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Gene Freudenburg

Publications and source records attributed to Gene Freudenburg.

18 recordsLinked to original sources
arXiv2025

Presentations, embeddings and automorphisms of homogeneous spaces for SL(2,C)

For an algebraically closed field $k$ of characteristic zero and a linear algebraic $k$-group $G$, it is well known that every affine $G$-variety admits a $G$-equivariant closed embedding into a finite-dimensional $G$-module. Such an embedding is a presentation of the $G$-variety, and a minimal presentation is one for which the dimension the $G$-module is minimal. The problem of finding a minimal presentation generalizes the problem of determining whether a group action on affine space is linearizable. We give a minimal presentation for each homogeneous space for $SL_2(k)$. This constitutes the paper's main work. Of particular interest are the surfaces $Y=SL_2(k)/T$ and $X=SL_2(k)/N$ where $T$ is the one-dimensional torus and $N$ is its normalizer. We show that the minimal presentation of $X$ has dimension 5, the embedding dimension of $X$ is 4, and there does not exist a closed $SL_2$-equivariant embedding of $X$ in $A_k^4$. Thus, the $SL_2$-action on $X$ is absolutely nonextendable to $A_k^4$. We give two other examples of surfaces with absolutely nonextendable group actions. In addition, $X$ is noncancelative, that is, there exists a surface $Z$ such that $X\times A_k^1\cong_k Z\times A_k^1$ and $X\not\cong_kZ$. Finally, we settle the long-standing open question of whether there exist inequivalent closed embeddings of $Y$ in $A_k^3$ by constructing inequivalent embeddings.

math.AG↗
arXiv2025

A Criterion for the Algebraic Density Property of Affine $SL_2$-Manifolds

Let $B$ be an affine $k$-domain which admits a nontrivial fundamental pair $(D,U)$ of locally nilpotent derivations, i.e., if $E=[D,U]$ then $(D,U,E)$ is an $\mathfrak{sl}_2$-triple. We prove an algebraic criterion, characterizing under which conditions the fundamental pair $(D,U)$ resp. the triple $(D,U,E)$ is compatible in a technical sense that allows us to construct many vector fields on the spectrum of $B$ from the complete ones. This criterion enables us to prove the algebraic density property for the following widely studied classes of $\mathrm{SL}_2$-varieties arising in physics: Classical Calogero--Moser spaces, Calogero--Moser spaces with "inner degrees of freedom'' and a smooth cyclic quiver variety.

math.AC↗
arXiv2024

A note on smooth $SL_2$-surfaces

Working over a field $k$ of characteristic zero, we study the ring $\mathfrak{R}=\mathfrak{D}^{\mathbb{Z}_2}$ where $\mathfrak{D}=k[x_0,x_1,x_2]/(2x_0x_2-x_1^2-1)$ and $\mathbb{Z}_2$ acts by $x_i\to -x_i$. $\mathfrak{D}$ admits an algebraic $SL_2(k)$-action which restricts to $\mathfrak{R}$. Our results include the following. (1) If $k$ is algebraically closed, the smooth $SL_2$-surface $X={\rm Spec}(\mathfrak{R})$ admits an algebraic embedding in $\mathbb{A}_k^4$, and for any such embedding the $SL_2(k)$-action on $X$ does not extend to $\mathbb{A}_k^4$. In addition, there is no algebraic embedding of $X$ in $\mathbb{A}_k^3$. (2) The automorphism group ${\rm Aut}_k(\mathfrak{R})$ acts transitively on the set of irreducible locally nilpotent derivations of $\mathfrak{R}$. (3) Every automorphism of $\mathfrak{R}$ extends to $\mathfrak{D}$, and ${\rm Aut}_k(\mathfrak{R})=PSL_2(k)\ast_HT$ where $T$ is its triangular subgroup. (4) $\mathfrak{R}$ is non-cancellative, i.e., there exists a ring $\mathfrak{S}$ such that $\mathfrak{R}^{[1]}\cong_k\mathfrak{S}^{[1]}$ but $\mathfrak{R}\not\cong_k\mathfrak{S}$. In order to distinguish $\mathfrak{R}$ from $\mathfrak{S}$, we calculate the plinth invariant for $\mathfrak{R}$.

math.AG↗
arXiv2024

Automorphisms of the ring of invariants of the binary quintic representation of SL2

Let k^[6] denote a polynomial ring in 6 variables over an algebraically closed field k of characteristic zero and consider the action of SL2(k) on k^[6] induced by the irreducible representation of SL2 of degree 5 (the binary quintic representation). We consider the ring Q = (k^[6])^SL2 of invariant polynomials and show that Aut_k(Q) = u(k), the unit group of k, where Aut_k(Q) is the group of k-algebra automorphisms of Q. Based on this result, we show that the group of SL2-equivariant polynomial automorphisms of k^[6] is isomorphic to u(k).

