On the canonical systems of certain algebraic surfaces

1937 ◽  
Vol 33 (3) ◽  
pp. 311-314
Author(s):  
D. Pedoe
Keyword(s):  
Algebraic Surface ◽  
Base Point ◽  
Canonical System ◽  
Algebraic Surfaces ◽  
Simple Point ◽  
Canonical Systems ◽  
Base Points ◽  
Complete Linear System ◽  
Pass Through ◽  
Simple Points

A complete linear system of curves on an algebraic surface may have assigned base points. The canonical system, from its definition, has no assigned base points at simple points of the surface. But we may construct surfaces on which, all the same, the canonical system has “accidental base points” at simple points of the surface. The classical example, due to Castelnuovo, is a quintic surface with two tacnodes. On this surface the canonical system is cut out by the planes passing through the two tacnodes. These planes also pass through the simple point in which the join of the two tacnodes meets the surface again. This point is the accidental base point of the canonical system on the quintic surface.

The fixed part of the canonical system on an algebraic surface

1938 ◽  
Vol 34 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Patrick Du Val
Keyword(s):  
Algebraic Surface ◽  
Canonical System ◽  
Rational Curve ◽  
Variable Part ◽  
Exceptional Curve ◽  
Base Points ◽  
Fixed Part ◽  
The Given ◽  
Simple Points

It is familiar that if on an algebraic surface there is an exceptional curve, that is an irreducible rational curve of virtual grade − 1 when no points of it are assigned as base points, and if there is on the surface a canonical system containing some actual curves, so that pg ≥ 1, then the exceptional curve is a fixed constituent of every curve of the canonical system, generally a simple constituent, and in that case has no intersections with the residual constituent. More generally, if there is on the surface a reducible exceptional curve, i.e. a set of curves which can be transformed into the neighbourhoods of a family of simple points (some of which are in the neighbourhoods of others) on a surface birationally equivalent to the given one, then the canonical system has as a fixed constituent of all its curves at least that combination of the curves which corresponds to the sum of the total neighbourhoods of the points, and generally just this combination, in which case this fixed part has no intersection with the residual variable part.

scholarly journals Birational transformations with isolated fundamental points

1938 ◽  
Vol 5 (3) ◽  
pp. 117-124 ◽  
Author(s):  
J. A. Todd
Keyword(s):  
Algebraic Surface ◽  
Canonical System ◽  
The Other ◽  
Canonical Systems ◽  
Birational Transformations ◽  
Birational Transformation ◽  
Image Position ◽  
Simple Points

It is well known that the canonical system of curves on an algebraic surface is only relatively invariant under birational transformations of the surface. That is, if we have a birational transformation T between two surfaces F and F′, and if K and K′ denote curves of the unreduced canonical systems on F and F′, thenwhere E and E′ denote the sets of curves, on F and F′ respectively which are transformed into the neighbourhoods of simple points on the other surface.

scholarly journals Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180761 ◽  
Author(s):  
Matteo Petrera ◽  
Jennifer Smirin ◽  
Yuri B. Suris
Keyword(s):  
Vector Field ◽  
Base Point ◽  
Hamiltonian Vector ◽  
Geometric Condition ◽  
Hamiltonian Vector Field ◽  
Cubic Curves ◽  
Birational Map ◽  
Base Points ◽  
Finite Base ◽  
Pass Through

Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic canonical Hamiltonian vector field, this map is known to be integrable and to preserve a pencil of cubic curves. Generically, the nine base points of this pencil include three points at infinity (corresponding to the asymptotic directions of cubic curves) and six finite points lying on a conic. We show that the Kahan discretization map can be represented in six different ways as a composition of two Manin involutions, corresponding to an infinite base point and to a finite base point. As a consequence, the finite base points can be ordered so that the resulting hexagon has three pairs of parallel sides which pass through the three base points at infinity. Moreover, this geometric condition on the base points turns out to be characteristic: if it is satisfied, then the cubic curves of the corresponding pencil are invariant under the Kahan discretization of a planar quadratic canonical Hamiltonian vector field.

scholarly journals Complete algebraic vector fields on affine surfaces

2020 ◽  
Vol 31 (03) ◽  
pp. 2050018
Author(s):  
Shulim Kaliman ◽  
Frank Kutzschebauch ◽  
Matthias Leuenberger
Keyword(s):  
Algebraic Surface ◽  
Vector Fields ◽  
Algebraic Surfaces ◽  
Algebraic Vector ◽  
Dual Graphs ◽  
Holomorphic Automorphisms

Let [Formula: see text] be the subgroup of the group [Formula: see text] of holomorphic automorphisms of a normal affine algebraic surface [Formula: see text] generated by elements of flows associated with complete algebraic vector fields. Our main result is a classification of all normal affine algebraic surfaces [Formula: see text] quasi-homogeneous under [Formula: see text] in terms of the dual graphs of the boundaries [Formula: see text] of their SNC-completions [Formula: see text].

scholarly journals ALGEBRAIC SURFACES WITH INFINITELY MANY TWISTOR LINES

2019 ◽  
Vol 101 (1) ◽  
pp. 61-70
Author(s):  
A. ALTAVILLA ◽  
E. BALLICO
Keyword(s):  
Algebraic Surface ◽  
Existence Results ◽  
Algebraic Surfaces ◽  
Odd Degree

We prove that a reduced and irreducible algebraic surface in $\mathbb{CP}^{3}$ containing infinitely many twistor lines cannot have odd degree. Then, exploiting the theory of quaternionic slice regularity and the normalisation map of a surface, we give constructive existence results for even degrees.

