161 research outputs found
Smooth affine surfaces with non-unique C*-actions
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 of C*. If a smooth affine surface V 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 our previous paper 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
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Embeddings of ℂ*-surfaces into weighted projective spaces
Let V be a normal affine surface which admits a C*- and a C+-action. In this note we show that in many cases V can be embedded as a principal Zariski open subset into a hypersurface of a weighted projective space. In particular, we recover a result of D. Daigle and P. Russell
Algebraic volume density property of affine algebraic manifolds
We introduce the notion of algebraic volume density property for affine
algebraic manifolds and prove some important basic facts about it, in
particular that it implies the volume density property. The main results of the
paper are producing two big classes of examples of Stein manifolds with volume
density property. One class consists of certain affine modifications of \C^n
equipped with a canonical volume form, the other is the class of all Linear
Algebraic Groups equipped with the left invariant volume form.Comment: 35 page
Flexible varieties and automorphism groups
Given an affine algebraic variety X of dimension at least 2, we let SAut (X)
denote the special automorphism group of X i.e., 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 of X then it is
infinitely transitive on this locus. In turn, the transitivity is equivalent to
the flexibility of X. The latter means that for every smooth point x of X the
tangent space at x is spanned by the velocity vectors of one-parameter
unipotent subgroups of Aut (X). We provide also different variations and
applications.Comment: Final version; to appear in Duke Math.
Flexibility properties in Complex Analysis and Affine Algebraic Geometry
In the last decades affine algebraic varieties and Stein manifolds with big
(infinite-dimensional) automorphism groups have been intensively studied.
Several notions expressing that the automorphisms group is big have been
proposed. All of them imply that the manifold in question is an
Oka-Forstneri\v{c} manifold. This important notion has also recently merged
from the intensive studies around the homotopy principle in Complex Analysis.
This homotopy principle, which goes back to the 1930's, has had an enormous
impact on the development of the area of Several Complex Variables and the
number of its applications is constantly growing. In this overview article we
present 3 classes of properties: 1. density property, 2. flexibility 3.
Oka-Forstneri\v{c}. For each class we give the relevant definitions, its most
significant features and explain the known implications between all these
properties. Many difficult mathematical problems could be solved by applying
the developed theory, we indicate some of the most spectacular ones.Comment: thanks added, minor correction
Varieties covered by affine spaces and their cones
It was shown in arXiv:2303.02036 that the affine cones over flag manifolds
and rational smooth projective surfaces are elliptic in the sense of Gromov.
The latter remains true after successive blowups of points on these varieties.
In the present note we extend this to smooth projective spherical varieties (in
particular, toric varieties) successively blown up along linear subvarieties.
The same also holds, more generally, for projective varieties covered by affine
spaces.Comment: 10 page
Affine modifications and affine hypersurfaces with a very transitive automorphism group
We study a kind of modification of an affine domain which produces another
affine domain. First appeared in passing in the basic paper of O. Zariski
(1942), it was further considered by E.D. Davis (1967). The first named author
applied its geometric counterpart to construct contractible smooth affine
varieties non-isomorphic to Euclidean spaces. Here we provide certain
conditions which guarantee preservation of the topology under a modification.
As an application, we show that the group of biregular automorphisms of the
affine hypersurface given by the equation
where acts transitively on the
smooth part reg of for any We present examples of such
hypersurfaces diffeomorphic to Euclidean spaces.Comment: 39 Pages, LaTeX; a revised version with minor changes and correction
Unipotent group actions on affine varieties
Algebraic actions of unipotent groups actions on affine varieties
( an algebraically closed field of characteristic 0) for which the algebraic
quotient has small dimension are considered In case is factorial,
and is one-dimensional, it is shown that
=, and if some point in has trivial isotropy, then is
equivariantly isomorphic to The main results are given
distinct geometric and algebraic proofs. Links to the Abhyankar-Sathaye
conjecture and a new equivalent formulation of the Sathaye conjecture are made.Comment: 10 pages. This submission comes out of an older submission ("A
commuting derivations theorem on UFDs") and contains part of i
Semi-Automatic Cell Correspondence Analysis Using Iterative Point Cloud Registration
In the field of biophysics, deformation of in-vitro model tissues is an experimental technique to explore the response of tissue to a mechanical stimulus. However, automated registration before and after deformation is an ongoing obstacle for measuring the tissue response on the cellular level. Here, we propose to use an iterative point cloud registration (IPCR) method, for this problem. We apply the registration method on point clouds representing the cellular centers of mass, which are evaluated with aWatershed based segmentation of phase-contrast images of living tissue, acquired before and after deformation. Preliminary evaluation of this method on three data sets shows high accuracy, with 82% - 92% correctly registered cells, which outperforms coherent point drift (CPD). Hence, we propose the application of the IPCR method on the problem of cell correspondence analysis
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