Forms of the affine line

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.

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On the Twisted Cohomology of Affine Line Arrangements

Springer INdAM Series - Configuration Spaces ◽

10.1007/978-3-319-31580-5_11 ◽

2016 ◽

pp. 275-290 ◽

Cited By ~ 1

Author(s):

Mario Salvetti ◽

Matteo Serventi

Keyword(s):

Line Arrangements ◽

Twisted Cohomology ◽

Affine Line

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Algebraic vector bundles over the product of an affine curve and the affine line

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1959-0164972-1 ◽

1959 ◽

Vol 10 (5) ◽

pp. 670-670 ◽

Cited By ~ 7

Author(s):

C. S. Seshadri

Keyword(s):

Vector Bundles ◽

Algebraic Vector ◽

Affine Line

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Equidistant Loci and the Minkowskian Geometries

Canadian Journal of Mathematics ◽

10.4153/cjm-1972-026-7 ◽

1972 ◽

Vol 24 (2) ◽

pp. 312-327 ◽

Cited By ~ 5

Author(s):

B. B. Phadke

Keyword(s):

Metric Spaces ◽

Open Convex ◽

Minkowskian Space ◽

Geometric Conditions ◽

Finite Dimensional ◽

Projective Metric ◽

Affine Line ◽

Finite Dimensional Banach Space ◽

Homogeneous Norm ◽

Symmetric Distance

The spaced of this paper is a metrization, with a not necessarily symmetric distance xy, of an open convex set D in the n-dimensional affine space An such that xy + yz = xz if and only if x, y, z lie on an affine line with y between x and z and such that all the balls px ≦ p are compact. These spaces are called straight desarguesian G-spaces or sometimes open projective metric spaces. The hyperbolic geometry is an example; a large variety of other examples is studied by contributors to Hilbert's problem IV. When D = An and all the affine translations are isometries for the metric xy, the space is called a Minkowskian space or sometimes a finite dimensional Banach space, the (not necessarily symmetric) distance of a Minkowskian space being a (positive homogeneous) norm. In this paper geometric conditions in terms of equidistant loci are given for the space R to be a Minkowskian space.

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Uniqueness of embeddings of the affine line into algebraic groups

Journal of Algebraic Geometry ◽

10.1090/jag/725 ◽

2019 ◽

Vol 28 (4) ◽

pp. 649-698 ◽

Cited By ~ 1

Author(s):

Peter Feller ◽

Immanuel van Santen

Keyword(s):

Algebraic Groups ◽

Affine Line

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Symmetric homogeneous convex domains

Nagoya Mathematical Journal ◽

10.1017/s0027763000020699 ◽

1984 ◽

Vol 93 ◽

pp. 1-17

Author(s):

Tadashi Tsuji

Keyword(s):

Real Number ◽

Convex Domain ◽

Convex Cone ◽

Affine Transformations ◽

Convex Domains ◽

Affine Line ◽

Homogeneous Convex

Let D be a convex domain in the n-dimensional real number space Rn, not containing any affine line and A(D) the group of all affine transformations of Rn leaving D invariant. If the group A(D) acts transitively on D, then the domain D is said to be homogeneous. From a homogeneous convex domain D in Rn, a homogeneous convex cone V = V(D) in Rn+1 = Rn × R is constructed as follows (cf. Vinberg [11]):which is called the cone fitted on the convex domain D.

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