A cutting machine for the transversal cutting of paper material logs, including: a structure on which are moved the logs; a cutting unit including a blade; at least one sharpening unit with two grinding wheels; a positioning device for positioning the grinding wheels with respect to the blade. The positioning device includes: a primary carriage and two secondary carriages supported by the primary carriage; a sensor for detecting the radius of the blade and configured to stop the primary carriage when the grinding wheels are arranged with the respective axes at a given distance from the axis of the blade. In a second phase of positioning the grinding wheels, the secondary carriages are controlled so as to bring the abrasive side of the wheels into contact with the blade.