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Isac Hedén

Publications and source records attributed to Isac Hedén.

9 recordsLinked to original sources
arXiv2026

The derived length of subgroups of the Cremona group

We prove that the maximal derived length of a solvable subgroup of the plane Cremona group over $\mathbb C$ is equal to $5$. We also show that the maximal derived length of a finite solvable subgroup is $4$, and give a short proof of the known sharp bound $5$ for bounded solvable subgroups. A major ingredient is a centraliser theorem: the centraliser of a finite solvable subgroup of derived length $4$ contains no loxodromic element. The proof relies on Blanc's classification of maximal algebraic subgroups and then uses equivariant Sarkisov theory, orbit estimates, superrigidity, and fixed curves of genus at least two.

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arXiv2022

Borel subgroups of the plane Cremona group

It is well known that all Borel subgroups of a linear algebraic group are conjugate. This result also holds for the automorphism group ${\mathrm{Aut}} (\mathbb A^2)$ of the affine plane \cite{BerestEshmatovEshmatov2016} (see also \cite{FurterPoloni2018}). In this paper, we describe all Borel subgroups of the complex Cremona group ${{\rm Bir}({\mathbb P}^2)}$ up to conjugation, proving in particular that they are not necessarily conjugate. More precisely, we prove that ${{\rm Bir}({\mathbb P}^2)}$ admits Borel subgroups of any rank $r \in \{ 0,1,2 \}$ and that all Borel subgroups of rank $r \in \{ 1,2 \}$ are conjugate. In rank $0$, there is a $1-1$ correspondence between conjugacy classes of Borel subgroups of rank $0$ and hyperelliptic curves of genus $g \geq 1$. Hence, the conjugacy class of a rank $0$ Borel subgroup admits two invariants: a discrete one, the genus $g$, and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus $g$. This latter space is of dimension $2g-1$.

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arXiv2019

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. 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.

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arXiv2017

Equivariant extensions of Ga-torsors over punctured surfaces

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 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 Ga-actions, whose algebraic quotient morphisms $π$ : 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.

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arXiv2017

The decomposition groups of plane conics and plane rational cubics

The decomposition group of an irreducible plane curve $X\subset\mathbb P^2$ is the subgroup $\mathrm{Dec}(X)\subset\mathrm{Bir}(\mathbb P^2)$ of birational maps which restrict to a birational map of $X$. We show that $\mathrm{Dec}(X)$ is generated by its elements of degree $\leq2$ when $X$ is either a conic or rational cubic curve.

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arXiv2016

The decomposition group of a line in the plane

We show that the decomposition group of a line $L$ in the plane, i.e. the subgroup of plane birational transformations that send $L$ to itself birationally, is generated by its elements of degree 1 and one element of degree 2, and that it does not decompose as a non-trivial amalgamated product.

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arXiv2015

Affine extensions of principal additive bundles over a punctured surface

The aim of this article is to make a first step towards the classification of complex normal affine $\mathbb G_a$-threefolds $X$. We consider the case where the restriction of the quotient morphism $π\colon X\to S$ to $π^{-1}(S_*)$, where $S_*$ denotes the complement of some regular closed point in $S$, is a principal $\mathbb G_a$-bundle. The variety $\mathrm{SL}_2$ will be of special interest and a source of many examples. It has a natural right $\mathbb G_a$-action such that the quotient morphism $\mathrm{SL}_2\to\mathbb A^2$ restricts to a principal $\mathbb G_a$-bundle over the punctured plane $\mathbb A^2_*$.

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arXiv2015

The group of Cremona transformations generated by linear maps and the standard involution

This article studies the group generated by automorphisms of the projective space of dimension $n$ and by the standard birational involution of degree $n$. Every element of this group only contracts rational hypersurfaces, but in odd dimension, there are simple elements having this property which do not belong to the group. Geometric properties of the elements of the group are given, as well as a description of its intersection with monomial transformations.

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