Fibers of rational maps and Jacobian matrices

Abstract A rational map ϕ : P k m ⇢ P k n is defined by homogeneous polynomials of a common degree d . We establish a linear bound in terms of d for the number of ( m − 1 ) -dimensional fibers of ϕ , by using ideals of minors of the Jacobian matrix. In particular, we answer affirmatively Question 11 in [9] .