Algebraic surfaces whose canonical image has a pencil of rational curves of degree two

1992 ◽  
Vol 209 (1) ◽  
pp. 67-74
Author(s):  
Sun Xia Tao
Keyword(s):  
Rational Curves ◽  
Algebraic Surfaces ◽  
Canonical Image

The central sphere of an ALE space

2020 ◽  
Author(s):  
Nigel Hitchin
Keyword(s):  
Fixed Point ◽  
Algebraic Geometry ◽  
Rational Curves ◽  
Algebraic Surfaces ◽  
Circle Action ◽  
Fixed Point Set ◽  
Point Set ◽  
Ale Space ◽  
Central Sphere

Abstract We consider the induced metric on the spherical fixed point set of a circle action on an ALE space and describe it by using the algebraic geometry of rational curves on algebraic surfaces, in particular the lines on a cubic.

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)

scholarly journals Non Kählerian surfaces with a cycle of rational curves

2021 ◽  
Vol 8 (1) ◽  
pp. 208-222
Author(s):  
Georges Dloussky
Keyword(s):  
Deformation Theory ◽  
Complex Surface ◽  
Rational Curves ◽  
Intersection Form ◽  
Connected Component ◽  
Hirzebruch Surface ◽  
Compact Complex Surface ◽  
A Chain

Abstract Let S be a compact complex surface in class VII0 + containing a cycle of rational curves C = ∑Dj . Let D = C + A be the maximal connected divisor containing C. If there is another connected component of curves C ′ then C ′ is a cycle of rational curves, A = 0 and S is a Inoue-Hirzebruch surface. If there is only one connected component D then each connected component Ai of A is a chain of rational curves which intersects a curve Dj of the cycle and for each curve Dj of the cycle there at most one chain which meets Dj . In other words, we do not prove the existence of curves other those of the cycle C, but if some other curves exist the maximal divisor looks like the maximal divisor of a Kato surface with perhaps missing curves. The proof of this topological result is an application of Donaldson theorem on trivialization of the intersection form and of deformation theory. We apply this result to show that a twisted logarithmic 1-form has a trivial vanishing divisor.

scholarly journals Two algebraic deformations of a K3 surface

1981 ◽  
Vol 82 ◽  
pp. 1-26
Author(s):  
Daniel Comenetz
Keyword(s):  
Hyperelliptic Curve ◽  
Double Cover ◽  
Rational Curves ◽  
K3 Surface ◽  
Quartic Surface ◽  
Quadric Cone ◽  
Birational Model ◽  
Affine Line ◽  
Characteristic 2 ◽  
General Member

Let X be a nonsingular algebraic K3 surface carrying a nonsingular hyperelliptic curve of genus 3 and no rational curves. Our purpose is to study two algebraic deformations of X, viz. one specialization and one generalization. We assume the characteristic ≠ 2. The generalization of X is a nonsingular quartic surface Q in P3 : we wish to show in § 1 that there is an irreducible algebraic family of surfaces over the affine line, in which X is a member and in which Q is a general member. The specialization of X is a surface Y having a birational model which is a ramified double cover of a quadric cone in P3.

Comments on ‘Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces’

2014 ◽  
Vol 266 ◽  
pp. 80-82 ◽  
Author(s):  
Antonio Algaba ◽  
Fernando Fernández-Sánchez ◽  
Manuel Merino ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Algebraic Surfaces ◽  
Global Dynamics ◽  
Invariant Algebraic Surfaces

scholarly journals Geometrically rational real conic bundles and very transitive actions

2010 ◽  
Vol 147 (1) ◽  
pp. 161-187 ◽  
Author(s):  
Jérémy Blanc ◽  
Frédéric Mangolte
Keyword(s):  
Automorphism Groups ◽  
Algebraic Surfaces ◽  
Algebraic Models ◽  
Transitive Automorphism ◽  
Real Locus ◽  
Real Algebraic Surfaces ◽  
Conic Bundles ◽  
Transitive Actions ◽  
Insight Into

AbstractIn this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.

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