On the algebraic hypersurfaces invariant by weighted projective foliations

TL;DR

This paper investigates algebraic hypersurfaces invariant under foliations in weighted projective spaces, extending classical results to these generalized settings, including singularity counts, Poincaré problem, and invariant hypersurface enumeration.

Contribution

It generalizes known results for projective spaces to weighted projective spaces, addressing invariant hypersurfaces, singularities, and foliation properties.

Findings

Counted singularities with multiplicities on invariant hypersurfaces

Addressed the Poincaré problem in weighted projective planes

Estimated the number of invariant hypersurfaces of fixed degree

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.