arXiv subjects

Mikhail Zaidenberg

Publications and source records attributed to Mikhail Zaidenberg.

At least 37 records · Page 2Linked to original sources

arXiv2018

On automorphism groups of affine surfaces

This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic groups. At the same time, they can be studied from the viewpoint of the combinatorial group theory, so we put a special accent on group-theoretical aspects (ind-groups, amalgams, etc.). We provide different approaches to classification, prove certain new results, and attract attention to several open problems.

math.AG↗

arXiv2018

Cancellation for surfaces revisited. II

Let $X$ and $X'$ be affine algebraic varieties over a field $\mathbb{k}$. The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism $X\times\mathbb{A}^n\cong X'\times\mathbb{A}^n$ implies $X\cong X'$. In Part I of this paper (arXiv:1610.01805) we provided a criterion for cancellation in the case where $X$ is a normal affine surface admitting an $\mathbb{A}^1$-fibration $X\to B$ over a smooth affine curve $B$. If $X$ does not admit such an $\mathbb{A}^1$-fibration then the cancellation by the affine line is known to hold for $X$ by a result of Bandman and Makar-Limanov. In the present Part II we classify all pairs $(X,X')$ of smooth affine surfaces $\mathbb{A}^1$-fibered over $B$ with only reduced fibers whose cylinders $X\times\mathbb{A}^1$, $X'\times\mathbb{A}^1$ are isomorphic over $B$. Our criterion of isomorphism of cylinders over $B$ is expressed in terms of linear equivalence of certain divisors on the Danielewski-Fieseler quotient of $X$ over $B$. Under a mild restriction we construct a coarse moduli of such surfaces.

math.AG↗

arXiv2018

Fano-Mukai fourfolds of genus $10$ as compactifications of $\mathbb{C}^4$

It is known that the moduli space of smooth Fano-Mukai fourfolds $V_{18}$ of genus $10$ has dimension one. We show that any such fourfold is a completion of $\mathbb{C}^4$ in two different ways. Up to isomorphism, there is a unique fourfold $V_{18}^{\mathrm s}$ acted upon by $\operatorname{SL}_2(\mathbb{C})$. The group $\operatorname{Aut}(V_{18}^{\mathrm s})$ is a semidirect product $\operatorname{GL}_2(\mathbb{C})\rtimes(\mathbb{Z}/2\mathbb{Z})$. Furthermore, $V_{18}^{\mathrm s}$ is a $\operatorname{GL}_2(\mathbb{C})$-equivariant completion of $\mathbb{C}^4$, and as well of $\operatorname{GL}_2(\mathbb{C})$. The restriction of the $\operatorname{GL}_2(\mathbb{C})$-action on $V_{18}^{\mathrm s}$ to $\mathbb{C}^4\hookrightarrow V_{18}^{\mathrm s}$ yields a faithful representation with an open orbit. There is also a unique, up to isomorphism, fourfold $V_{18}^{\mathrm a}$ such that the group $\operatorname{Aut}(V_{18}^{\mathrm a})$ is a semidirect product $({\mathbb G}_{\mathrm{a}}\times{\mathbb G}_{\mathrm{m}})\rtimes (\mathbb{Z}/2\mathbb{Z})$. For a Fano-Mukai fourfold $V_{18}$ neither isomorphic to $V_{18}^{\mathrm s}$, nor to $V_{18}^{\mathrm a}$, one has $\operatorname{Aut}^0 (V_{18})\cong ({\mathbb G}_{\mathrm{m}})^2$, and $\operatorname{Aut}(V_{18})$ is a semidirect product of $\operatorname{Aut}^0(V_{18})$ and a finite cyclic group whose order is a factor of $6$.

math.AG↗

arXiv2017

Cancellation for surfaces revisited. I

The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism $X\times\mathbb{A}^n\cong X'\times\mathbb{A}^n$ for (affine) algebraic varieties $X$ and $X'$ implies that $X\cong X'$. In this paper we provide a criterion for cancellation by the affine line (that is, $n=1$) in the case where $X$ is a normal affine surface admitting an $\mathbb{A}^1$-fibration $X\to B$ over a smooth affine curve $B$. If $X$ does not admit such an $\mathbb{A}^1$-fibration then the cancellation by the affine line is known to hold for $X$ by a result of Bandman and Makar-Limanov. It occurs that for a smooth $\mathbb{A}^1$-fibered affine surface $X$ over $B$ the cancellation by an affine line holds if and only if $X\to B$ is a line bundle, and, for a normal such $X$, if and only if $X\to B$ is a cyclic quotient of a line bundle (an orbifold line bundle). When the cancellation does not hold for $X$ we include $X$ in a non-isotrivial deformation family $X_λ\to B$, $λ\inΛ$, of $\mathbb{A}^1$-fibered surfaces with cylinders $X_λ\times\mathbb{A}^1$ isomorphic over $B$. This gives large families of examples of non-cancellation for surfaces which extend the known examples constructed by Danielewski, tom Dieck, Wilkens, Masuda and Miyanishi, e.a.

