arXiv subjects

Luis Cid

Publications and source records attributed to Luis Cid.

5 recordsLinked to original sources

arXiv2026

On the Tame Isotropy Group of Locally Finite Derivations of K[X,Y]

Let K be an algebraically closed field of characteristic zero. We study the tame isotropy group Tame_D(K[X,Y]) of locally finite derivations of the polynomial ring K[X,Y], using Van den Essen's classification up to conjugation. For each normal form, we explicitly determine the corresponding tame isotropy group. We then compare Tame_D(K[X,Y]) with the tame isotropy group of the associated exponential automorphism exp(D), and prove that these groups always coincide. This stands in contrast to the behaviour of the full automorphism group, where such an equality may fail for derivations with a nontrivial semisimple part.

math.AG↗

arXiv2026

On nice $\mathbb{G}_m$-actions arising from locally nilpotent derivations with slice

Let $k$ be an algebraically closed field of characteristic zero and $B$ a finitely generated $k$-domain. Given a locally nilpotent derivation $D$ on $B$ admitting a slice $s$, the derivation $\partial=NsD$ ($N\in\mathbb{Z}\setminus\{0\}$) is semisimple and defines a regular $\mathbb{G}_m$-action on $\mathrm{Spec}(B)$. We show that this derivation provides a new explicit description of the $\mathbb{G}_m$-action introduced by Freudenburg in terms of the infinitesimal generator $\partial=NsD$. In the nice case ($D^2(x_i)=0$ for all generators), we prove a linearizability criterion: the associated $\mathbb{G}_m$-action is linearizable if and only if $D$ is automorphically conjugate to $\partial/\partial x_n$ and the slice becomes affine-linear in the distinguished variable; moreover, this criterion is independent of the choice of slice.

math.AG↗

arXiv2026

On isotropy group of locally finite derivations on $\mathbb{K}[X,Y]$

In this paper, we study the isotropy groups of locally finite derivations of the polynomial ring $\mathbb{K}[X,Y]$, using Van den Essen's classification of locally finite derivations in two variables. We compare the isotropy group of a locally finite derivation with that of its associated exponential automorphism, showing that they coincide in the locally nilpotent case, whereas they may differ when the semisimple part is nontrivial. We also prove that every nonzero locally finite derivation has a nontrivial isotropy group.

math.AC↗

arXiv2026

Semisimple derivations, rational slice and kernels over affine domains

Let k be an algebraically closed field of characteristic zero and let B be a finitely generated k-domain. We study semisimple derivations on B, with special emphasis on those whose eigenvalues are integers. For such derivations, after passing to the field of fractions and choosing a rational slice s with D(s) = s, we describe the kernel of D explicitly in terms of semi-invariant generators. We also obtain descriptions of the kernel on suitable localizations of B and on B itself by intersection. Several basic properties of semisimple derivations and their behavior under conjugation are also discussed

math.AG↗

arXiv2022

On rational multiplicative group actions

We establish a one-to-one correspondence between rational multiplicative group actions on an algebraic variety $X$ and derivations $\partial\colon K_X\to K_X$ of the field of fractions $K_X$ of $X$ satisfying that there exists a generating set $\{a_i\}_{i\in I}$ of $K_X$ as a field such that $\partial(a_i)=\lambda_i a_i$ with $\lambda_i \in \mathbb{Z}$ for all $i\in I$. We call such derivations rational semisimple. Furthermore, we also prove the existence of a rational slice for every rational semisimple derivation, i.e., an element $s\in K_X$ such that $\partial(s)=s$. By analogy with the case of additive group actions case, we prove that $K_X\simeq K_X^{\mathbb{G}_m}(s)$ and that under this isomorphism the derivation $\partial$ is given by $\partial=s\frac{d}{ds}$. Here, $K_X^{\mathbb{G}_m}$ is the field of invariant of the $\mathbb{G}_m$-action.

math.AG↗