Search results for "rational maps"
showing 3 items of 3 documents
Geometry of the projectivization of ideals and applications to problems of birationality
2018
In this thesis, we interpret geometrically the torsion of the symmetric algebra of the ideal sheaf I_Z of a scheme Z defined by n+1 equations in an n-dimensional variety. This is equivalent to study the geometry of the projectivization of I_Z. The applications of this point of view concern, in particular, the topic of birational maps of the projective space of dimension 3 for which we construct explicit birational maps that have the same algebraic degree as their inverse, free and nearly-free curves for which we generalise a characterization of free curves by extending the notion of Milnor and Tjurina numbers. We tackle also the topic of homaloidal hypersurfaces, our original motivation, fo…
Invariant Jordan curves of Sierpinski carpet rational maps
2015
In this paper, we prove that if $R\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpi\'nski carpet, then there is an integer $n_0$, such that, for any $n\ge n_0$, there exists an $R^n$-invariant Jordan curve $\Gamma$ containing the postcritical set of $R$.
Bifurcations in the elementary Desboves family
2017
International audience; We give an example of a family of endomorphisms of $\mathbb{P}^2(\mathbb{C})$ whose Julia set depends continuously on the parameter and whose bifurcation locus has non-empty interior.