scholarly journals Base Point Freeness, Uniqueness of Decompositions and Double Points for Veronese and Segre Varieties

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2344
Author(s):  
Edoardo Ballico
Keyword(s):  
Projective Plane ◽  
Linear Systems ◽  
Tangent Vector ◽  
Base Point ◽  
Uniqueness Problem ◽  
Segre Variety ◽  
Double Points ◽  
Segre Varieties ◽  
Tangent Vectors

We prove a base point freeness result for linear systems of forms vanishing at general double points of the projective plane. For tensors we study the uniqueness problem for the representation of a tensor as a sum of terms corresponding to points and tangent vectors of the Segre variety associated with the format of the tensor. We give complete results for unions of one point and one tangent vector.

scholarly journals Tangent vectors to sets in the theory of geodesics

1987 ◽  
Vol 106 ◽  
pp. 29-47 ◽  
Author(s):  
Dumitru Motreanu
Keyword(s):  
Closed Subset ◽  
Tangent Vector ◽  
General Concept ◽  
Recent Book ◽  
Closed Sets ◽  
Banach Manifolds ◽  
Flow Invariance ◽  
Tangent Vectors

In the setting of Banach manifolds the notion of tangent vector to an arbitrary closed subset has been introduced in [11] by the author and N. H. Pavel, and it has been used in flow-invariance and optimization ([11], [12], [13]). For detailed informations on tangent vectors to closed sets (including historical comments) we refer to the recent book of N. H. Pavel [17].The aim of this paper is to apply this general concept of tangency in the study of geodesies. We are concerned with geodesies which have either the endpoints in given closed subsets or the same angle for a fixed closed subset. This approach allows one to extend important results due to K. Grove [4] and T. Kurogi ([6], [7]).

The sheaf of relative differentials of a fibred surface

1993 ◽  
Vol 114 (3) ◽  
pp. 461-470
Author(s):  
Fernando Serrano
Keyword(s):  
Linear Systems ◽  
Smooth Curve ◽  
Base Point ◽  
Projective Surface ◽  
Positive Part ◽  
Rational Singularities ◽  
Smooth Projective Surface ◽  
Genus 2

AbstractLet Φ: S → C denote a fibration from a smooth projective surface onto a smooth curve, with fibres of genus ≥2. The double dual of the sheaf of relative differentials has been studied by F. Serrano [14]. There, it was proved that dim grows asymptotically as the square of n in case Φ is not isotrivial (i.e. fibres vary in modulus), and the converse holds true in most cases, in a way that can be made precise. In the non-isotrivial case, the present paper provides further information about by analysing the linear systems for large n. If P denotes the positive part of in its Zariski decomposition, then it is shown that |rP| is eventually base-point free for some r > 0. Furthermore, Proj is a normal projective surface, fibred over C, birational to S, and with only rational singularities.

Loci Conversion and Corner Smoothing with PH Curves in CNC System

2009 ◽  
Vol 419-420 ◽  
pp. 161-164
Author(s):  
Qi Kui Wang ◽  
You Dong Chen ◽  
Wei Li ◽  
Tian Miao Wang ◽  
Hong Xing Wei
Keyword(s):  
Tangent Vector ◽  
Machining Process ◽  
Free Form ◽  
Surface Interpolation ◽  
Pythagorean Hodograph ◽  
Linear Elements ◽  
Smoothing Process ◽  
Ph Curve ◽  
Interpolation Functions ◽  
Tangent Vectors

Free-form surface interpolation functions give more advantages in machining than the traditional line and circle functions. A method is developed to convert lines and circles into Pythagorean-hodograph (PH) curves. In order to get smooth machining process the PH curve is used to replace the joints of the circular/linear elements by the connection situation. The slope of line is used to get the tangent vector in the line conversion. When converting a circle to a PH curve, points of the divided circle are introduced to compute the vectors. The methods of computing tangent vectors are proposed according to the slope of the line and the quadrant of the circle. The transformation errors from lines and circles to PH curves are computed. In the corner smoothing process the tangent vectors are computed by the connection between lines and circles. Replacement errors at the joints are computed for the use of PH curve. The results demonstrate the feasibility of the conversion from line and circle to PH curve. The PH curves at the joints of the circular and linear elements show continuous trajectory.

