On the Quasi - Ordinary Cuspidal Foliations

i,j cijxjdxi, cij ∈ C. The integrability condition ω ∧ dω = 0 implies that ω1 ∧ dω1 = 0. Let C be the matrix (cij)i,j ∈ Mn×n(C). Writing down explicitly the integrability condition, the coefficient of dxi ∧ dxj ∧ dxk (i < j < k) in ω1 ∧ dω1 is ci(ckj − cjk)− cj(cki − cik) + ck(cji − cij), where ci = ∑n j=1 cijxj . Two cases appear: (1) C is a symmetric matrix. (2) C is not symmetric. So, it exists (j, k) with ckj 6= cjk. In the last case, the polynomials ci, cj, ck are linearly dependent for every i, j, k, and so rk(C) ≤ 2. Moreover, dω1(0) = dω(0) 6= 0, so we are in presence of a Kupka-type phenomenon and, in fact, it exists a biholomorphism f such that fω ∧ η = 0, where η is a form in 2-variables. For bidimensional phenomena, lots of work have been done. We then focus on the symmetric case. A linear change of coordinates changes C in P CP , P invertible, so we can suppose C diagonal and moreover