On the algebraic hypersurfaces invariant by weighted projective foliations

Abstract

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.