Minimal invariant varieties and first integrals for algebraic foliations
| dc.creator | Bonnet, Philippe | |
| dc.date | 2006-02-13 | |
| dc.date.accessioned | 2026-07-07T07:03:22Z | |
| dc.date.available | 2026-07-07T07:03:22Z | |
| dc.description | Let $X$ be an irreducible algebraic variety over $\mathbb{C}$, endowed with an algebraic foliation ${\cal{F}}$. In this paper, we introduce the notion of minimal invariant variety $V({\cal{F}},Y)$ with respect to $({\cal{F}},Y)$, where $Y$ is a subvariety of $X$. If $Y=\{x\}$ is a smooth point where the foliation is regular, its minimal invariant variety is simply the Zariski closure of the leaf passing through $x$. First we prove that for very generic $x$, the varieties $V({\cal{F}},x)$ have the same dimension $p$. Second we generalize a result due to X. Gomez-Mont. More precisely, we prove the existence of a dominant rational map $F:X\to Z$, where $Z$ has dimension $(n-p)$, such that for every very generic $x$, the Zariski closure of $F^{-1}(F(x))$ is one and only one minimal invariant variety of a point. We end up with an example illustrating both results. | |
| dc.description | 15 pages | |
| dc.identifier | https://arxiv.org/abs/math/0602274 | |
| dc.identifier | http://arxiv.org/abs/math/0602274 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/108949 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 13N15; 13N99; 14R99 | |
| dc.title | Minimal invariant varieties and first integrals for algebraic foliations | |
| dc.type | text |