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Hubert Flenner

Publications and source records attributed to Hubert Flenner.

At least 19 recordsLinked to original sources
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↗
arXiv2016

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 $X_\lambda\to B$, $\lambda\in\Lambda$, of $\mathbb{A}^1$-fibered surfaces with cylinders $X_\lambda\times\mathbb{A}^1$ 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↗
arXiv2013

Strong global dimension of commutative rings and schemes

The strong global dimension of a ring is the supremum of the length of perfect complexes that are indecomposable in the derived category. In this note we characterize the noetherian commutative rings that have finite strong global dimension. We also give a similar characterization for arbitrary noetherian schemes.

math.AC↗
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↗
arXiv2008

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.

math.AG↗
arXiv2008

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.

math.AG↗
arXiv2008

On the Danilov-Gizatullin Isomorphism Theorem

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.

math.AG↗
arXiv2007

Uniqueness of $\bf C^*$- and $\bf C_+$-actions on Gizatullin surfaces

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 $\bf A^1$.

math.AG↗
arXiv2006

The global decomposition theorem for Hochschild (co-)homology of singular spaces via the Atiyah-Chern character

We generalize the decomposition theorem of Hochschild, Kostant and Rosenberg for Hochschild (co-)homology to arbitrary morphisms between complex spaces or schemes over a field of characteristic zero. To be precise, we show that for each such morphism, the Hochschild complex, as introduced in math.AG/0606593, decomposes naturally in the derived category into the direct sum of the derived symmetric powers of the shifted cotangent complex, a result due to Quillen in the affine case. Even in the affine case, our proof is new and provides further information. It shows that the decomposition is given explicitly and naturally by the universal Atiyah-Chern character, the exponential of the universal Atiyah class. We further use the decomposition theorem to show that the semiregularity map for perfect complexes factors through Hochschild homology and, in turn, factors the Atiyah-Hochschild character through the characteristic homomorphism from Hochschild cohomology to the graded centre of the derived category.

math.AG↗
arXiv2005

Birational transformations of weighted graphs

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.

math.AG↗
arXiv2005

Completions of $\C^*$-surfaces

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.

math.AG↗
arXiv2005

On a result of Miyanishi-Masuda

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 $m>1$.

math.AG↗
arXiv2005

Codimension and connectedness of degeneracy loci over local rings

We deduce results on the dimension and connectedness of degeneracy loci of maps of finite modules $f:M\to N$ over a local noetherian ring $(A,{\mathfrak m})$. We show for instance that the expected determinantal bounds on the dimension of the t-$th$ degeneracy locus of $f$ hold if $f\in {\mathfrak m} Hom (M,N)$, and that this degeneracy locus is connected in the expected dimension provided $\hat A$ is a domain.

math.AC↗
arXiv2004

On the uniqueness of ${\bf C}^*$-actions on affine surfaces

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.

math.AG↗
arXiv2004

Locally nilpotent derivations on affine surfaces with a $\C^*$-action

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.

math.AG↗
arXiv2002

Normal affine surfaces with $\bf C^*$-actions

A classification of normal affine surfaces admitting a $\bf C^*$-action was given in the work of Białynicki-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.

math.AG↗