摘要 :
We introduce a new family of affine metrics on a locally strictly convex surface M in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if M is immersed in ...
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We introduce a new family of affine metrics on a locally strictly convex surface M in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if M is immersed in a locally strictly convex hyperquadric, then the symmetric and the antisymmetric planes coincide and contain the affine normal of the hyperquadric. In particular, any surface immersed in a locally strictly convex hyperquadric is affine semiumbilical with respect to the symmetric or antisymmetric equiaffine planes.
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摘要 :
We consider locally strictly convex surfaces M in affine 4-space. By using the metric of the transversal vector field on M we introduce a new affine normal plane and the familly of affine distance functions on M. We show that the ...
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We consider locally strictly convex surfaces M in affine 4-space. By using the metric of the transversal vector field on M we introduce a new affine normal plane and the familly of affine distance functions on M. We show that the singularities of the family of affine distance functions appear at points on the affine normal plane and the affine focal points correspond to degenerate singularities of this family. Moreover we show that if M is immersed in a locally strictly convex hypersurface, then the affine normal plane contains the affine normal vector to the hypersurface and conclude that any surface immersed in a locally strictly convex hypersphere is affine semiumbilical.
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A Gizatullin surface is a normal affine surface V over C, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of C*-actions and A(1)-fibra...
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A Gizatullin surface is a normal affine surface V over C, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of C*-actions and A(1)-fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with C+-actions on V considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one A(1)-fibration V -> S up to an isomorphism of the base S. Moreover, an effective C*-action on them, if it does exist, is unique up to conjugation and inversion t -> t(-1) of C*. Obviously, uniqueness of C*-actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov{Gizatullin surfaces, where there are in general several conjugacy classes of C*-actions and A(1)-fibrations, see, e.g., [FKZ(1)]. In the present paper we obtain a criterion as to when A(1)-fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base S congruent to A(1). We exhibit as well large subclasses of Gizatullin C*-surfaces for which a C*-action is essentially unique and for which there are at most two conjugacy classes of A(1)-fibrations over A(1).
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For non-degenerate surfaces in R-4, a distinguished transversal bundle called affine normal plane bundle was proposed in [K. Nomizu and L. Vrancken, A new equiaffine theory for surfaces in R-4, Internat. J. Math. 4(1) (1993) 127-1...
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For non-degenerate surfaces in R-4, a distinguished transversal bundle called affine normal plane bundle was proposed in [K. Nomizu and L. Vrancken, A new equiaffine theory for surfaces in R-4, Internat. J. Math. 4(1) (1993) 127-165]. Lagrangian surfaces have remarkable properties with respect to this normal bundle, like for example, the normal bundle being Lagrangian. In this paper, we characterize those surfaces which are Lagrangian with respect to some parallel symplectic form in R-4.
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We give the classification of normal del Pezzo surfaces of rank one with at most log canonical singularities containing the affine plane defined over an algebraically nonclosed field of characteristic zero. As an application, we h...
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We give the classification of normal del Pezzo surfaces of rank one with at most log canonical singularities containing the affine plane defined over an algebraically nonclosed field of characteristic zero. As an application, we have a criterion for del Pezzo fibrations with canonical singularities whose generic fibers are not smooth to contain vertical A2-cylinders.
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