# Hubert FlennerRuhr-Universität Bochum | RUB

Hubert Flenner

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90

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## Publications

Publications (90)

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 norm...

We deduce results on the dimension and connectedness of degeneracy loci of maps of finite modules f:M→N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:M\rightarrow N$...

The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism X × A n ≅ X ′ × A n X\times \mathbb {A}^n\cong X’\times \mathbb {A}^n for (affine) algebraic varieties X X and X ′ X’ implies that X ≅ X ′ X\cong X’ . In this paper we provide a criterion for cancellation by the affine line (that is, n = 1 n=1 ) in the case...

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.

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 nor...

A smooth family $\varphi:\mathcal V\to S$ 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 ${\m...

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.

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,...

Let V be a normal affine surface which admits \({\mathbb{C}^*}\) - and \({\mathbb{C}_+}\) -actions. Such surfaces were classified e.g., in: Flenner and Zaidenberg (Osaka J Math 40:981–1009, 2003; 42:931–974), see also the references therein. In this note we show that in many cases V can be embedded as a principal Zariski open subset into a hypersur...

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. Comment: Subj-class: math.AG 4 pages

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 previou...

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.

A Gizatullin surface is a normal affine surface V over
$ \mathbb{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
$ \mathbb{C}^{ * } $
-actions and
$ \mathbb{A}^{{\text{1}}} $
-fibrations on such a surface V up to automorphisms. The latter...

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 X→Y, the Hochschild complex HX/Y, as introduced in [R.-O. Buchweitz, H. Flenner, Global Hochschild...

Affine geometry deals with algebro-geometric questions of affine varieties that are treated with methods coming from various areas of mathematics like commutative and non-commutative algebra, algebraic, complex analytic and differential geometry, singularity theory and topology. The conference had several main topics. One of them was the famous Jac...

Affine geometry deals with algebro-geometric questions of affine varieties. In the last decades this area has developed into a systematic discipline with a sizeable international group of researchers, and with methods coming from commutative and non-commutative algebra, algebraic, complex analytic and differential geometry, singularity theory and t...

We show that a formal power series ring A[[X]] over a noetherian ring A is not a projective module unless A is artinian. However, if (A,
) is any local ring, then A[[X]] behaves like a projective module in the sense that Extp
A
(A[[X]], M)=0 for all
-adically complete A-modules. The latter result is shown more generally for any flat A-module B inst...

We introduce Hochschild (co-)homology of morphisms of schemes or analytic spaces and study its fundamental properties. In analogy with the cotangent complex we introduce the so-called (derived) Hochschild complex of a morphism; the Hochschild cohomology and homology groups are then the Ext and Tor groups of that complex. We prove that these objects...

Let X be an affine surface admitting a unique affine ruling and a
\mathbbC*\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 t...

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...

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 tran...

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’...

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'...

surfaces V = SpecA endowed with a C -action. It is well known that a C -action gives rise to a grading A = L i2Z Ai of the coordinate ring A. Such a surface can be elliptic, parabolic or hyperbolic. It is called elliptic, if it has an attractive fixed point or, equivalently, if A0 = C and A = A 0. It is parabolic if it has an attractive curve of fi...

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 surfa...

We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a ℂ*-action. For varieties with an isolated singularity, covered by a family of rational curves with a general member not passing
through the singular point, we show that this singularity is rational. In particular, this provides an explanation of cl...

A classification of normal affine surfaces admitting a $\bf C^*$-action was given in the work of Bia{\l}ynicki-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 Dolg...

Let $X \subseteq {\bf P}^N ={\bf P}^{2n}_K$ be a subvariety of dimension $n$ and $P \in {\bf P}^N$ a generic point. If the tangent variety Tan$ X$ is equal to ${\bf P}^N$ then for generic points $x$, $y$ of $X$ the projective tangent spaces $t_xX$ and $t_yX$ meet in one point $P=P(x,y)$. The main result of this paper is that the rational map $(x,y)...

To an arbitrary ideal I in a local ring (A,m) one can associate a multiplicity j(I,A) that generalizes the classical Hilbert–Samuel multiplicity of an m-primary ideal and which plays an important role in intersection theory. If the ideal is strongly Cohen–Macaulay in A and satisfies a suitable Artin–Nagata condition then our main result states that...

A contact singularity is a normal singularity (V,0) together with a holomorphic contact form η on V\ Sing V in a neighbourhood of 0, i.e. η∧ (dη)r
has no zero, where dim V=2r+1. The main result of this paper is that there are no isolated contact singularities.

Let X be a complex space and F a coherent O
x
-module, A F-(co)framed sheaf on X is a pair (ε, ϕ) with a coherent O
x
-module ε and a morphism of coherent sheaves ϕ: F
F ε (resp. ϕ: ε → F). Two such pairs (ε, ϕ) and (ε′,ϕ′) are said to be isomorphic if there exists an isomorphism of sheaves α: ε →ε′ with α° ϕ = ϕ′ (resp. ϕ′° α = ϕ). A pair (α, ϕ) i...

For a normal subvariety $V$ of ${\bf C}^n$ with a good ${\bf C}^*$-action we give a simple characterization for when it has only log canonical, log terminal or rational singularities. Moreover we are able to give formulas for the plurigenera of isolated singular points of such varieties and of the logarithmic Kodaira dimension of $V\backslash \{0\}...

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.

This paper contains the details and complete proofs of our earlier announcement in math.AG/9907004 . We construct a general semiregularity map for algebraic cycles as asked for by S. Bloch in 1972. The existence of such a semiregularity map has well known consequences for the structure of the Hilbert scheme and for the variational Hodge conjecture....

