arXiv subjects
Publications and source records attributed to Luis Loeza.
We classify algebraic curves in $\mathbb{CP}^{n}$ ($n \geq 2$) that are invariant under an infinite subgroup of $\operatorname{PGL}(n+1,\mathbb{C})$. In particular, we prove that any irreducible, non-degenerate, one-dimensional algebraic set in $\mathbb{CP}^{n}$ invariant under an infinite subgroup of $\operatorname{Aut}(\mathbb{CP}^{n})$ must be projectively equivalent to a monomial curve.
In this note, we show that the exceptional algebraic set of an infinite discrete group in $PSL(3,\Bbb{C})$ should be a finite union of complex lines, copies of the Veronese curve or copies of the cubic $xy^2-z^3$.
In this article we characterize the complex hyperbolic groups that leave invariant a copy of the Veronese curve in $\Bbb{P}^2_{\Bbb{C}}$. As a corollary we get that every discrete compact surface group in $\PO^+(2,1)$ admits a deformation in $\PSL(3,\Bbb{C})$ with a non-empty region of discontinuity which is not conjugate to a complex hyperbolic subgroup. This provides a way to construct new examples of Kleinian groups acting on $\Bbb{P}^2_\Bbb{C}$.
In this article we provide a classification of the projective transformations in $PSL(n+1,\Bbb{C})$ considered as automorphisms of the complex projective space $\Bbb{P}^n$. Our classification is an interplay between algebra and dynamics, which just as in the case of isometries of CAT(0)-spaces, can be given by means of tree three types, namely: elliptic, parabolic and loxodromic. We carefully describe the dynamic in each case, more precisely we determine the corresponding Kulkarni's limit set, the equicontinuity region, the discontinuity region and in some cases we provide families of maximal regions where the respective cyclic group acts properly discontinuously. Also we provide, in each case, some equivalents ways to classify the projective transformations.
In this article we show that Bers' simultaneous uniformization as well as the Köebe's retrosection theorem are not longer true for discrete groups of projective transformations acting on the complex projective space.