Affine ind-varieties are infinite dimensional generalizations of algebraic varieties which appear naturally in many different contexts, in particular in the study of automorphism groups of affine spaces. In this article we introduce and develop the b
asic algebraic theory of topologically integrable derivations of complete topological rings. We establish a bijective algebro-geometric correspondence between additive group actions on affine ind-varieties and topologically integrable derivations of their coordinate pro-rings which extends the classical fruitful correspondence between additive group actions on affine varieties and locally nilpotent derivations of their coordinate rings.
We give a general structure theorem for affine A 1-fibrations on smooth quasi-projective surfaces. As an application, we show that every smooth A 1-fibered affine surface non-isomorphic to the total space of a line bundle over a smooth affine curve f
ails the Zariski Cancellation Problem. The present note is an expanded version of a talk given at the Kinosaki Algebraic Geometry Symposium in October 2019.
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. We illustrate these properties on several examples which played important roles in the development of the algebraic theory of locally nilpotent derivations and give some applications to the construction of families of affine threefolds with Ga-actions.
We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group $mathbb{S}^1$ up to equivariant isomorphisms. As an application, we show that every compact diffe
rentiable surface endowed with an action of the circle $S^1$ admits a unique smooth rational real quasi-projective model up to $mathbb{S}^1$-equivariant birational diffeomorphism.
Every deformed Koras-Russell threefold of the first kind $Y = left{ x^{n}z=y^{m}-t^{r} + xh(x,y,t)right}$ in $mathbb{A}^{4}$ is the algebraic quotient of proper Zariski locally trivial $mathbb{G}_a$-action on $mathrm{SL}_2 times mathbb{A}^1$.
We construct smooth rational real algebraic varieties of every dimension $ge$ 4 which admit infinitely many pairwise non-isomorphic real forms.
Motivated by the general question of existence of open A1-cylinders in higher dimensional pro-jective varieties, we consider the case of Mori Fiber Spaces of relative dimension three, whose general closed fibers are isomorphic to the quintic del Pezz
o threefold V5 , the smooth Fano threefold of index two and degree five. We show that the total spaces of these Mori Fiber Spaces always contain relative A2-cylinders, and we characterize those admitting relative A3-cylinders in terms of the existence of certain special lines in their generic fibers.
An algebraic variety is called $mathbb{A}^{1}$-cylindrical if it contains an $mathbb{A}^{1}$-cylinder, i.e. a Zariski open subset of the form $Ztimesmathbb{A}^{1}$ for some algebraic variety Z. We show that the generic fiber of a family $f:Xrightarro
w S$ of normal $mathbb{A}^{1}$-cylindrical varieties becomes $mathbb{A}^{1}$-cylindrical after a finite extension of the base. Our second result is a criterion for existence of an $mathbb{A}^{1}$-cylinder in X which we derive from a careful inspection of a relative Minimal Model Program ran from a suitable smooth relative projective model of X over S.
Motivated by the study of the structure of algebraic actions the additive group on affine threefolds X, we consider a special class of such varieties whose algebraic quotient morphisms X $rightarrow$ X//Ga restrict to principal homogeneous bundles ov
er 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 Ga-actions, whose algebraic quotient morphisms $pi$ : X $rightarrow$ X//Ga are surjective with only isolated degenerate fibers, all isomorphic to the affine plane A 2 when equipped with their reduced structures.
We give a criterion for a flat fibration with affine plane fibers over a smooth scheme defined over a field of characteristic zero to be a Zariski locally trivial $mathbb{A}^2$-bundle. An application is a positive answer to a version of the Dolgachev
-Weisfeiler Conjecture for such fibrations: a flat fibration $mathbb{A}^m$ $rightarrow$ $mathbb{A}^n$ with all fibers isomorphic to $mathbb{A}^2$ is the trivial $mathbb{A}^2$-bundle.
