arXiv subjects

M. Zaidenberg

Publications and source records attributed to M. Zaidenberg.

13 recordsLinked to original sources

arXiv2023

Varieties covered by affine spaces, uniformly rational varieties and their cones

It was shown in [S. Kaliman, M. Zaidenberg, Gromov ellipticity of cones over projective manifolds, Math. Res. Lett. (to appear), arXiv:2303.02036 (2023)] 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 smooth subvarieties. The same also holds, more generally, for uniformly rational projective varieties, in particular, for projective varieties covered by affine spaces. It occurs also that stably uniformly rational complete varieties are elliptic.

math.AG↗

arXiv2020

Tits type alternative for groups acting on toric affine varieties

Given a toric affine algebraic variety $X$ and a collection of one-parameter unipotent subgroups $U_1,\ldots,U_s$ of $\mathop{\rm Aut}(X)$ which are normalized by the torus acting on $X$, we show that the group $G$ generated by $U_1,\ldots,U_s$ verifies the following alternative of Tits' type: either $G$ is a unipotent algebraic group, or it contains a non-abelian free subgroup. We deduce that if $G$ is $2$-transitive on a $G$-orbit in $X$, then $G$ contains a non-abelian free subgroup, and so, is of exponential growth.

math.AG↗

arXiv2019

A remark on the intersection of plane curves

Let $D$ be a very general curve of degree $d=2\ell-ε$ in $\mathbb{P}^2$, with $ε\in \{0,1\}$. Let $Γ\subset \mathbb{P}^2$ be an integral curve of geometric genus $g$ and degree $m$, $Γ\neq D$, and let $ν: C\to Γ$ be the normalization. Let $δ$ be the degree of the \emph{reduction modulo 2} of the divisor $ν^*(D)$ of $C$. In this paper we prove the inequality $4g+δ\geqslant m(d-8+2ε)+5$. We compare this with similar inequalities due to Geng Xu and Xi Chen. Besides, we provide a brief account on genera of subvarieties in projective hypersurfaces.

math.AG↗

arXiv2012

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.

math.AG↗

arXiv2012

Infinite transitivity on affine varieties

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.

math.AG↗

arXiv2012

Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity

We say that a group G acts infinitely transitively on a set X if for every integer m the induced diagonal action of G is transitive on the cartesian mth power of X with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of affine cones over flag varieties, the second of non-degenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups of a reinforced type.

math.AG↗

arXiv2011

Acyclic curves and group actions on affine toric surfaces

We show that every irreducible, simply connected curve on a toric affine surface X over the field of complex numbers is an orbit closure of a multiplicative group action on X. It follows that up to the action of the automorphism group Aut(X) there are only finitely many non-equivalent embeddings of the affine line in X. A similar description is given for simply connected curves in the quotients of the affine plane by small finite linear groups. We provide also an analog of the Jung-van der Kulk theorem for affine toric surfaces, and apply this to study actions of algebraic groups on such surfaces.

math.AG↗

arXiv2001

Simple birational extensions of the polynomial ring $\C^{[3]}$

The Abhyankar-Sathaye Problem asks whether any biregular embedding of affine spaces $A^m_k\to A^n_k$ can be rectified, that is, is equivalent to a linear embedding up to an automorphism of the target space. Here we study this problem for the embeddings $C^3 \to C^4$ whose image $X$ is given in $C^4$ by an equation $p=f(x,y)u+g(x,y,z)=0$, where $f\in C[x,y],$ $f\neq 0$ and $g\in C[x,y,z]$. Under certain additional assumptions we show that, indeed, the polynomial $p$ is a variable of the polynomial ring $C[x,y,z,u]$ (i.e., a coordinate of a polynomial automorphism of $C^4$). This is an analog of a theorem due to Sathaye which concerns the case of embeddings $C^2\to C^3$. Besides, we generalize a theorem of Miyanishi giving, for a polynomial $p$ as above, a criterion for as when $X$ is isomorphic to $C^3$.

math.AG↗

arXiv1997

Rational cuspidal plane curves of type (d, d-3)

In the previous paper [E-print alg-geom/9507004] we classified the rational cuspidal plane curves C with a cusp of multiplicity deg C - 2. In particular, we showed that any such curve can be transformed into a line by Cremona transformations. Here we do the same for the rational cuspidal plane curves C with a cusp of multiplicity deg C - 3.

alg-geom↗

arXiv1997

On fundamental groups of elliptically connected surfaces

A compact complex manifold $X$ is called elliptically connected if any pair of points in $X$ can be connected by a chain of elliptic or rational curves. We prove that the fundamental group of an elliptically connected compact complex surface is almost abelian. This confirms a conjecture which states that the fundamental group of an elliptically connected Kähler manifold must be almost abelian.

alg-geom↗

arXiv1997

Plane curves with a big fundamental group of the complement

Let $C \s \pr^2$ be an irreducible plane curve whose dual $C^* \s \pr^{2*}$ is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincaré group $π_1(\pr^2 \se C)$ contains a free group with two generators. If the geometric genus $g$ of $C$ is at least 2, then a subgroup of $G$ can be mapped epimorphically onto the fundamental group of the normalization of $C$, and the result follows. To handle the cases $g=0,1$, we construct universal families of immersed plane curves and their Picard bundles. This allows us to reduce the consideration to the case of Plücker curves. Such a curve $C$ can be regarded as a plane section of the corresponding discriminant hypersurface (cf. [Zar, DoLib]). Applying Zariski--Lefschetz type arguments we deduce the result from `the bigness' of the $d$-th braid group $B_{d,g}$ of the Riemann surface of $C$.

alg-geom↗

arXiv1995

On a class of rational cuspidal plane curves

We obtain new examples and the complete list of the rational cuspidal plane curves $C$ with at least three cusps, one of which has multiplicity ${\rm deg}\,C - 2$. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational cuspidal plane curves.

alg-geom↗

arXiv1995

On the number of singular points of plane curves

This is an extended, renovated and updated report on a joint work which the second named author presented at the Conference on Algebraic Geometry held at Saitama University, 15-17 of March, 1995. The main result is an inequality for the numerical type of singularities of a plane curve, which involves the degree of the curve, the multiplicities and the Milnor numbers of its singular points. It is a corollary of the logarithmic Bogomolov-Miyaoka-Yau's type inequality due to Miyaoka. It was first proven by F. Sakai at 1990 and rediscovered by the authors independently in the particular case of an irreducible cuspidal curve at 1992. Our proof is based on the localization, the local Zariski--Fujita decomposition and uses a graph discriminant calculus. The key point is a local analog of the BMY-inequality for a plane curve germ. As a corollary, a boundedness criterium for a family of plane curves has been obtained. Another application of our methods is the following fact: a rigid rational cuspidal plane curve cannot have more than 9 cusps.

alg-geom↗