arXiv subjects
Publications and source records attributed to Immanuel van Santen.
Ramanujam's theorem states that any connected finite-dimensional subgroup of the automorphism group $\mathrm{Aut}(X)$ of an irreducible variety $X$ is an algebraic group, in a natural way. In this note, we discuss the notion of dimension and extend Ramanujam's theorem to arbitrary (not necessarily irreducible) varieties.
Let $X$ be a variety of dimension $n$, and let $\mathrm{Aut}(X)$ be its automorphism group. When $X$ is quasi-affine, we prove that a solvable subgroup of $\mathrm{Aut}(X)$ that is generated by an irreducible family of automorphisms containing the identity is an algebraic subgroup. Our main applications concern arbitrary varieties. First, every connected solvable subgroup of $\mathrm{Aut}(X)$ is contained in a Borel subgroup and its derived length is $\leq n+1$. Second, the notion of solvable and unipotent radicals are well defined for any subgroup of $\mathrm{Aut}(X)$. Third, if $X$ is quasi-affine and connected and $\mathcal{B} \subset \mathrm{Aut}(X)$ is a Borel subgroup of derived length $n+1$, then $X$ is isomorphic to the affine $n$-space $\mathbb{A}^n$ and $\mathcal{B}$ is conjugate to the Jonquières subgroup.
The symmetric subrank of homogeneous polynomial is the largest number of terms in a diagonal form to which it can be specialized by a (typically non-invertible) linear variable substitution. Building on earlier work by Derksen-Makam-Zuiddam and Biaggi-Chang-Draisma-Rupniewski for ordinary tensors, we determine the asymptotic behavior of symmetric subrank and symmetric border subrank of degree-d forms as the number of variables tends to infinity. Furthermore, by using results from geometric invariant theory we show that for cubic (resp. quartic) forms the symmetric subrank and symmetric border subrank coincide if the latter is at most three (resp. two).
Suppose $X$ is an irreducible complex variety. We show that when $X$ is ruled, the group of birational transformations $Bir(X)$, as a group, determines $X$ up to birational transformations and automorphisms of the base field. In contrast, we demonstrate that this same property never holds for non-uniruled varieties.
We show that a ring closely related to the Grothendieck ring of varieties has nilpotent elements, provided that the characteristic of the ground field is equal to $11$ or at least $17$.
Let X be an irreducible variety and Bir(X) its group of birational transformations. We show that the group structure of Bir(X) determines whether X is rational and whether X is ruled. Additionally, we prove that any Borel subgroup of Bir(X) has derived length at most twice the dimension of X, with equality occurring if and only if X is rational and the Borel subgroup is standard. We also provide examples of non-standard Borel subgroups of Bir(P^n) and Aut(A^n), thereby resolving conjectures by Popov and Furter-Poloni.
We give a description of the algebraic families of birational transformations of an algebraic variety X. As an application, we show that the morphisms to Bir(X) given by algebraic families satisfy a Chevalley type result and a certain fibre-dimension formula. Moreover, we show that the algebraic subgroups of Bir(X) are exactly the closed finite-dimensional subgroups with finitely many components. We also study algebraic families of birational transformations preserving a fibration. This builds on previous work of Blanc-Furter, Hanamura, and Ramanujam.
Let $X$ be an affine toric variety and let $D(X)$ be the set of weights of all root subgroups. It is known that $D(X)$ together with its embedding into the character group determines $X$ as a toric variety. In this article we prove that $X$ is already determined by the abstract set $D(X)$ together with some additional combinatorial data.
In this article we give two explicit families of automorphisms of degree $\leq 3$ of the affine $3$-space $\mathbb{A}^3$ such that each automorphism of degree $\leq 3$ of $\mathbb{A}^3$ is a member of one of these families up to composition of affine automorphisms at the source and target; this shows in particular that all of them are tame. As an application, we give the list of all dynamical degrees of automorphisms of degree $\leq 3$ of $\mathbb{A}^3$; this is a set of $3$ integers and $9$ quadratic integers. Moreover, we also describe up to compositions with affine automorphisms for $n\geq 1$ all morphisms $\mathbb{A}^3 \to \mathbb{A}^n$ of degree $\leq 3$ with the property that the preimage of every affine hyperplane in $\mathbb{A}^n$ is isomorphic to $\mathbb{A}^2$.
We describe maximal commutative unipotent subgroups of the automorphism group $\mathrm{Aut}(X)$ of an irreducible affine variety $X$. Further we show that a group isomorphism $\mathrm{Aut}(X) \to \mathrm{Aut}(Y)$ maps unipotent elements to unipotent elements, where $Y$ is irreducible and affine. Using this result, we show that the automorphism group detects sphericity and the weight-monoid. As an application, we show that an affine toric variety different from an algebraic torus is determined by its automorphism group among normal irreducible affine varieties and we show that a smooth affine spherical variety different from an algebraic torus is determined by its automorphism group (up to an automorphism of the base field) among smooth irreducible affine varieties.
We describe the Zariski-closure of sets of torsion points in connected algebraic groups. This is a generalization of the Manin-Mumford conjecture for commutative algebraic groups proved by Hindry. He proved that every subset with Zariski-dense torsion points is the finite union of torsion-translates of algebraic subgroups. We formulate and prove an analogous theorem for arbitrary connected algebraic groups. We also define a canonical height on connected algebraic groups that coincides with a Néron-Tate height if $G$ is a (semi-) abelian variety. This motivates a generalization of the Bogomolov conjecture to arbitrary connected algebraic groups defined over a number field. We prove such a generalization as well.
We prove that every smooth affine variety of dimension $d$ embeds into every simple algebraic group of dimension at least $2d+2$. We do this by establishing the existence of embeddings of smooth affine varieties into the total space of certain principal bundles. For the latter we employ and build upon parametric transversality results for flexible affine varieties due to Kaliman. By adapting a Chow-group-based argument due to Bloch, Murthy, and Szpiro, we show that our result is optimal up to a possible improvement of the bound to $2d+1$. In order to study the limits of our embedding method, we use rational homology group calculations of homogeneous spaces and we establish a domination result for rational homology of complex smooth varieties.
We study the possible dynamical degrees of automorphisms of the affine space $\mathbb{A}^n$. In dimension $n=3$, we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalises the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space $\mathbb{A}^n$ for some $n$, and we give the best possible $n$ for quadratic integers, which is either $3$ or $4$.
Let $G$ be a connected reductive algebraic group. In this note we prove that for a quasi-affine $G$-spherical variety the weight monoid is determined by the weights of its non-trivial $\mathbb{G}_a$-actions that are homogeneous with respect to a Borel subgroup of $G$. As an application we get that a smooth affine $G$-spherical variety that is non-isomorphic to a torus is determined by its automorphism group inside the category of smooth affine irreducible varieties.