Completely reducible hypersurfaces in a pencil (Q944325)

Consider two relatively prime polynomials \(F\) and \(G\) of degree \(d\) in the variables \(x_0, \dots, x_n\) and with coefficients in the field of complex numbers. A natural question to pose is about the number \(k\) of polynomials in the pencil \({\mathcal P}=\{aF+bG: [a:b] \in\mathbb{P}^1\}\) which are products of linear forms. Or geometrically the number of hypersurfaces in a pencil with general irreducible element which are in fact the union of hyperplanes. The paper under review provides upper bounds for this number \(k\). To be precise: if \(k>5\) then \({\mathcal P}\) is a pencil of hyperplanes; if \(k>3\), it is a linear pull-back of a pencil on the projective plane and if \(k>2\) then \(\mathcal{P}\) is the linear pull-back of a pencil on the \(4\)-dimensional linear space. The proof consists of reducing the problem to a question of non-vanishing of a certain determinant \(D\) and then using the theory of foliation on the \(n\)-dimensional projective space and some properties of hyperplane multi-arrangements to get \(D \neq 0\). Some comments on the sharpness of the bounds are also provided.