A bound on the plurigenera of projective varieties

We exhibit a sharp Castelnuovo bound for the i-th plurigenus of a smooth variety of given dimension n and degree d in the projective space P ^r , and classify the varieties attaining the bound, when n2, r2n+1, d>>r and i>>r. When n=2 and r=5, or n=3 and r=7, we give a complete classification, i.e. for any i1. In certain cases, the varieties with maximal plurigenus are not Castelnuovo varieties, i.e. varieties with maximal geometric genus. For example, a Castelnuovo variety complete intersection on a variety of dimension n+1 and minimal degree in P ^r , with r>(n ² +3n)/(n–1), has not maximal i-th plurigenus, for i>>r. As a consequence of the bound on the plurigenera, we obtain an upper bound for the self-intersection of the canonical bundle of a smooth projective variety, whose canonical bundle is big and nef. Mathematics Subject Classification (2000):Primary 14J99; Secondary 14N99     

Non-normal surfaces of degree 5 and 6 in P n

The purpose of this paper is to describe the non-normal surfaces of degree 5 and 6 embedded in the complex projective space P ⁿ , with n4. The idea is to study the normalization of such a surface, and then to find how a non-normal surface can be obtained from its normalization.     

Self-dual sets of type (m, n) in projective spaces

In this paper we introduce and analyze the notion of self-dual k-sets of type (m, n). We show that in a non-square order projective space such sets exist only if the dimension is odd. We prove that, in a projective space of odd dimension and order q, self-dual k-sets of type (m, n), with , are of elliptic and hyperbolic type, respectively. As a corollary we obtain a new characterization of the non-singular elliptic and hyperbolic quadrics.     

Dimensional linear spaces whose automorphism group is (line,hyperplane)-flag transitive

We classify the pairs (S, G) where S is a finite n-dimensional linear space with n 4 and G is an automorphism group of S acting transitively on the (line, hyperplane)-flags. We show in particular that S must be either a Desarguesian projective or affine space provided with its subspaces of dimension n - 1, or a Mathieu-Witt design provided with its blocks and its subsets of size n - 1. Our proof uses a recent classification of the flag transitive 2-(v, k, 1) designs, which in turn heavily depends on the classification of all finite simple groups. The case n = 3 has been settled in another paper.     

A Gap in GRM Code Weight Distributions

The weight distribution of GRM (generalized Reed-Muller) codes is unknown in general. This article describes and applies some new techniques to the codes over F3. Specifically, we decompose GRM codewords into words from smaller codes and use this decomposition, along with a projective geometry technique, to relate weights occurring in one code with weights occurring in simpler codes. In doing so, we discover a new gap in the weight distribution of many codes. In particular, we show there is no word of weight 3^m–2 in GRM₃(4,m) for m>6, and for even-order codes over the ternary field, we show that under certain conditions, there is no word of weight d+, where d is the minimum distance and is the largest integer dividing all weights occurring in the code.     

On the Structure of Submanifolds with Degenerate Gauss Maps

An n-dimensional submanifold X of a projective space P ^N (C) is called tangentially degenerate if the rank of its Gauss mapping gamma;; X G(n, N) satisfies 0 < rank < n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space P ^N (C). By means of the focal images, three basic types of submanifolds are discovered: cones, tangentially degenerate hypersurfaces, and torsal submanifolds. Moreover, for tangentially degenerate submanifolds, a structural theorem is proven. By this theorem, tangentially degenerate submanifolds that do not belong to one of the basic types are foliated into submanifolds of basic types. In the proof the authors introduce irreducible, reducible, and completely reducible tangentially degenerate submanifolds. It is found that cones and tangentially degenerate hypersurfaces are irreducible, and torsal submanifolds are completely reducible while all other tangentially degenerate submanifolds not belonging to basic types are reducible.     

A Note on the Robust 1-Center Problem on Trees

We consider the robust 1-center problem on trees with uncertainty in vertex weights and edge lengths. The weights of the vertices and the lengths of the edges can take any value in prespecified intervals with unknown distribution. We show that this problem can be solved in O(n ³ logn) time thus improving on Averbakh and Berman's algorithm with time complexity O(n ⁶). For the case when the vertices of the tree have weights equal to 1 we show that the robust 1-center problem can be solved in O(nlogn) time, again improving on Averbakh and Berman's time complexity of O(n ² logn).     

On rational embeddings of curves in the second Garcia–Stichtenoth tower

In 1995, Garcia and Stichtenoth explicitly constructed a tower of projective curves over a finite field with q² elements which reaches the Drinfeld–Vlăduţ bound. These curves are given recursively by covers of Artin–Schreier type where the curve on the nth level of the tower has a natural model in . In this paper, for q an even prime power, we use point projections in order to embed these curves into projective space of the lowest possible dimension.     

Dual linear connections on fitted manifolds of a space with projective connection

This paper presents an investigation of dual linear connections (projective and affine), induced by different fittings of a space with a projective connection P_n,n, a regular hypersurface V_n-1P _n,n , and a regular hyperbandH _m P _n,n .Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 8, pp. 25–46, 1977.     

On the complement of the set of n-collinear points in PG(3, n)

Let ${S = (\mathcal{P}, \mathcal{L}, \mathcal{H})}$ be the finite planar space obtained from the 3-dimensional projective space PG(3, n) of order n by deleting a set of n-collinear points. Then, for every point ${p\in S}$ , the quotient geometry S/p is either a projective plane or a punctured projective plane, and every line of S has size n or n + 1. In this paper, we prove that a finite planar space with lines of size n + 1 ? s and n + 1, (s ≥ 1), and such that for every point ${p\in S}$ , the quotient geometry S/p is either a projective plane of order n or a punctured projective plane of order n, is obtained from PG(3, n) by deleting either a point, or a line or a set of n-collinear points.     

Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank

We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m5) of positive sectional curvature on which an (m–1)-torus acts isometrically is homeomorphic to a complex projective space if and only if its Euler characteristic is not 2. By [Wi], these results imply a homeomorphism classification for positively curved n-manifolds (n8) of almost maximal symmetry rank Supported by CNPq of Brazil, NSFC Grant 19741002, RFDP and Qiu-Shi Foundation of China.Supported partially by NSF Grant DMS 0203164 and a research grant from Capital normal university.     

Projective change between two classes of -metrics

In this paper, we find equations to characterize projective change between (α,β)-metric and Randers metric on a manifold with dimension n3, where α and are two Riemannian metrics, β and are two nonzero one forms. Moreover, we consider this projective change when F has some special curvature properties.     

Uniformization and the Poincaré metric on the leaves of a foliation by curves

In this paper we prove that a holomorphic foliation by curves, on a complex compact manifoldM, whose singularities are non degenerated and whose tangent line bundle admits a metric of negative curvature, satisfies the following properties:(a): All leaves are hyperbolic.(b): The Poincaré metric on the leaves is continuous.(c): The set of uniformizations of the leaves by the Poincaré disc D is normal. Moreover, if ( _n ) _n 1 is a sequence of uniformizations which converges to a map : D, then either is a constant map (a singularity), or is an uniformization of some leaf. This result generalizes Theorem B of [LN], in which we prove the same facts for foliations of degree 2 on projective spaces.This research was partially supported by Pronex-Dynamical Systems, FINEP-CNPq.     

Manifolds polarized by vector bundles

LetX be a complex projective manifold of dimension n and let ε be an ample vector bundle of rank r. Let also τ = τ (X,ε) = min {t ∈ ℝ : K_X + t det ε is nef} be the nef value of the pair (X, ε). In this paper we classify the pairs (X, ε) such that{ Mathematics Subject Classification (2000)14J60; 14J40; 14E30     

On Certain Schur Rings of Dimension 4

In this paper, we consider Schur rings on a finite group G of ordern(n-1) suchthat G has a partition with . Then Gis characterized as follows. (a) G has subgroups E andH of order n andn-1 respectively, and , or(b)G has subgroupsK andH( K) of order 2(n-1) and n-1 respectively,and . In addition assume that G has a subsetR of sizen-1 satisfying in the groupalgebraC[G]. Then G is characterized as a collineation groupof a projective plane of order n such that G has five orbits ofpoints of lengthsn(n-1), n, n-1, 1 and 1. In particular, we characterize projective planesof ordern admitting a quasiregular collineation group of order n(n-1)as the case that E and H are normal subgroups ofG.     

Set-theoretic complete intersection monomial curves in $${\mathbb{P}^n}$$

In this paper, we give a sufficient numerical criterion for a monomial curve in a projective space to be a set-theoretic complete intersection. Our main result generalizes a similar statement proven by Keum for monomial curves in three-dimensional projective space. We also prove that there are infinitely many set-theoretic complete intersection monomial curves in the projective n?space for any suitably chosen n ? 1 integers. In particular, for any positive integers p, q, where gcd(p, q) = 1, the monomial curve defined by p, q, r is a set-theoretic complete intersection for every \({r \geq pq( q - 1)}\).     

Tactical Decompositions of Steiner Systems and Orbits of Projective Groups

Block's lemma states that the numbers m of point-classes and n of block-classes in a tactical decomposition of a 2-(v, k, ) design with b blocks satisfy m n m + b – v. We present a strengthening of the upper bound for the case of Steiner systems (2-designs with = 1), together with results concerning the structure of the block-classes in both extreme cases. Applying the results to the Steiner systems of points and lines of projective space PG(N, q), we obtain a complete classification of the groups inducing decompositions satisfying the upper bound; answering the analog of a question raised by Cameron and Liebler (P.J. Cameron and R.A. Liebler, Lin. Alg. Appl. 46 (1982), 91–102) (and still open).     

Polynomial relations among principal minors of a -matrix

The image of the principal minor map for n×n-matrices is shown to be closed. In the 19th century, Nanson and Muir studied the implicitization problem of finding all relations among principal minors when n=4. We complete their partial results by constructing explicit polynomials of degree 12 that scheme-theoretically define this affine variety and also its projective closure in . The latter is the main component in the singular locus of the 2×2×2×2-hyperdeterminant.     

The threshold of effective damping for semilinear wave equations

In this paper, we obtain the global existence of small data solutions to the Cauchy problem in space dimension n ≥ 1, for p > 1 + 2 ∕ n, where μ is sufficiently large. We obtain estimates for the solution and its energy with the same decay rate of the linear problem. In particular, for μ ≥ 2 + n, the damping term is effective with respect to the L¹ ? L² low‐frequency estimates for the solution and its energy. In this case, we may prove global existence in any space dimension n ≥ 3, by assuming smallness of the initial data in some weighted energy space. In space dimension n = 1,2, we only assume smallness of the data in some Sobolev spaces, and we introduce an approach based on fractional Sobolev embedding to improve the threshold for global existence to μ ≥ 5 ∕ 3 in space dimension n = 1 and to μ ≥ 3 in space dimension n = 2. Copyright © 2014 John Wiley & Sons, Ltd.