arXiv subjects

Immanuel Stampfli

Publications and source records attributed to Immanuel Stampfli.

8 recordsLinked to original sources

arXiv2017

On Maximal Subalgebras

Let $\textbf{k}$ be an algebraically closed field. We classify all maximal $\textbf{k}$-subalgebras of any one-dimensional finitely generated $\textbf{k}$-domain. In dimension two, we classify all maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, t^{-1}, y]$. To the authors' knowledge, this is the first such classification result for an algebra of dimension $> 1$. In the course of this study, we classify also all maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, y]$ that contain a coordinate. Furthermore, we give examples of maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, y]$ that do not contain a coordinate.

math.AC↗

arXiv2016

Algebraic embeddings of $\mathbb{C}$ into $\textrm{SL}_n(\mathbb{C})$

We prove that any two algebraic embeddings of $\mathbb{C}$ into $\textrm{SL}_n(\mathbb{C})$ are the same up to an algebraic automorphism of $\textrm{SL}_n(\mathbb{C})$, provided that $n$ is at least $3$. Moreover, we prove that two algebraic embeddings of $\mathbb{C}$ into $\textrm{SL}_2(\mathbb{C})$ are the same up to a holomorphic automorphism of $\textrm{SL}_2(\mathbb{C})$.

math.AG↗

arXiv2016

On the Topologies on ind-Varieties and related Irreducibility Questions

In the literature there are two ways of endowing an affine ind-variety with a topology. One possibility is due to Shafarevich and the other to Kambayashi. In this paper we specify a large class of affine ind-varieties where these two topologies differ. We give an example of an affine ind-variety that is reducible with respect to Shafarevich's topology, but irreducible with respect to Kambayashi's topology. Moreover, we give a counter-example of a supposed irreducibility criterion of Shafarevich which is different from a counter-example given by Homma. We finish the paper with an irreducibility criterion similar to the one given by Shafarevich.

math.AG↗

arXiv2016

Automorphisms of $\mathbb{C}^3$ Commuting with a $\mathbb{C}^+$-Action

Let $ρ$ be an algebraic action of the additive group $\mathbb{C}^+$ on the three-dimensional affine space $\mathbb{C}^3$. We describe the group $\textrm{Cent}(ρ)$ of polynomial automorphisms of $\mathbb{C}^3$ that commute with $ρ$. A particular emphasis lies in the description of the automorphisms in $\textrm{Cent}(ρ)$ coming from algebraic $\mathbb{C}^+$-actions. As an application we prove that the automorphisms in $\textrm{Cent}(ρ)$ that are the identity on the algebraic quotient of $ρ$ form a characteristic subgroup of $\textrm{Cent}(ρ)$.

math.AG↗

arXiv2016

Automorphisms of the plane preserving a curve

We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the groups of positive dimension occuring is also given in the case where the curve is geometrically irreducible and the field is perfect.

math.AG↗

arXiv2016

A note on Automorphisms of the Affine Cremona Group

Let $\mathcal{G}$ be an ind-group and let $\mathcal{U} \subseteq \mathcal{G}$ be a unipotent ind-subgroup. We prove that an abstract group automorphism $θ\colon \mathcal{G} \to \mathcal{G}$ maps $\mathcal{U}$ isomorphically onto a unipotent ind-subgroup of $\mathcal{G}$, provided that $θ$ fixes a closed torus $T \subseteq \mathcal{G}$, which normalizes $\mathcal{U}$ and the action of $T$ on $\mathcal{U}$ by conjugation fixes only the neutral element. As an application we generalize a result by Hanspeter Kraft and the author as follows: If an abstract group automorphism of the affine Cremona group $\mathcal{G}_3$ in dimension 3 fixes the subgroup of tame automorphisms $T{\mathcal G}_3$, then it also fixes a whole family of non-tame automorphisms (including the Nagata automorphism).

math.AG↗

arXiv2016

On Automorphisms of the Affine Cremona Group

We show that every automorphism of the group $\mathcal{G}_n:= \textrm{Aut}(\mathbb{A}^n)$ of polynomial automorphisms of complex affine $n$-space $\mathbb{A}^n=\mathbb{C}^n$ is inner up to field automorphisms when restricted to the subgroup $T \mathcal{G}_n$ of tame automorphisms. This generalizes a result of \textsc{Julie Deserti} who proved this in dimension $n=2$ where all automorphisms are tame: $T \mathcal{G}_2 = \mathcal{G}_2$.

math.AG↗

arXiv2014

Holomorphically Equivalent Algebraic Embeddings

We prove that two algebraic embeddings of a smooth variety $X$ in $\mathbb{C}^m$ are the same up to a holomorphic coordinate change, provided that $2 \dim X + 1$ is smaller than or equal to $m$. This improves an algebraic result of Nori and Srinivas. For the proof we extend a technique of Kaliman using generic linear projections of $\mathbb{C}^m$.

math.AG↗