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 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 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.

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.

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.

scholarly journals Algebraic surfaces determine analyticity of functions

2021 ◽  
Author(s):  
Jacek Bochnak ◽  
Wojciech Kucharz
Keyword(s):  
Algebraic Surface ◽  
Algebraic Curves ◽  
Algebraic Surfaces ◽  
Connected Component ◽  
Algebraic Set ◽  
Real Algebraic Set

AbstractLet $$f :X \rightarrow \mathbb {R}$$ f : X → R be a function defined on a nonsingular real algebraic set X of dimension at least 3. We prove that f is an analytic (resp. a Nash) function whenever the restriction $$f|_{S}$$ f | S is an analytic (resp. a Nash) function for every nonsingular algebraic surface $$S \subset X$$ S ⊂ X whose each connected component is homeomorphic to the unit 2-sphere. Furthermore, the surfaces S can be replaced by compact nonsingular algebraic curves in X, provided that dim$$X \ge 2$$ X ≥ 2 and f is of class $$\mathcal {C}^{\infty }$$ C ∞ .

Adaptive Slicing of Parametrizable Algebraic Surfaces for Layered Manufacturing

1995 ◽  
Author(s):  
Prashant Kulkarni ◽  
Debasish Dutta
Keyword(s):  
Variable Thickness ◽  
Algebraic Surface ◽  
Layered Manufacturing ◽  
Algebraic Surfaces ◽  
Geometric Accuracy ◽  
Adaptive Slicing ◽  
Staircase Effect

Abstract As the various applications of Layered Manufacturing (LM) expand from just prototyping, the geometric accuracy issues become more prominent. Variable thickness, or adaptive, slicing aides in reducing a major source of geometric inaccuracy, the staircase effect. This paper develops a procedure for the adaptive slicing of a parametrizable algebraic surface to be manufactured by an LM process. An implemented example of the procedure is presented.

Topological characterization of C2 via open algebraic surface theory

2020 ◽  
Vol 31 (07) ◽  
pp. 2050055
Author(s):  
R. V. Gurjar ◽  
Sudarshan Gurjar
Keyword(s):  
Algebraic Surface ◽  
Algebraic Surfaces ◽  
Topological Characterization ◽  
Surface Theory ◽  
Affine Line

We will give a new proof of Ramanujam’s topological characterization of the affine [Formula: see text]-space [Formula: see text] using the Abhyankar–Moh–Suzuki theorem on embeddings of the affine line in [Formula: see text] and some ideas from the theory of open algebraic surfaces.

Algebraic surfaces with nef and big anti-canonical divisor

1995 ◽  
Vol 117 (1) ◽  
pp. 161-163 ◽  
Author(s):  
D.-Q. Zhang
Keyword(s):  
Algebraic Surface ◽  
Algebraic Surfaces ◽  
Pezzo Surface ◽  
Canonical Divisor ◽  
Del Pezzo Surface ◽  
Projective Algebraic Surface ◽  
Quotient Singularities ◽  
Del Pezzo

Let S be a normal projective algebraic surface over C with at worst quotient singularities. S is a quasi-log del Pezzo surface if the anti-canonical divisor — Ks is nef (= numerically effective) and big, i.e. — Ks. C ≥ 0 for all curves C on S and (−Ks)2 > 0. Further, if — Ks is ample we say S is a log del Pezzo surface.

scholarly journals Q_l-cohomology projective planes and singular Enriques surfaces in characteristic two

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Matthias Schütt
Keyword(s):  
Projective Planes ◽  
Rational Curves ◽  
Existence Results ◽  
Algebraic Surfaces ◽  
Prime Field ◽  
One Dimensional ◽  
Enriques Surfaces ◽  
Characteristic Two ◽  
The Given ◽  
Root Types

We classify singular Enriques surfaces in characteristic two supporting a rank nine configuration of smooth rational curves. They come in one-dimensional families defined over the prime field, paralleling the situation in other characteristics, but featuring novel aspects. Contracting the given rational curves, one can derive algebraic surfaces with isolated ADE-singularities and trivial canonical bundle whose Q_l-cohomology equals that of a projective plane. Similar existence results are developed for classical Enriques surfaces. We also work out an application to integral models of Enriques surfaces (and K3 surfaces). Comment: 24 pages; v3: journal version, correcting 20 root types to 19 and ruling out the remaining type 4A_2+A_1 (in new section 11)

Invariant groups associated with an algebraic surface

1940 ◽  
Vol 36 (4) ◽  
pp. 414-423 ◽  
Author(s):  
D. B. Scott
Keyword(s):  
Fundamental Group ◽  
Algebraic Curve ◽  
Algebraic Surface ◽  
Algebraic Surfaces ◽  
Topological Invariants ◽  
Period Matrix ◽  
Birational Transformations

Alexander (1, 2) has introduced certain topological invariants of a manifold which arise from the intersections of cycles of non-complementary dimensions, and he points out that they are not derivable from the Betti and torsion numbers, nor from the fundamental group. In the present paper we consider some topological invariants of this type on an algebraic surface, and, although we cannot define them completely, we show that they are intimately connected with the multiplications of the period matrix of the simple integrals of the first kind. We are then led to a concept which we call the “intersection group” of the surface, which is, by its definition, topologically invariant, and we show that it is also invariant under birational transformations. The proofs are based on Lefschetz's theory of cycles for an algebraic surface (4) and some simple properties of the period matrix of an algebraic curve. The results obtained here have a number of applications to the theory of ∞3 correspondences between algebraic surfaces, as we propose to show in a later paper.

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