Mauricio Correa | Searcharxiv

arXiv2009

On the algebraic hypersurfaces invariant by weighted projective foliations

In this work we study some problems related with algebraic hypersurfaces invariant by foliations on weighted projective spaces $\mathbb{P}_{\mathbb{C}}(\varpi_0,...,\varpi_n)$ generalizing some results known for $\p$, as for example: the number of singularities, with multiplicities, contained in the invariant quasi-smooth hypersurfaces; Poincare problem on weighted projective plane and the number of the hypersurfaces, of a degree fixed, invariant by a foliation on $\mathbb{P}_{\mathbb{C}}(\varpi_0,...,\varpi_n)$ which does not admit a rational first integral.

math.GT↗

arXiv2009

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

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 $\mathbb{P}_{\mathbb{C}}^2$, whose the number of invariants curves is greater that ${k+2\choose k}$, we show that there exist a number $\mathcal{M}(d,k)$ such that if $k>\mathcal{M}(d,k),$ then $\F$ admits a rational first integral of degree $\leq k$. Moreover, there exist a number $\mathscr{G}(d,k)$, such that if $\F$ has an algebraic solution of degree $k$ and genus smaller than $\mathscr{G}(d,k)$, then it has a rational first integral of degree $\leq k$.

math.GT↗