Cutting machine for the transversal cutting of paper material logs, comprising: - a structure on which are moved the logs to be transversely cut in order to obtain rolls of shorter length; - a cutting unit arranged at a predetermined point of said structure and comprising a support plate for a blade that is removably connectable to a respective rotary actuator able to determine the rotation of the same blade about its own axis (x-x) at a predetermined speed, said plate being, in turn, constrained to a further actuator which drags it into rotation with a predetermined angular speed about an axis parallel to the rotation axis (x-x) of the blade, the blade being arranged along a lying plane perpendicular to said rotation axis (x-x) at a predetermined position in the cutting unit; - at least one sharpening unit with two grinding wheels suitably arranged to sharpen the blade on opposite sides with respect to said plane and provided with an abrasive side, said grinding wheels being circular wheels of predetermined radius; - a positioning device for positioning said grinding wheels with respect to the blade arranged on each sharpening unit, by means of which each grinding wheel is placed in a position of contact with the blade in a sharpening step of the latter starting from an initial inoperative position; characterized in that - said positioning device comprises a primary carriage (40, 41; 400) which can be moved along a primary direction radially with respect to the blade, starting from an initial waiting position, by means of one or more primary actuators (MO, Ml), and two secondary carriages (42, 43) each of which is supported by the primary carriage (40, 41; 400) and can be moved along a secondary direction parallel to the rotation axis of the blade by means of two secondary actuators (M2, M3); and the machine is controlled such that: - in a first step of positioning the grinding wheels, said one or more actuators (MO, Ml) moving the primary carriage (40, 41; 400) are controlled by a sensor which detects the radius of the blade and stops the primary carriage along the primary direction when the grinding wheels are arranged with the respective axes, with respect to the axis of the blade, at a distance (h) equal to the radius of the blade increased by the radius of the grinding wheels and decreased by a predetermined value (b); - in a second phase of positioning the grinding wheels, the secondary carriages (42, 43) are controlled so as to bring the abrasive side of the wheels into contact with the blade. - a structure on which are moved the logs to be transversely cut in order to obtain rolls of shorter length; - a cutting unit arranged at a predetermined point of said structure and comprising a support plate for a blade that is removably connectable to a respective rotary actuator able to determine the rotation of the same blade about its own axis (x-x) at a predetermined speed, said plate being, in turn, constrained to a further actuator which drags it into rotation with a predetermined angular speed about an axis parallel to the rotation axis (x-x) of the blade, the blade being arranged along a lying plane perpendicular to said rotation axis (x-x) at a predetermined position in the cutting unit; - at least one sharpening unit with two grinding wheels suitably arranged to sharpen the blade on opposite sides with respect to said plane and provided with an abrasive side, said grinding wheels being circular wheels of predetermined radius; - a positioning device for positioning said grinding wheels with respect to the blade arranged on each sharpening unit, by means of which each grinding wheel is placed in a position of contact with the blade in a sharpening step of the latter starting from an initial inoperative position; characterized in that - said positioning device comprises a primary carriage (40, 41; 400) which can be moved along a primary direction radially with respect to the blade, starting from an initial waiting position, by means of one or more primary actuators (MO, Ml), and two secondary carriages (42, 43) each of which is supported by the primary carriage (40, 41; 400) and can be moved along a secondary direction parallel to the rotation axis of the blade by means of two secondary actuators (M2, M3); - in a first step of positioning the grinding wheels, said one or more actuators (MO, Ml) moving the primary carriage (40, 41; 400) are controlled by a sensor which detects the radius of the blade and stops the primary carriage along the primary direction when the grinding wheels are arranged with the respective axes, with respect to the axis of the blade, at a distance (h) equal to the radius of the blade increased by the radius of the grinding wheels and decreased by a predetermined value (b); - in a second phase of positioning the grinding wheels, the secondary carriages (42, 43) are controlled so as to bring the abrasive side of the wheels into contact with the blade.