math.AC↗
arXiv2023

A Criterion for the Density Property of Stein Manifolds

We generalize a criterion for the density property of Stein manifolds. As an application, we give a new, simple proof of the fact that the Danielewski surfaces have the algebraic density property. Furthermore, we have found new examples of Stein manifolds with the density property.

math.CV↗
arXiv2023

Affine and Unirational unique factorial domains with unmixed gradings

This paper studies the class of unique factorial domains $B$ over an algebraically closed field $k$ which are affine or unirational over $k$ and which admit an effective unmixed $\mathbb{Z}^{d-1}$-grading with $B_0=k$, where $d$ is the dimension of $B$. Geometrically, these correspond to factorial affine $k$-varieties with an unmixed torus action of complexity one and trivial invariants. Our main result shows that this class is identical to the class of rings defined by trinomial data, thus generalizing earlier work of Mori, of Ishida, and of Hausen, Herrppich and S\"uss.

math.AG↗
arXiv2021

Actions of $SL_2(k)$ on affine $k$-domains and fundamental pairs

Working over a field $k$ of characteristic zero, this paper studies algebraic actions of $SL_2(k)$ on affine $k$-domains by defining and investigating fundamental pairs of derivations. There are three main results: (1) The Structure Theorem for Fundamental derivations (Theorem 3.4) describes the kernel of a fundamental derivation, together with its degree modules and image ideals. (2) The Classification Theorem (Theorem 4.5) lists all normal affine $SL_2(k)$-surfaces with trivial units, generalizing the classification given by Gizatullin and Popov for complex $SL_2(C)$-surfaces [16]. (3) The Extension Theorem (Theorem 7.6) describes the extension of a fundamental derivation of a $k$-domain $B$ to $B[t]$ by an invariant function. The Classification Theorem is used to describe three-dimensional UFDs which admit a certain kind of $SL_2(k)$-action (Theorem 6.2). This description is used to show that any $SL_2(k)$-action on $A_k^3$ is linearizable, which was proved by Kraft and Popov in the case $k$ is algebraically closed. This description is also used, together with Panyushev's theorem on linearization of $SL_2(k)$-actions on $A_k^4$, to show a cancelation property for threefolds $X$: If $k$ is algebraically closed, $X\times A_k^1\cong A_k^4$ and $X$ admits a notrivial action of $SL_2(k)$, then $X\cong A_k^3$ (Theorem 6.6). The Extension Theorem is used to investigate free $G_a$-actions on $A_k^n$ of the type first constructed by Winkelmann.

math.AG↗
arXiv2019

Smooth factorial affine surfaces of logarithmic Kodaira dimension zero with trivial units

This paper considers the family $\mathscr{S}_0$ of smooth affine factorial surfaces of logarithmic Kodaira dimension 0 with trivial units over an algebraically closed field $k$. Our main result (Theorem 4.1) is that the number of isomorphism classes represented in $\mathscr{S}_0$ is at least countably infinite. This contradicts the earlier classification of Gurjar and Miyanishi [5] which asserted that $\mathscr{S}_0$ has at most two elements up to isomorphism when $k=\mathbb{C}$. Thus, the classification of surfaces in $\mathscr{S}_0$ for the field $\mathbb{C}$, long thought to have been settled, is an open problem.

math.AG↗
arXiv2018

Factorial rational varieties which admit or fail to admit an elliptic $\mathbb{G}_m$-action

Over a field $k$, we study rational UFDs of finite transcendence degree $n$ over $k$. We classify such UFDs $B$ when $n=2$, $k$ is algebraically closed, and $B$ admits a positive $\mathbb{Z}$-grading, showing in particular that $B$ is affine over $k$. We also consider the Russell cubic threefold over $\mathbb{C}$, and the Asanuma threefolds over a field of positive characterstic, showing that these threefolds admit no elliptic $\mathbb{G}_m$-action. Finally, we show that, if $X$ is an affine $k$-variety and $X\times\mathbb{A}^m_k\cong_k\mathbb{A}^{n+m}_k$, then $X\cong_k\mathbb{A}^n_k$ if and only if $X$ admits an elliptic $\mathbb{G}_m$-action.

math.AG↗
arXiv2018

The polar group of a real form of an affine or projective $\mathbb{C}$-variety

A general problem is to classify the real forms of a complex variety up to isomorphism. This paper introduces the polar group of a real form $X$ of a complex variety $Y$ as a tool to distinguish such real forms. This group is an invariant of $X$ which encodes information about the residual divisors in the coordinate ring of $Y$ over the coordinate ring of $X$. We calculate polar groups for various curves, including the real line and the algebraic 1-sphere.