The purpose of this paper is to give a new approach to some classical results on equimultiplicity. Our proofs are based on [FVo] in which there was described in a very precise way the behaviour of Hilbert function and multiplicities in an exact sequence and, in particular, under hyperplane sections. In [FOV], Section 1.1, this approach was used to...

We construct a general semiregularity map for cycles on a complex analytic or algebraic manifold and show that such semiregularity map can be obtained from the classical tool of the Atiyah-Chern character. The first part of the paper is fairly detailed, deducing the existence and explicit form of a generalized semiregularity map from known results...

This initial chapter is devoted to setting in place certain basic pieces of theory which will be used many times throughout the book, and to establishing certain themes in a classical setting which will subsequently be developed more generally. Sections 1.1 and 1.2 treat the fundamental notions of degree and multiplicity. A classical notion in Inte...

This chapter focuses on the converse to Bezout’s Theorem in the general, locally Cohen-Macaulay and arithmetically Cohen-Macaulay cases respectively. More precisely, we examine the following question. Let X, Y be subvarieties of projective space and assume that
where the sum is taken over the irreducible components of X ∩ Y. Is it true that then th...

This chapter concentrates on the connectedness and Bertini theorems that will be needed in Chapters 4 and 5 to prove a result on the existence of components of the Stückrad-Vogel cycle in all possible dimensions and a converse to Bezout’s Theorem. Section 3.1 introduces the suitably inductive type of connectedness, namely connectedness ‘in a given...

In this chapter we will focus on the study of multiplicities and intersection numbers. We will introduce in Section 6.1 a version of the j-multiplicity for arbitrary local rings which was first studied by Achilles and Manaresi [AMa2]. It assigns a new multiplicity j(a, M) to every finite module M over a local ring A, where a is an arbitrary (not ne...

The central topic in this chapter is the Refined Bezout Theorem, which applies to possibly improper intersections. To treat this, we need to refine our considerations of the process of reduction to the diagonal. This is done in general in Section 2.1 in a scheme-theoretic version of the Intersection Algorithm; a more concrete version for treating i...

In this final chapter we will give a further interpretation of the intersection algorithm for the self-intersection of subvarieties X of ℙN
. Assume that X is smooth of dimension n and
is a generic linear projection, i.e. p
m
= [λ0 : ... : λm
] where λi
= ΣNj=0u
ij
x
j
are sufficiently generic linear forms. Let R(pm
) be the ramification locus of p...

In this chapter we apply the theory of linkage (i.e. liaison) or, more generally, that of residual intersections to the intersection cycle of a self-intersection, to embedded join and secant varieties, and to limit of join and secant varieties, especially in the case of low dimension or of small deviation. Once again, coning allows us to pass betwe...

Classical Intersection Theory (see for example Weil [Wei]) treats the case of proper intersections, where geometrical objects (usually subvarieties of a nonsingular variety) intersect with the expected dimension. In 1984, two books appeared which surveyed and developed work by the individual authors, coworkers and others on a refined version of Int...

Dedicated to the memory of Wolfgang Classical Intersection Theory (see for example Wei! [Wei]) treats the case of proper intersections, where geometrical objects (usually subvarieties of a non singular variety) intersect with the expected dimension. In 1984, two books appeared which surveyed and developed work by the individual authors, co worker...

Let X C PN = P be a subvariety of dimension n and ∧ PN be a generic linear subspace of dimension N k - 1 with k ≥ n. Then the linear projection π∧: X → Pk is a finite map. Let R(π∧) be its ramification locus. In this paper we study the map from the Grassmannian G(N - k - 1, N) of planes of dimension N - k - 1 in PN to the Hilbert moduli space given...

Let X, Y ⊆ ℙN be closed subvarieties of dimensions n and m respectively. Proving a Bezout theorem for improper intersections Stückrad and Vogel [SVo] introduced cycles vk = vk(X, Y) of dimension k on X∩Y and βk on the ruled join variety J := J(X, Y) of X and Y which are obtained by a simple algorithm. ·In this paper we give an interpretation of the...

We obtain new examples and the complete list of the rational cuspidal plane curvesC with at least three cusps, one of which has multiplicitydegC-2. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational
cuspidal plane curves.

The aim of this note is to study multiplicities of local rings under (iterated) hyperplane sections, and Bezout-type theorems.
An important application is a local version of a converse to Bezout's theorem. Even in the projective case this improves known
results for arithmetically Cohen-Macaulay schemes. For the proofs our key result is a descriptio...

Let f: X → S be a morphism of complex or real spaces, and P a property of homomorphisms of local rings. Consider the set ℙ(f) of points x∈X for which the induced map of local rings O
S,f(x)
→ O
X,x
has property P. In this chapter we give a criterion for ℙ(f) being constructible (resp., Zariski open) in X. Moreover, we verify this criterion for a wi...

In this chapter, we develop the theme of estimating dimensions of joins and sets of tangencies, often under assumptions of connectedness, using results from Chapter 3. In Section 4.1 we study the v-cycle again and show as an application of the connectedness theorem that it has no gaps, and this is interpreted in terms of normal cones. We apply this...

Introduction. In this paper we will make some contributions to the theory of stratifications of real and complex analytic spaces. One main goal is the generalization of a result of Verdier, which says that for a proper map of complex analytic spaces the direct image sheaves of constructible sheaves are again constructible. This result is generalize...

Introduction. Over the last 10 years there was a very interesting development in affine algebraic geometry, in particular in questions related to group actions.

Thesis (doctoral)--Ruhr-Universität Bochum, 1974.