The effective monoids of some monoidal transformations of Hirzebruch surfaces Brenda Leticia De La Rosa Navarro Facultad de Ciencias Universidad Autónoma de Baja California Ensenada, Baja California, Mexico [email protected] In this work, we study families of smooth projective rational surfaces with only a finite number of (−1)-curves and (−2)-curves. These surfaces are obtained by blowing-up points in Hirzebruch surfaces. We also compute the number of (−1)-curves and (−2)-curves for a special case. The base field is assumed to be algebraically closed of any characteristic.