Variedades Involutivas e Aplicações Enumerativas

Abstract

In this work are introduced the concepts of involutive affine and projective varieties. Taking into account that every projective variety in P2n-1 has dimension greater than or equal to n-1 and that every hypersurface is involutive, we put our focus on the study of involutive curves in P3, noting that a curve in P3 contained in a plane will be involutive if and only if it is a union of lines passing through the point associated to the suported by plane the correspondence between points and planes determined by the standard symplectic form in P3. We started using the Poisson bracelete invariance of the definition ideal of a varity criterion to determine the involutive lines and conics in P3. Moreover, we exhibit a family of involutive twisted curves. Finally, having in mind that the parameters spaces for involutive lines and conics are 3 and 5 dimensional spaces, respectively. We find how many involutive lines and conic meet 3 and 5 given lines in P3, respectively.