycles (independent because they cut independent 1-cycleson the curve C), which may be completed, to form a basis forsuch cycles on F, by a further set ₁', ..., _2q–2r–pof independent 3-cycles, each of which meets C in a cycle homologousto zero on C. The cycles ₁^*, ..., ^* are invariant cycles andare independent on F so that, if > 2r + ß, thereis a non-trivial linear combination ^* of these having zero intersectionon C with each of the cycles ₁^*, ..., _2r^*, ₁^*, ..., _ß^*.Thus we have. (^* ._k^*)c = 0 = (^* ._i^*)c i.e. (^* ._k^*) = 0 = (^* ._i^* on F (1 k 2r; 1 i ß). Furthermore, (_j . C) 0 on C and we have (^* ._j .C)_C = 0 i.e. (^* ._j) = 0 on F (1 j 2q – 2r – ß). It now follows that ^* 0 on F (for it has zero intersectionwith every member of a basic set of 3-cycles on F). But thiscondradicts the assumption that ^* is a non-trivial linear combinationof the independent cycles ₁^*, ...,^*; and hence < 2r + ß.     

Partial Flocks of Non-Singular Quadrics in PG(2r + 1, q)

We generalise the definition and many properties of partial flocks of non-singular quadrics in PG(3, q) to partial flocks of non-singular quadrics in PG(2r + 1, q).     

Intersection domains and the sum of two complex hyperbolic metrics

In this article, we prove that the intersection of the unit ball in ℂⁿ with an affine transformation of it is a negatively curved domain. The two-dimensional case is due to Cheung and Wu.     

Intersection of Curves and Crossing Number of C m × C n on Surfaces

Let be two families of closed curves on a surface , such that , each curve in intersects each curve in , and no point of is covered three times. When is the plane, the projective plane, or the Klein bottle, we prove that the total number of intersections in is at least 10mn/9 , 12mn/11 , and mn+10 ^-13 m ² , respectively. Moreover, when m is close to n , the constants are improved. For instance, the constant for the plane, 10/9 , is improved to 8/5 , for n ≤ 5(m-1)/4 . Consequently, we prove lower bounds on the crossing number of the Cartesian product of two cycles, in the plane, projective plane, and the Klein bottle. All lower bounds are within small multiplicative factors from easily derived upper bounds. No general lower bound has been previously known, even on the plane. Received January 20, 1996, and in revised form October 21, 1996.     

关于亏格g的超椭圆曲线同构等价类数目的估计

研究了对应于椭圆曲线同构的变换类,并利用计数统计法获得了没有重根且第二项系数为常数的首一多项式的数目计算公式,在此基础上给出了奇数特征的有限域上亏格g的超椭圆曲线同构等价类的数目的估计.