math.AG↗

arXiv2016

Gaps for geometric genera

We investigate the possible values for geometric genera of subvarieties in a smooth projective variety. Values which are not attained are called gaps. For curves on a very general surface in $\mathbb{P}^3$, the initial gap interval was found by Xu (see [7] in References), and the next one in our previous paper (see [4] in References), where also the finiteness of the set of gaps was established and an asymptotic upper bound of this set was found. In the present paper we extend some of these results to smooth projective varieties of arbitrary dimension using a different approach.

math.AG↗

arXiv2015

New examples of cylindrical Fano fourfolds

We construct new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form $Z\times A^1$, where $Z$ is a quasiprojective variety. The affine cones over such a fourfold admit effective $G_a$-actions. Similar constructions of cylindrical Fano threefolds and fourfolds were done previously in [KPZ11, KPZ14, PZ15].

math.AG↗

arXiv2015

Automorphism Groups of Configuration Spaces and Discriminant Varieties

The configuration space $\mathcal{C}^n(X)$ of an algebraic curve $X$ is the algebraic variety consisting of all $n$-point subsets $Q\subset X$. We describe the automorphisms of $\mathcal{C}^n(\mathbb{C})$, deduce that the (infinite dimensional) group Aut$\,\mathcal{C}^n(\mathbb{C})$ is solvable, and obtain an analog of the Mostow decomposition in this group. The Lie algebra and the Makar-Limanov invariant of $\mathcal{C}^n(\mathbb{C})$ are also computed. We obtain similar results for the level hypersurfaces of the discriminant, including its singular zero level. This is an extended version of our paper \cite{Lin-Zaidenberg14}. We strengthened the results concerning the automorphism groups of cylinders over rigid bases, replacing the rigidity assumption by the weaker assumption of tightness. We also added alternative proofs of two auxiliary results cited in \cite{Lin-Zaidenberg14} and due to Zinde and to the first author. This allowed us to provide the optimal dimension bounds in our theorems.

math.AG↗

arXiv2014

Plane Curves with Hyperbolic and C-hyperbolic Complements

The general problem which initiated this work is: What are the quasiprojective varieties which can be uniformized by means of bounded domains in $\cz^n$ ? Such a variety should be, in particular, C--hyperbolic, i.e. it should have a Carathéodory hyperbolic covering. We study here the plane projective curves whose complements are C--hyperbolic. For instance, we show that most of the curves whose duals are nodal or, more generally, immersed curves, belong to this class. We also give explicit examples of irreducible such curves of any even degree d greater or equal 6.

alg-geom↗

arXiv2014

Genera of curves on a very general surface in $P^3$

In this paper we consider the question of determining the geometric genera of irreducible curves lying on a very general surface $S$ of degree $d$ at least 5 in $\mathbb{P}^3$ (the cases $d \leqslant 4$ are well known). We introduce the set $Gaps(d)$ of all non-negative integers which are not realized as geometric genera of irreducible curves on $S$. We prove that $Gaps(d)$ is finite and, in particular, that $Gaps(5)= \{0,1,2\}$. The set $Gaps(d)$ is the union of finitely many disjoint and separated integer intervals. The first of them, according to a theorem of Xu, is $Gaps_0(d):=[0, \frac{d(d-3)}{2} - 3]$. We show that the next one is $Gaps_1(d):= [\frac{d^2-3d+4}{2}, d^2-2d-9]$ for all $d \geqslant 6$.

math.AG↗

arXiv2014

Examples of cylindrical Fano fourfolds

We construct 4 di erent families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form Z x A1, where Z is a quasiprojective variety. The affi ne cones over such a fourfold admit eff ective Ga-actions. Similar constructions of cylindrical Fano threefolds were done previously in our papers jointly with Takashi Kishimoto.

math.AG↗

arXiv2014

Ga-actions on affine cones

We give a criterion of existence of a unipotent group action on the affine cone over a projective variety or, more generally, on the affine quasicone over a variety which is projective over another affine variety.

