A rational map ϕ:P_k^m--->P_kⁿ is defined by homogeneous polynomials of a common degree d. In [4, 9], the authors have established some bounds in terms of d for the number of (m-1)-dimensional fibers of ϕ. An interesting question is whether the number of points in P_k³ with one-dimensional fibers under a rational map ϕ:P_k²--->P_k³ can be arbitrarily large.  In this paper, we show that the number of one-dimensional fibers can reach the bound 2d-2, thereby proving the sharpness of previously known estimates.