arXiv subjects
Publications and source records attributed to Mikhail Zaidenberg.
We give a corrected version of Corollary 3.33 in: H. Flenner, S. Kaliman, and M. Zaidenberg, Birational transformations of weighted graphs. Affine algebraic geometry. Osaka Univ. Press, 2007, 107-147.
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.
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.
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.
This survey addresses pluri-periodic harmonic functions on lattices with values in a positive characteristic field. We mention, as a motivation, the game "Lights Out" following the work of Sutner, Goldwasser-Klostermeyer-Ware, Barua-Ramakrishnan-Sarkar, Hunzikel-Machiavello-Park e.a.; see also 2 previous author's preprints for a more detailed account. Our approach explores harmonic analysis and algebraic geometry over a positive characteristic field. The Fourier transform allows us to interpret pluri-periods of harmonic functions on lattices as torsion multi-orders of points on the corresponding affine algebraic variety.
We present two methods of constructing low degree Kobayashi hyperbolic hypersurfaces in the projective space: the projection method and the deformation method. The talk is based on joint works of the speaker with B. Shiffman and C. Ciliberto.
We modify the deformation method explored previously in a joint work of B. Shiffman and the author, in order to construct further examples of Kobayashi hyperbolic surfaces in the projective 3-space of any even degree starting with degree 8.
A Gizatullin surface is a normal affine surface $V$ over $\bf C$, 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 $\bf C^*$-actions and $\bf A^1$-fibrations on such a surface $V$ up to automorphisms. The latter fibrations are in one to one correspondence with $\bf C_+$-actions on $V$ considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one $\bf A^1$-fibration $V\to S$ up to an isomorphism of the base $S$. Moreover an effective $\bf C^{*}$-action on them, if it does exist, is unique up to conjugation and inversion $t\mapsto t^{-1}$ of $\bf C^*$. Obviously uniqueness of $\bf C^*$-actions fails for affine toric surfaces; however we show in this case that there are at most two conjugacy classes of $\bf A^1$-fibrations. There is a further interesting family of non-toric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $\bf C^*$-actions and $\bf A^1$-fibrations. In the present paper we obtain a criterion as to when $\bf A^1$-fibrations of Gizatullin surfaces are conjugate up to an automorphism of $V$ and the base $S$. We exhibit as well a large subclasses of Gizatullin $\bf C^{*}$-surfaces for which a $\bf C^*$-action is essentially unique and for which there are at most two conjugacy classes of $\bf A^1$-fibrations over $\bf A^1$.
These notes are inspired by the theory of cellular automata. A linear cellular automaton on a lattice of finite rank or on a toric grid is a discrete dinamical system generated by a convolution operator with kernel concentrated in the nearest neighborhood of the origin. In the present paper we deal with general convolution operators. We propose an approach via harmonic analysis which works over a field of positive characteristic. It occurs that a standard spectral problem for a convolution operator is equivalent to counting points on an associate algebraic hypersurface in a torus according to the torsion orders of their coordinates.
This note is a supplement to the papers: R. V. Gurjar, K. Masuda, M. Miyanishi and P. Russell, Affine lines on affine surfaces and the Makar-Limanov invariant, preprint, 2005, 42p. and T. Kishimoto and H. Kojima, Affine lines on {\bf Q}-homology planes with logarithmic Kodaira dimension $-\infty$, Transform. Groups 11 (2006), no. 4 (to appear). We show the role of group actions in classification of affine lines on {\bf Q}-homology planes.
A function on a (generally infinite) graph $\G$ with values in a field $K$ of characteristic 2 will be called {\it harmonic} if its value at every vertex of $\G$ is the sum of its values over all adjacent vertices. We consider binary pluri-periodic harmonic functions $f: \Z^s\to\F_2=\GF(2)$ on integer lattices, and address the problem of describing the set of possible multi-periods $\bar n=(n_1,...,n_s)\in\N^s$ of such functions. Actually this problem arises in the theory of cellular automata. It occurs to be equivalent to determining, for a certain affine algebraic hypersurface $V_s$ in $\A_{\bar\F_2}^s$, the torsion multi-orders of the points on $V_s$ in the multiplicative group $(\bar\F_2^\times)^s$. In particular $V_2$ is an elliptic cubic curve. In this special case we provide a more thorough treatment. A major part of the paper is devoted to a survey of the subject.
We introduce the notion of a standard weighted graph and show that every weighted graph has an essentially unique standard model. Moreover we classify birational transformations between such models. Our central result shows that these are composed of elementary transformations. The latter ones are defined similarly to the well known elementary transformations of ruled surfaces. In a forthcoming paper, we apply these results in the geometric setup to obtain standard equivariant completions of affine surfaces with an action of certain algebraic groups. We show that these completions are unique up to equivariant elementary transformations.
Following an approach of Dolgachev, Pinkham and Demazure, we classified in math.AG/0210153 normal affine surfaces with hyperbolic $\C^{*}$-actions in terms of pairs of $\Q$-divisors $(D_+,D_-)$ on a smooth affine curve. In the present paper we show how to obtain from this description a natural equivariant completion of these $\C^*$-surfaces. Using elementary transformations we deduce also natural completions for which the boundary divisor is a standard graph in the sense of math.AG/0511063 and show in certain cases their uniqueness. This description is especially precise in the case of normal affine surfaces completable by a zigzag i.e., by a linear chain of smooth rational curves. As an application we classify all zigzags that appear as boundaries of smooth or normal $\C^*$-surfaces.
Let $X$ be an affine surface admitting a unique affine ruling and a $\mathbb C^*$-action. Assume that the ruling has a unique degenerate fibre and that this fibre is irreducible. In this paper we give a short proof of the following result of Miyanishi and Masuda: the universal covering of $X$ is a hypersurface in the affine 3-space given by the equation $x^my=z^d-1$, where $m>1$.
This is a renovated list of open problems, to appear in: "Affine Algebraic Geometry" conference Proceedings volume in Contemporary Mathematics series of the Amer. Math. Soc. Ed. by Jaime Gutierrez, Vladimir Shpilrain, and Jie-Tai Yu.
We prove that a normal affine surface $V$ over $\bf C$ admits an effective action of a maximal torus ${\bf T}={\bf C}^{*n}$ ($n\le 2$) such that any other effective ${\bf C}^*$-action is conjugate to a subtorus of $\bf T$ in Aut $(V)$, in the following particular cases: (a) the Makar-Limanov invariant ML$(V)$ is nontrivial, (b) $V$ is a toric surface, (c) $V={\bf P}^1\times {\bf P}^1\backslash Δ$, where $Δ$ is the diagonal, and (d) $V={\bf P}^2\backslash Q$, where $Q$ is a nonsingular quadric. In case (a) this generalizes a result of Bertin for smooth surfaces, whereas (b) was previously known for the case of the affine plane (Gutwirth) and (d) is a result of Danilov-Gizatullin and Doebeli.
This is a recent conference report on the Kobayashi Problem on hyperbolicity of generic projective hypersurfaces. As an appendix, a (non-updated) author's survey article of 1992 on the same subject, published in an edition with a limited distribution, is added.
We give a classification of normal affine surfaces admitting an algebraic group action with an open orbit. In particular an explicit algebraic description of the affine coordinate rings and the defining equations of such varieties is given. By our methods we recover many known results, e.g. the classification of normal affine surfaces with a `big' open orbit of Gizatullin and Popov or some of the classification results of Danilov-Gizatullin, Bertin and others.