Hubert Flenner

Flexible varieties and automorphism groups

Given an irreducible affine algebraic variety X of dimension n≥2 , we let SAut(X) denote the special automorphism group of X , that is, the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus X reg , then it is infinitely transitive on X reg . In turn, the transitivity is equivalent to the flexibility of X . The latter means that for every smooth point x∈X reg the tangent space T x X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X) . We also provide various modifications and applications.

A Semiregularity Map for Modules and Applications to Deformations

We construct a general semiregularity map for algebraic cycles as asked for by S.Bloch in 1972. The existence of such a semiregularity map has well known consequences for the structure of the Hilbert scheme and for the variational Hodge conjecture. Aside from generalizing and extending considerably previously known results in this direction, we give new applications to deformations of modules that encompass, for example, results of Artamkin and Mukai. The formation of the semiregularity map here involves powers of the cotangent complex, Atiyah classes, and trace maps, and is defined not only for subspaces of manifolds but for perfect complexes on arbitrary complex spaces. It generalizes in particular Illusies treatment of the Chern character to the analytic context and specializes to Blochs earlier description of the semiregularity map for locally complete intersections as well as to the infinitesimal Abel–Jacobi map for submanifolds.

Normal affine surfaces with C^*-actions

A classification of normal affine surfaces admitting a

On a class of rational cuspidal plane curves

\bf C^*

On the Danilov-Gizatullin Isomorphism Theorem

-action was given in the work of Bia{\l}ynicki-Birula, Fieseler and L. Kaup, Orlik and Wagreich, Rynes and others. We provide a simple alternative description of such surfaces in terms of their graded rings as well as by defining equations. This is based on a generalization of the Dolgachev-Pinkham-Demazure construction in the case of a hyperbolic grading. As an apllication we determine the structure of singularities, of the orbits and the divisor class groups for such surfaces.

On multiplicities of local rings

We obtain new examples and the complete list of the rational cuspidal plane curvesC with at least three cusps, one of which has multiplicitydegC-2. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational cuspidal plane curves.

Smooth affine surfaces with non-unique ℂ*-actions

A Danilov-Gizatullin surface is a normal affine surface V, which is a complement to an ample section S in a Hirzebruch surface of index d. By a surprising result due to Danilov and Gizatullin, V depends only on the self-intersection number of S and neither on d nor on S. In this note we provide a new and simple proof of this Isomorphism Theorem.

Infinite transitivity on affine varieties

The aim of this note is to study multiplicities of local rings under (iterated) hyperplane sections, and Bezout-type theorems. An important application is a local version of a converse to Bezouts theorem. Even in the projective case this improves known results for arithmetically Cohen-Macaulay schemes. For the proofs our key result is a description of the behaviour of the Hilbert-Samuel polynomial in a short exact sequence. Using a generalization of this to the case of complexes we are also able to give an extended version of a theorem of Auslander-Buchsbaum and Serre expressing multiplicities by using Koszul-homology.

A length formula for the multiplicity of distinguished components of intersections

In this paper we complete the classification of effective C � -actions on smooth affine surfaces up to conjugation in the full automorphism group and up to inversion � 7→ � 1 of C � . If a smooth affine surfaceV admits more than one C � -action then it is known to be Gizatullin i.e., it can be completed by a linear chain of smooth rational curves. In (FKZ3) we gave a sufficient condition, in terms of the Dolgachev- Pinkham-Demazure (or DPD) presentation, for the uniqueness of a C � -action on a Gizatullin surface. In the present paper we show that this condition is also necessary, at least in the smooth case. In fact, if the uniqueness fails for a smooth Gizatullin surface V which is neither toric nor Danilov-Gizatullin, then V admits a continuous family of pairwise non-conjugated C � -actions depending on one or two parameters. We give an explicit description of all such surfaces and their C � -actions in terms of DPD presentations. We also show that for every k > 0 one can find a Danilov- Gizatullin surface V (n) of index n = n(k) with a family of pairwise non-conjugate C+-actions depending on k parameters.

On the Fibers of Analytic Mappings

In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.