scholarly journals Towards the weighted Bounded Negativity Conjecture for blow-ups of algebraic surfaces

2019 ◽  
Vol 163 (3-4) ◽  
pp. 361-373
Author(s):  
Roberto Laface ◽  
Piotr Pokora
Keyword(s):  
Line Bundle ◽  
Characteristic Zero ◽  
Algebraic Surface ◽  
Intersection Number ◽  
Algebraic Surfaces ◽  
The Self ◽  
Projective Surface ◽  
Intersection Numbers ◽  
Weighted Version ◽  
Smooth Projective Surface

AbstractIn the present paper we focus on a weighted version of the Bounded Negativity Conjecture, which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a function depending on the intesection of curve with an arbitrary big and nef line bundle that is positive on the curve. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.

scholarly journals Contextuality in canonical systems of random variables

2017 ◽  
Vol 375 (2106) ◽  
pp. 20160389 ◽  
Author(s):  
Ehtibar N. Dzhafarov ◽  
Víctor H. Cervantes ◽  
Janne V. Kujala
Keyword(s):  
Joint Distribution ◽  
Random Variables ◽  
Canonical System ◽  
Canonical Representation ◽  
Single Pair ◽  
Canonical Systems ◽  
Joint Distributions ◽  
Content Sharing ◽  
Maximal Coupling ◽  
Quantum Revolution

Random variables representing measurements, broadly understood to include any responses to any inputs, form a system in which each of them is uniquely identified by its content (that which it measures) and its context (the conditions under which it is recorded). Two random variables are jointly distributed if and only if they share a context. In a canonical representation of a system, all random variables are binary, and every content-sharing pair of random variables has a unique maximal coupling (the joint distribution imposed on them so that they coincide with maximal possible probability). The system is contextual if these maximal couplings are incompatible with the joint distributions of the context-sharing random variables. We propose to represent any system of measurements in a canonical form and to consider the system contextual if and only if its canonical representation is contextual. As an illustration, we establish a criterion for contextuality of the canonical system consisting of all dichotomizations of a single pair of content-sharing categorical random variables. This article is part of the themed issue ‘Second quantum revolution: foundational questions’.

scholarly journals Base Point Split Algorithm for Generating Polygon Skeleton Lines on the Example of Lakes

2020 ◽  
Vol 9 (11) ◽  
pp. 680
Author(s):  
Elżbieta Lewandowicz ◽  
Paweł Flisek
Keyword(s):  
Base Point ◽  
Vector Data ◽  
Triangulated Irregular Network ◽  
Base Points ◽  
Complex Polygon ◽  
Selection Of

This article presents the Base Point Split (BPSplit) algorithm to generate a complex polygon skeleton based on sets of vector data describing lakes and rivers. A key feature of the BPSplit algorithm is that it is dependent on base points representing the source or mouth of a river or a stream. The input values of base points determine the shape of the resulting skeleton of complex polygons. Various skeletons can be generated with the use of different base points. Base points are applied to divide complex polygon boundaries into segments. Segmentation supports the selection of triangulated irregular network (TIN) edges inside complex polygons. The midpoints of the selected TIN edges constitute a basis for generating a skeleton. The algorithm handles complex polygons with numerous holes, and it accounts for all holes. This article proposes a method for modifying a complex skeleton with numerous holes. In the discussed approach, skeleton edges that do not meet the preset criteria (e.g., that the skeleton is to be located between holes in the center of the polygon) are automatically removed. An algorithm for smoothing zigzag lines was proposed.

Algebraic Surfaces and the Miyaoka-Yau Inequality

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Differential Geometry ◽  
Algebraic Surface ◽  
Smooth Surface ◽  
General Type ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Smooth Complex ◽  
Dimension 2

This chapter discusses complex algebraic surfaces, with particular emphasis on the Miyaoka-Yau inequality and the rough classification of surfaces. Every complex algebraic surface is birationally equivalent to a smooth surface containing no exceptional curves. The latter is known as a minimal surface. Two related birational invariants, the plurigenus and the Kodaira dimension, play an important role in distinguishing between complex surfaces. The chapter first provides an overview of the rough classification of (smooth complex connected compact algebraic) surfaces before presenting two approaches that, in dimension 2, give the Miyaoka-Yau inequality. The first, due to Miyaoka, uses algebraic geometry, whereas the second, due to Aubin and Yau, uses analysis and differential geometry. The chapter also explains why equality in the Miyaoka-Yau inequality characterizes surfaces of general type that are free quotients of the complex 2-ball.

Dependence of the Weyl coefficient on singular interface conditions

2009 ◽  
Vol 52 (2) ◽  
pp. 445-487 ◽  
Author(s):  
Matthias Langer ◽  
Harald Woracek
Keyword(s):  
Hamiltonian Function ◽  
Positive Definite Function ◽  
Canonical System ◽  
Positive Definite ◽  
Definite Function ◽  
Interface Conditions ◽  
Canonical Systems ◽  
Bessel Equation ◽  
Continuation Problem ◽  
Sturm Liouville

AbstractWe investigate the influence of interface conditions at a singularity of an indefinite canonical system on its Weyl coefficient. An explicit formula which parametrizes all possible Weyl coefficients of indefinite canonical systems with fixed Hamiltonian function is derived. This result is illustrated with two examples: the Bessel equation, which has a singular end point, and a Sturm–Liouville equation whose potential has an inner singularity, which arises from a continuation problem for a positive definite function.

Sign in / Sign up

Export Citation Format

Share Document