math.AG↗

arXiv2013

The Gromov-Winkelmann theorem for flexible varieties

An affine variety $X$ of dimension $\ge 2$ is called {\em flexible} if its special automorphism group SAut$(X)$ acts transitively on the smooth locus $X_{reg}$ \cite{AKZ}. Recall that the special automorphism group SAut$(X)$ is the subgroup of the automorphism group Aut$(X)$ generated by all one-parameter unipotent subgroups \cite{AKZ}. Given a normal, flexible, affine variety $X$ and a closed subvariety $Y$ in $X$ of codimension at least 2, we show that the pointwise stabilizer subgroup of $Y$ in the group SAut$(X)$ acts infinitely transitively on the complement $X\backslash Y$, that is, $m$-transitively for any $m\ge 1$. More generally we show such a result for any quasi-affine variety $X$ and codimension $\ge 2$ subset $Y$ of $X$. In the particular case of $X=Å^n$, $n\ge 2$, this yields a Theorem of Gromov and Winkelmann \cite{Gr1}, \cite{Wi}.

math.AG↗

arXiv2013

Deformation equivalence of affine ruled surfaces

A smooth family $φ:\mathcal V\to S$ of surfaces will be called {\em completable} if there is a logarithmic deformation $(\bar {\mathcal V},{\mathcal D})$ over $S$ so that ${\mathcal V}=\bar{\mathcal V}\backslash {\mathcal D}$. Two smooth surfaces $V$ and $V'$ are said to be deformations of each other if there is a completable flat family ${\mathcal V}\to S$ of smooth surfaces over a connected base so that $V$ and $V'$ are fibers over suitable points $s,s'\in S$. This relation generates an equivalence relation called {\em deformation equivalence}. In this paper we give a complete combinatorial description of this relation in the case of affine ruled surfaces, which by definition are surfaces that admit an affine ruling $V\to B$ over an affine base with possibly degenerate fibers. In particular we construct complete families of such affine ruled surfaces. In a few particular cases we can also deduce the existence of a coarse moduli space.

math.AG↗

arXiv2013

Unipotent Group Actions on Del Pezzo Cones

In a previous paper we established that for any del Pezzo surface Y of degree at least 4, the affine cone X over Y embedded via a pluri-anticanonical linear system admits an effective Ga-action. In particular, the group Aut(X) is infinite dimensional. In contrast, we show in this note that for a del Pezzo surface Y of degree at most 2 the generalized cones X as above do not admit any non-trivial action of a unipotent algebraic group.

math.AG↗

arXiv2011

Affine cones over Fano threefolds and additive group actions

We address the following question: When an affine cone over a smooth Fano threefold admits an effective action of the additive group? In this paper we deal with Fano threefolds of index 1 and Picard number 1. Our approach is based on a geometric criterion from our previous paper, which relates the existence of an additive group action on the cone over a smooth projective variety X with the existence of an open polar cylinder in X. Non-trivial families of Fano threefolds carrying a cylinder were found in loc. cit. Here we provide new such examples.

math.AG↗

arXiv2011

Scrolls and hyperbolicity

Using degeneration to scrolls, we give an easy proof of non-existence of curves of low genera on general surfaces in P3 of degree d >=5. We show, along the same lines, boundedness of families of curves of small enough genera on general surfaces in P3. We also show that there exist Kobayashi hyperbolic surfaces in P3 of degree d = 7 (a result so far unknown), and give a new construction of such surfaces of degree d = 6. Finally we provide some new lower bounds for geometric genera of surfaces lying on general hypersurfaces of degree 3d > 15 in P4.

math.AG↗

arXiv2010

Discrete convolution operators in positive characteristic: a variation on the Floquet-Bloch Theory

The classical Floquet theory deals with Floquet-Bloch solutions of periodic PDEs (see e.g., P. Kuchment. Floquet Theory for Partial Differential Equations. Basel: Birkhauser, 1993). Peter Kuchment developed as well a discrete version of this theory for difference vector equations on lattices, including the Floquet theory on infinite periodic graphs. Here we propose a variation on this theory for matrix convolution operators acting on vector functions on lattices with values in a field of positive characteristic.

math.AP↗

arXiv2010

Group actions on affine cones

We address the following question: Determine the affine cones over smooth projective varieties which admit an action of a connected algebraic group different from the standard C*-action by scalar matrices and its inverse action. We show in particular that the affine cones over anticanonically embedded smooth del Pezzo surfaces of degree at least 4 possess such an action. A question by Flenner and the third author whether this is also true for cubic surfaces, occurs to be out of reach for our methods. Nevertheless, we provide a general geometric criterion that could be helpful also in this case.

math.AG↗