Abstract: Cutting-off machine for the transversal cutting of logs of paper material, comprising: a structure (SC) on which are moved the logs; a cutting unit (CU) with a blade (2); a grinding unit for sharpening the blade; a device for positioning the grinding wheels with respect to the blade (2); wherein the positioning device comprises a primary carriage (4) and two secondary carriages (42, 43) driven by respective actuators; there is an optical sensor (100) that detects a cutting edge (200) of the blade (2); in a phase of operative positioning of the grinding wheels which involves sharpening the blade following the contact of the grinding wheels with the blade, the abrasive side of the grinding wheels is pushed against the blade with a thrust having a predetermined value.