Affine Threefolds Research Articles

Rees algebras of additive group actions

We establish basic properties of a sheaf of graded algebras canonically associated to every relative affine scheme \(f:X\rightarrow S\) endowed with an action of the additive group scheme \({\mathbb {G}}_{a,S}\) over a base scheme or algebraic space S, which we call the (relative) Rees algebra of the \({\mathbb {G}}_{a,S}\)-action. In the case of affine algebraic varieties defined over a field of characteristic zero, we establish further properties of the Rees algebra of a \({\mathbb {G}}_{a}\)-action in terms of its associated locally nilpotent derivation. We give an algebro-geometric characterization of pairs consisting of an affine algebraic variety and a \({\mathbb {G}}_{a}\)-action on it whose associated Rees algebras are finitely generated and provide an algorithm extending van den Essen’s kernel algorithm for locally nilpotent derivations to compute generators of these Rees algebras. We illustrate these properties on several examples which played important historical roles in the development of the algebraic theory of locally nilpotent derivations and give applications to the construction of new families of affine threefolds with \({\mathbb {G}}_{a}\)-actions.

Equivariant extensions of $\mathbb{G}_a$-torsors over punctured surfaces

Motivated by the study of the structure of algebraic actions of the additive group on affine threefolds X, we consider a special class of such varieties whose algebraic quotient morphisms X -> X//G(a) restrict to principal homogeneous bundles over the complement of a smooth point of the quotient. We establish basic general properties of these varieties and construct families of examples illustrating their rich geometry. In particular, we give a complete classification of a natural subclass consisting of threefolds X endowed with proper G(a)-actions, whose algebraic quotient morphisms pi : X -> X//G(a) are surjective with only isolated degenerate fibers, all isomorphic to the affine plane A(2) when equipped with their reduced structures.

Affine threefolds admitting $$G_a$$ G a -actions

The structures of affine varieties of dimension greater than two can be explored with the help of fibrations by the affine line or plane and quotient morphisms by $$G_a$$ -actions. We consider $$G_a$$ -actions on affine threefolds and discuss the structure and the singularities of the quotient surface as well as the singular fibers of the quotient morphism.

Families of $\mathbb{A}^1$-contractible affine threefolds

We provide families of affine threefolds which are $\mathbb A^1$-contractible (that is, contractible in the unstable $\mathbb A^1$-homotopy category of Morel-Voevodsky) and pairwise non-isomorphic, thus answering a conjecture of Asok and Doran. As a particular case, we show that the Koras-Russell threefolds of the first kind are $\mathbb A^1$-contractible, extending results of Hoyois, Krishna and Ostvaer.

Families of $\mathbb{A}^1$-contractible affine threefolds

We provide families of affine threefolds which are contractible in the unstable $\mathbb A^1$-homotopy category of Morel-Voevodsky and pairwise non-isomorphic, thus answering a question of A. Asok and B. Doran. As a particular case, we show that the Koras-Russell threefolds of the first kind are contractible, extending results of M. Hoyois, A. Krishna and P. A. Ostvaer.

Multigraded linear series and recollement

Given a scheme Y equipped with a collection of globally generated vector bundles E_1, ldots , E_n, we study the universal morphism from Y to a fine moduli space {mathcal {M}}(E) of cyclic modules over the endomorphism algebra of E:={mathcal {O}}_Yoplus E_1oplus cdots oplus E_n. This generalises the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We describe the image of the morphism and present necessary and sufficient conditions for surjectivity in terms of a recollement of a module category. When the morphism is surjective, this gives a fine moduli space interpretation of the image, and as an application we show that for a small, finite subgroup Gsubset {{mathrm{GL}}}(2,mathbb {k}), every sub-minimal partial resolution of {mathbb {A}}^2_mathbb {k}/G is isomorphic to a fine moduli space {mathcal {M}}(E_C) where E_C is a summand of the bundle E defining the reconstruction algebra. We also consider applications to Gorenstein affine threefolds, where Reid’s recipe sheds some light on the classes of algebra from which one can reconstruct a given crepant resolution.

Families of exotic affine 3-spheres

We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non-algebraically isomorphic to it. We show in particular that for every smooth topologically contractible affine surface S with trivial automorphism group, there exists a canonical smooth family of pairwise non-isomorphic exotic affine 3-spheres parametrized by the closed points of S.

FAMILIES OF AFFINE RULED SURFACES: EXISTENCE OF CYLINDERS

We show that the generic fiber of a family $f:X\rightarrow S$ of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base $S$. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking $S$, such a family actually factors through an $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over a certain $S$-scheme $Y\rightarrow S$ induced by the MRC-fibration of a relative smooth projective model of $X$ over $S$. For affine threefolds $X$ equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$-ruled surfaces over a smooth curve $B$, the induced $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$.

AFFINE THREEFOLDS WITH A $$ \mathbb{A} $$ 2-FIBRATIONS

An affine threefold containing an \( \mathbb{A} \)2-cylinder is studied. The existence of \( \mathbb{A} \)2-cylinders is almost equivalent to the existence of mutually commuting, independent Ga-actions σ1, σ2. A typical example of such affine threefolds is a hypersurface xmy = f(x, z, t), and we generalize such a hypersurface to define an affine pseudo-3-space. After I. Heden [He], we observe also a Ga-action on the hypersurface xmy = f(x, z, t) with m ≥ 2.

Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday

By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over a simple singularity is triangle equivalent to the $1$-cluster category of the path algebra of a Dynkin quiver (i.e., the orbit category of the derived category by the action of the Auslander-Reiten translation). In this paper we give a systematic method to construct a similar type of triangle equivalence between the stable category of Cohen-Macaulay modules over a Gorenstein isolated singularity $R$ and the generalized (higher) cluster category of a finite dimensional algebra $\Lambda$. The key role is played by a bimodule Calabi-Yau algebra, which is the higher Auslander algebra of $R$ as well as the higher preprojective algebra of an extension of $\Lambda$. As a byproduct, we give a triangle equivalence between the stable category of graded Cohen-Macaulay $R$-modules and the derived category of $\Lambda$. Our main results apply in particular to a class of cyclic quotient singularities and to certain toric affine threefolds associated with dimer models.

A cohomological classification of vector bundles on smooth affine threefolds

We give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2. As a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties. The proofs of these results involve three main ingredients. First, we give a description of the first nonstable A1-homotopy sheaf of the symplectic group. Second, these computations can be used in concert with F. Morel’s A1-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in A1-homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.

On the family of affine threefolds

AbstractLet $k$ be a field and $\mathbb{V}$ the affine threefold in $\mathbb{A}^4_k$ defined by $x^m y=F(x, z, t)$, $m \ge 2$. In this paper, we show that $\mathbb{V} \cong \mathbb{A}^3_k$ if and only if $f(z, t): = F(0, z, t)$ is a coordinate of $k[z, t]$. In particular, when $k$ is a field of positive characteristic and $f$ defines a non-trivial line in the affine plane $\mathbb{A}^2_k$ (we shall call such a $\mathbb{V}$ as an Asanuma threefold), then $\mathbb{V}\ncong \mathbb{A}^3_k$ although $\mathbb{V} \times \mathbb{A}^1_k \cong \mathbb{A}^4_k$, thereby providing a family of counter-examples to Zariski’s cancellation conjecture for the affine 3-space in positive characteristic. Our main result also proves a special case of the embedding conjecture of Abhyankar–Sathaye in arbitrary characteristic.

Formula omitted]-fibrations on affine threefolds

An A 1 -fibration on an algebraic surface is well understood and plays an important role in the classification of open algebraic surfaces. We consider A 1 -fibrations on higher-dimensional affine algebraic varieties, mostly in the case of affine threefolds and extend the results known in the case of surfaces. Special attention is paid to degenerate fibers of A 1 -fibrations and singularities of the quotient surface by a G a -action. We distinguish the quotient morphism by a G a -action from an A 1 -fibration. G a -actions will give more information on the A 1 -fibration induced by the actions.

Noncancellation for contractible affine threefolds

We construct two nonisomorphic contractible affine threefolds X X and Y Y with the property that their cylinders X × A 1 X\times \mathbb {A}^{1} and Y × A 1 Y\times \mathbb {A}^{1} are isomorphic, showing that the generalized Cancellation Problem has a negative answer in general for contractible affine threefolds. We also establish that X X and Y Y are actually biholomorphic as complex analytic varieties, providing the first example of a pair of biholomorphic but not isomorphic exotic A 3 \mathbb {A}^{3} ’s.

Stably free modules over smooth affine threefolds

We prove that the stably free modules over a smooth affine threefold over an algebraically closed field of characteristic different from 2 are free.

Affine threefolds whose log canonical bundles are not numerically effective

Let X ↪ ( T , D ) be a compactification of an affine 3-fold X into a smooth projective 3-fold T such that the (reduced) boundary divisor D is SNC. In this paper, as an affine counterpart to the work due to S. Mori (cf. [S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982) 133–176]), we shall classify ( K + D ) -negative extremal rays on T . In particular, if such an extremal ray R = R + [ C ] intersects K non-negatively, we shall describe the log flips and divisorial contractions appearing explicitly.

ON THE LOGARITHMIC KODAIRA DIMENSION OF AFFINE THREEFOLDS

In this article, we shall consider how to analyze affine threefolds associated to the log Kodaira dimension [Formula: see text] and make the framework for this purpose under a certain geometric condition. As a consequence of our main result, under this geometric condition, we can describe the construction of affine threefolds with [Formula: see text] fairly explicitly, and show that affine threefolds with [Formula: see text] have the structure of ℂ*-fibrations.

The Equivariant Grothendieck Groups of the Russell-Koras Threefolds

AbstractThe Russell-Koras contractible threefolds are the smooth affine threefolds having a hyperbolic *-action with quotient isomorphic to the corresponding quotient of the linear action on the tangent space at the unique fixed point. Koras and Russell gave a concrete description of all such threefolds and determined many interesting properties they possess. We use this description and these properties to compute the equivariant Grothendieck groups of these threefolds. In addition, we give certain equivariant invariants of these rings.

Witt groups of affine three-folds

Witt groups of affine three-folds

Algebraic cycles and vector bundles on real affine threefolds

LetX=Spec(B), whereB is a regular ring of Krull dimension 3 which is a finitely generatedR-algebra. We give necessary and sufficient conditions for the existence of a vector bundle of rank 3 onX with prescribed Chern classes. 1 Research supported by NSF Grant DMS-8602672