L. Makar-Limanov计算了由方程$x^{n}z-P(y)=0$定义的$\mathbb{C}^{3}$中曲面的自同构群，其中$n\geq1$和$P(y)$是非零多项式。A. Crachiola的表面定义的方程$x^{n}z-y^{2}-h(x)y=0$，其中$n\geq2$和$h(0)\neq0$，定义在一个任意的基本领域。这里我们考虑由方程$x^{n}z-Q(x,y)=0$定义的更一般的曲面，其中$n\geq2$和$Q(x,y)$是系数在任意基域$k$中的多项式。在这些曲面中，我们刻画了Danielewski曲面，并计算了它们的自同构群。我们研究了这些曲面在仿射三维空间中的闭嵌入。我们表明，在一般情况下，他们的自同构不延伸到周围的空间。最后，我们给出了$\mathbb{C}^{3}$中曲面上的$\mathbb{C}^{*}$ -作用的显式例子，这些作用可以全纯地但不能代数地推广到$\mathbb{C}^{3}$上的$\mathbb{C}^{*}$ -作用.
英文摘要：
L. Makar-Limanov computed the automorphisms groups of surfaces in $\mathbb{C}^{3}$ defined by the equations $x^{n}z-P(y)=0$, where $n\geq1$ and $P(y)$ is a nonzero polynomial. Similar results have been obtained by A. Crachiola for surfaces defined by the equations $x^{n}z-y^{2}-h(x)y=0$, where $n\geq2$ and $h(0)\neq0$, defined over an arbitrary base field. Here we consider the more general surfaces defined by the equations $x^{n}z-Q(x,y)=0$, where $n\geq2$ and $Q(x,y)$ is a polynomial with coefficients in an arbitrary base field $k$. Among these surfaces, we characterize the ones which are Danielewski surfaces and we compute their automorphism groups. We study closed embeddings of these surfaces in affine 3-space. We show that in general their automorphisms do not extend to the ambient space. Finally, we give explicit examples of $\mathbb{C}^{*}$-actions on a surface in $\mathbb{C}^{3}$ which can be extended holomorphically but not algebraically to a $\mathbb{C}^{*}$-action on $\mathbb{C}^{3}$.
A special Danielewski surface is an affine surface which is the total space of a principal $(\mathbb{C},+)$-bundle over an affine line with a multiple origin. Using a fiber product trick introduced by Danielewski, it is known that cylinders over two such surfaces are always isomorphic provided that both bases have the same number of origins. The goal of this note is to give an explicit method to find isomorphisms between cylinders over special Danielewski surfaces. The method is based on the construction of appropriate locally nilpotent derivations.
In this note we show that if the automorphism group of a normal affine surface $S$ is isomorphic to the automorphism group of a Danielewski surface, then $S$ is isomorphic to a Danielewski surface.
We study a two-dimensional family of affine surfaces which are counter-examples to the Cancellation Problem. We describe the Makar-Limanov invariant of these surfaces, determine their isomorphism classes and characterize the automorphisms of these surfaces.
In this note we prove that if $S$ is an affine non-toric $\mathbb{G}_m$-surface of hyperbolic type that admits a $\mathbb{G}_a$-action and $X$ is an affine irreducible variety such that $Aut(X)$ is isomorphic to $Aut(S)$ as an abstract group, then $X$ is a $\mathbb{G}_m$-surface of hyperbolic type. Further, we show that a smooth Danielewski surface $Dp = \{ xy = p(z) \} \subset \mathbb{A}^3$, where $p$ has no multiple roots, is determined by its automorphism group seen as an ind-group in the category of affine irreducible varieties.
We give the classification of all complete algebraic vector fields on Danielewski surfaces (smooth surfaces given by $xy=p(z)$). We use the fact that for each such vector field there exists a certain fibration that is preserved under its flow. In order to get the explicit list of vector fields a classification of regular function with general fiber $\mathbb{C}$ or $\mathbb{C}^*$ is required. In this text we present results about such fibrations on Gizatullin surfaces and we give a precise description of these fibrations for Danielewski surfaces.
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…
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 occu…
We show that the class of the affine line is a zero divisor in the Grothendieck ring of algebraic varieties over complex numbers. The argument is based on the Pfaffian-Grassmannian double mirror correspondence.
In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine commutative algebraic monoids. In this work, we extend this classification to noncommutative monoid structures on normal affine surfaces. We prove that two-dimensional algebraic monoids are toric. We also show how to find all monoid structures on a normal toric surface. Every such structure is induced by a comultiplication formula involving Demazure roots. We also give descriptions of opposite monoids, quotient monoids, and boundary divisors.
Lev A. Borisov has shown that the class of the affine line is a zero divisor in the Grothendieck ring of algebraic varieties over complex numbers. We improve the final formula by removing a factor.
We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n-1 or n on a normal affine variety of dimension n.
An enumerative problem on a variety $V$ is usually solved by reduction to intersection theory in the cohomology of a compactification of $V$. However, if the problem is invariant under a "nice" group action on $V$ (so that $V$ is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of $V$. We call this limit the affine cohomology of $V$ and construct affine characteristic classes of subvarieties of a complex torus, taking values in the affine cohomology of the torus. This allows us to make t…
We study the geometry of the stratification induced by an affine hyperplane arrangement H on the quotient of a complex affine space by the action of a discrete group preserving H. We give conditions ensuring normality or normality in codimension 1 of strata. As an application, we provide the list of those categorical quotients of closures of Jordan classes and of sheets in all complex simple algebraic groups that are normal. In the simply connected case, we show that normality of such a quotient is equivalent to its smoothness.
Affine Deligne-Lusztig varieties can be thought of as affine analogs of classical Deligne-Lusztig varieties, or Frobenius-twisted analogs of Schubert varieties. We provide a method for proving a non-emptiness statement for affine Deligne-Lusztig varieties inside the affine flag variety associated to affine Weyl group elements satisfying a certain length additivity hypothesis. In particular, we prove that non-emptiness holds whenever it is conjectured to do so for alcoves in the shrunken dominant Weyl chamber, providing a partial converse to the emptiness results of Goertz, Haines, Kottwitz, an…
Class polynomials attached to affine Hecke algebras were first introduced by X.~He in \cite{He1}. They play an important role in the study of affine Deligne-Lusztig varieties. Motivated by \cite{He2}, we compute the class polynomials attached to an affine Hecke algebra of type (twisted) $\widetilde{A}_2$. Using these class polynomials we prove a conjecture of Görtz-Haines-Kottwitz-Reuman for the general linear group, unitary group and division algebra of semisimple rank 2. Furthermore, we discuss some interesting patterns on affine Deligne-Lusztig varieties.
