Poincare problem for divisors invariant by one-dimensional foliations on smooth algebraic variety TeX Export

@article{citeulike:3860174, abstract = {In this paper we consider the question of bounding the degree of an divisor \$D\$ invariant by a \$\F\$ holomorphic foliation, without rational first integral, on smooth algebraic variety \$X\$ in terms of degree of \$\F\$ and some invariants of \$D\$ and \$X\$. Particularly, if \$\F\$ is a foliation, of degree \$d\$, on \$\mathbb{P}\_{\mathbb{C}}^2\$, we show that there exist a number \$\mathscr{G}(d,k)\$, such that if \$\F\$ has an algebraic solution of degree \$k\$ and genus than or equal to \$\mathscr{G}(d,k)\$, then it has a rational first integral of degree \$\leq k\$. Also, if the number of invariants curves is different of \$\frac{(k+2)(k+1)}{2}\$ then exist a number \$\mathcal{M}(d,k)\$ such that if \$k\>\mathcal{M}(d,k),\$ then \$\F\$ admits a rational first integral of degree \$\leq k\$.}, archivePrefix = {arXiv}, author = {Barros, Mauricio}, citeulike-article-id = {3860174}, citeulike-linkout-0 = {http://arxiv.org/abs/0901.0745}, citeulike-linkout-1 = {http://arxiv.org/pdf/0901.0745}, eprint = {0901.0745}, keywords = {foliations, integrability}, month = {Jan}, posted-at = {2009-01-08 14:42:10}, priority = {4}, title = {Poincare problem for divisors invariant by one-dimensional foliations on smooth algebraic variety}, url = {http://arxiv.org/abs/0901.0745}, year = {2009} }

TY - JOUR ID - citeulike:3860174 L3 - citeulike-article-id:3860174 N2 - In this paper we consider the question of bounding the degree of an divisor $D$ invariant by a $\F$ holomorphic foliation, without rational first integral, on smooth algebraic variety $X$ in terms of degree of $\F$ and some invariants of $D$ and $X$. Particularly, if $\F$ is a foliation, of degree $d$, on $\mathbb{P}_{\mathbb{C}}^2$, we show that there exist a number $\mathscr{G}(d,k)$, such that if $\F$ has an algebraic solution of degree $k$ and genus than or equal to $\mathscr{G}(d,k)$, then it has a rational first integral of degree $\leq k$. Also, if the number of invariants curves is different of $\frac{(k+2)(k+1)}{2}$ then exist a number $\mathcal{M}(d,k)$ such that if $k>\mathcal{M}(d,k),$ then $\F$ admits a rational first integral of degree $\leq k$. TI - Poincare problem for divisors invariant by one-dimensional foliations on smooth algebraic variety KW - foliations KW - integrability AU - Barros, Mauricio PY - 2009/Jan/07/ UR - http://arxiv.org/abs/0901.0745 ER -