Let
be an algebraically closed field. We prove that a polynomial
-derivation D in two variables is locally nilpotent if and only if the subgroup of polynomial
-automorphisms which commute with D ...admits elements whose degree is arbitrary big.
We consider differential rings of the form (kx,y,D), where k is an algebraically closed field of characteristic zero and D:kx,y→kx,y is a k-derivation. We study the Automorphism Group of such a ring ...and give criteria for deciding whether that group is an algebraic group. In most cases, from that study we deduce a primary classification of this type of differential ring up to conjugation with a polynomial automorphism.
We consider Cremona transformations ϕ:P4⤏P4 which factorize through projections of a smooth complete intersection of quadrics in P7. We prove there are three types of such transformations according ...to the relative position of the centers of projection. Moreover, in order to fix one of these three types, we give a geometric characterization of Cremona transformations of Pn which act birationally on the set of hyperplanes passing through a point.
We consider Cremona transformations
ϕ
:
P
4
⤏
P
4
which factorize through projections of a smooth complete intersection of quadrics in
P
7
. We prove there are three types of such transformations ...according to the relative position of the centers of projection. Moreover, in order to fix one of these three types, we give a geometric characterization of Cremona transformations of
P
n
which act birationally on the set of hyperplanes passing through a point.
We introduce a general notion of solution for a Noetherian differential k-algebra and study its relationship with simplicity, where k is an algebraically closed field; then we analyze conditions ...under which such solutions may exist and be unique, with special emphasis in the cases of k-algebras of finite type and formal series rings over k. Using that notion we generalize a criterion for simplicity due to Brumatti-Lequain-Levcovitz and give a geometric characterization of that; as an application we give a new proof of a classification theorem for local simplicity due to Hart and obtain a general result for simplicity of formal series rings over k.
This article deals with the study of the birational transformations of the projective complex plane which leave invariant an irreducible algebraic curve. We try to describe the state of the art and ...provide some new results on this subject.
On construit explicitement toutes les transformations de Cremona de \mathbf{P}^{n} qui satisfont à la propriété suivante: il existe P_{1},P_{2}\in\mathbf{P}^{n} tels que les droites par P_{1} sont ...envoyées sur les droites par P_{2}. On caractérise de plusieurs manières ces transformations et pour chaque entier non-négatif d on donne des formules pour la dimension de l'ensemble constitué de celles qui ont degré d. \vskip6pt \noindent \textsc{Abstract}. We construct the Cremona transformations of \mathbf{P}^{n} satisfying the following property: there exist P_{1},P_{2}\in\mathbf{P}^{n} such that the image of all straight lines through P_{1} are straight lines through P_{2}. We characterise these transformations, and for all non-negative integer d we give a formula for the dimension of the set of those whose degree is d.
We consider the subgroup Aut(D) consisting of automorphisms of Kx,y commuting with a derivation D, where K is an algebraically closed field of characteristic 0. We prove that if D is simple (i.e. D ...does not stabilize non-trivial ideals), then Aut(D)=1. In the case where D is of Shamsuddin type, this result was proven by R. Baltazar in 2014. Moreover, we show that general Shamsuddin type derivations D with Aut(D)=1 are simple.
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