Some Results on Surfaces of General Type

2005 ◽  
Vol 57 (4) ◽  
pp. 724-749 ◽  
Author(s):  
B. P. Purnaprajna
Keyword(s):  
Linear Systems ◽  
General Type ◽  
Base Point ◽  
Linear Series ◽  
Ample Divisor ◽  
Surfaces Of General Type ◽  
Projective Normality

AbstractIn this article we prove some new results on projective normality, normal presentation and higher syzygies for surfaces of general type, not necessarily smooth, embedded by adjoint linear series. Some of the corollaries of more general results include: results on property Np associated to KS ⊗B⊗n where B is base-point free and ample divisor with B⊗K* nef, results for pluricanonical linear systems and results giving effective bounds for adjoint linear series associated to ample bundles. Examples in the last section show that the results are optimal.

Maps generating normals on a manifold

2019 ◽  
pp. 110-125
Author(s):  
K. Polyakova
Keyword(s):  
Tangent Space ◽  
Affine Connection ◽  
Tangent Vector ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
Osculating Space ◽  
Curvature And Torsion ◽  
Order Tangent ◽  
Tangent Vectors

The first- and second-order canonical forms are considered on a smooth m-manifold. The approach connected with coordinate expressions of basic and fiber forms, and first- and second-order tangent vectors on a manifold is implemented. It is shown that the basic first- and secondorder tangent vectors are first- and second-order Pfaffian (generalized) differentiation operators of functions set on a manifold. Normals on a manifold are considered. Differentials of the basic first-order (second-order) tangent vectors are linear maps from the first-order tangent space to the second-order (third-order) tangent space. Differentials of the basic first-order tangent vectors (basic vectors of the first-order normal) set a map which takes a first-order tangent vector into the second-order normal for the first-order tangent space of (into the third-order normal for second-order tangent space). The map given by differentials of the basic first-order tangent vectors takes all tangent vectors to the vectors of the second-order normal. Such splitting the osculating space in the direct sum of the first-order tangent space and second-order normal defines the simplest (canonical) affine connection which horizontal subspace is the indicated normal. This connection is flat and symmetric. The second-order normal is the horizontal subspace for this connection. Derivatives of some basic vectors in the direction of other basic vectors are equal to values of differentials of the first vectors on the second vectors, and the order of differentials is equal to the order of vectors along which differentiation is made. The maps set by differentials of basic vectors of r-order normal for (r – 1)-order tangent space take the basic first-order tangent vectors to the basic vectors of (r + 1)-order normal for r-order tangent space. Horizontal vectors for the simplest second-order connection are constructed. Second-order curvature and torsion tensors vanish in this connection. For horizontal vectors of the simplest second-order connection there is the decomposition by means of the basic vectors of the secondand third-order normals.

A theorem on Newton's polygon

1934 ◽  
Vol 30 (3) ◽  
pp. 287-296
Author(s):  
R. Frith
Keyword(s):  
Linear System ◽  
A Priori ◽  
Newton Polygon ◽  
Base Point ◽  
Hyperelliptic Curves ◽  
Convex Polygons ◽  
Double Points ◽  
Base Points ◽  
Double Base

Castelnuovo has shown that the maximum freedom of a linear system of curves of genus p is 3p + 5, and that a system with this maximum freedom consists of hyperelliptic curves which can be transformed into a system of curves of order n with an (n − 2)-ple base point and a certain number of double base points; the only exceptions being that when p = 3 the system may be transformable into that of all quartics, and when p = 1 the system may be transformable into that of all cubics. Further, since, in the transformed system, the characteristic series is a it is non-special and hence the redundancy of the base points is zero, therefore each of the double points of this system reduces the freedom by exactly three; and hence if we remove all the double points we get a system of curves of genus p′ and freedom r′ = 3p′ + 5 with only one base point. If we take this point for origin the system of curves can be represented by a single Newton polygon containing in its interior exactly p points, collinear since the curves are hyperelliptic (p ≠ 3), and containing on its boundary 2p + 6 points. From this we can deduce immediately a theorem concerning convex polygons drawn on squared paper; I shall now give an a priori proof of this theorem.

scholarly journals Smooth Rational Surfaces Of $d=11$ And $\pi=8$ In $\mathbb{P}^5$

2016 ◽  
Vol 119 (2) ◽  
pp. 169
Author(s):  
Abdul Moeed Mohammad
Keyword(s):  
Projective Plane ◽  
Linear Systems ◽  
Rational Surface ◽  
Sectional Genus ◽  
Short List ◽  
Rational Surfaces ◽  
Numerical Invariants

We construct a linearly normal smooth rational surface $S$ of degree $11$ and sectional genus $8$ in the projective five space. Surfaces satisfying these numerical invariants are special, in the sense that $h^1(\mathscr{O}_S(1))>0$. Our construction is done via linear systems and we describe the configuration of points blown up in the projective plane. Using the theory of adjunction mappings, we present a short list of linear systems which are the only possibilities for other families of surfaces with the prescribed numerical invariants.

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