Sobre o grau de componentes dos espaços das folheações holomorfas de codimensão um em CPn

Abstract

Two of the main discrete invariants of a projective variety are its dimension and degree. The spaces of holomorphic foliations of codimension one and degree d in CPn, n 3 are subschemes of the projective space CP (H0(Pn; 1(d + 1))) defined by the equations ofconditon of integrability, w ^ dw = 0. We determine in this thesis the degrees of certain components of the spaces of holomorphicfoliations of codimension one in CPn, n 3. For each integer r 1, letR(2; 2r +1) denote the set of foliations induced by 1-forms of type 2FdG(2r +1)GdF, where F;G denote homogeneous polinomials of degrees 2; 2r + 1. X. Gomez-Mont e A. Lins Neto proved in [2] that R(2; 2r + 1) is an irreducible component of the space of holomorphic foliations of degree 2r + 1. After, J. V. Pereira, F. Cukierman and I. Vainsencher proved in [5] that it is a rational and generically reduced component. They found the degree of that component for r = 1; n 5 and conjectured a few more in higherdimensions.Our main result gives a closed formula for the degree of the component R(2; 2r + 1) for r 1 in arbitrary dimension n 2, to wit