math.AG↗
arXiv2018

On the uniqueness of polynomial embeddings of the real 1-sphere in the plane

This paper considers real forms of closed algebraic $\mathbb{C}^*$-embeddings in $\mathbb{C}^2$. The classification of such embeddings was recently completed by Cassou-Nogues, Koras, Palka and Russell. Based on their classification, this paper shows that, up to an algebraic change of coordinates, there is only one polynomial embedding of the real 1-sphere $\mathbb{S}^1$ in the affine plane $\mathbb{R}^2$.

math.AG↗
arXiv2016

Canonical factorization of the quotient morphism for an affine $\mathbb{G}_a$-variety

Working over a ground field of characteristic zero, this paper studies the quotient morphism $π:X\to Y$ for an affine $\mathbb{G}_a$-variety $X$ with affine quotient $Y$. It is shown that the degree modules associated to the $\mathbb{G}_a$-action give a uniquely determined sequence of dominant $\mathbb{G}_a$-equivariant morphisms, $X=X_r\to X_{r-1}\to\cdots\to X_1\to X_0=Y$, where $X_i$ is an affine $\mathbb{G}_a$-variety and $X_{i+1}\to X_i$ is birational for each $i\ge 1$. This is the canonical factorization of $π$. We give an algorithm for finding the degree modules associated to the given $\mathbb{G}_a$-action, and this yields the canonical factorization of the quotient morphism. The algorithm is applied to compute the canonical factorization for several examples, including the homogeneous $(2,5)$-action on $\mathbb{A}^3$. By a fundamental result of Kaliman and Zaidenberg, any birational morphism of affine varieties is an affine modification, and each mapping in these examples is presented as a $\mathbb{G}_a$-equivariant affine modification.

math.AG↗
arXiv2015

Cable algebras and rings of $G_a$-invariants

For a field $k$, the ring of invariants of an action of the unipotent $k$-group $G_a$ on an affine $k$-variety is quasi-affine, but not generally affine. Cable algebras are introduced as a framework for studying these invariant rings. It is shown that the ring of invariants for the $G_a$-action on $A^5_k$ constructed by Daigle and Freudenburg is a monogenetic cable algebra. A generating cable is constructed for this ring, and a complete set of relations is given as a prime ideal in the infinite polynomial ring over $k$. In addition, it is shown that the ring of invariants for the well-known $G_a$-action on $A^7_k$ due to Roberts is a cable algebra.

math.AG↗
arXiv2014

An affine version of a theorem of Nagata

Let R be an affine k-domain over the field k. The paper's main result is that, if R admits a non-trivial embedding in a polynomial ring K[s] for some field K containing k, then R can be embedded in a polynomial ring F[t] which extends R algebraically. This theorem can be applied to subrings of a ring which admits a non-zero locally nilpotent derivation. In this way, we obtain a concise new proof of the cancellation theorem for rings of transcendence degree one for fields of characteristic zero.

math.AC↗
arXiv2013

Laurent cancellation for rings of transcendence degree one

If $R$ is an integral domain and $A$ is an $R$-algebra, then $A$ has the {\it Laurent cancellation property over $R$} if $A^{[\pm n]}\cong_RB^{[\pm n]}$ implies $A\cong_RB$ ($n\ge 0$ and $B$ an $R$-algebra). Here, $A^{[\pm n]}$ denotes the ring of Laurent polynomials in $n$ variables over $A$. Our main result (Thm. 4.3) is that, if the transcendence degree of $A$ over $R$ is one, then $A$ has the Laurent cancellation property. The proof uses the characterization of Laurent polynomial rings given in Thm. 3.2.

math.AC↗
arXiv2009

Curves defined by Chebyshev polynomials

Working over a field $\kk$ of characteristic zero, this paper studies line embeddings of the form $ϕ= (T_i,T_j,T_k):\A^1\to\A^3$, where $T_n$ denotes the degree $n$ Chebyshev polynomial of the first kind. In {\it Section 4}, it is shown that (1) $ϕ$ is an embedding if and only if the pairwise greatest common divisor of $i,j,k$ is 1, and (2) for a fixed pair $i,j$ of relatively prime positive integers, the embeddings of the form $(T_i,T_j,T_k)$ represent a finite number of algebraic equivalence classes. {\it Section 2} gives an algebraic definition of the Chebyshev polynomials, where their basic identities are established, and {\it Section 3} studies the plane curves $(T_i,T_j)$. {\it Section 5} establishes the Parity Property for Nodal Curves, and uses this to parametrize the family of alternating $(i,j)$-knots over the real numbers.

math.AG↗