A cutting system for cutting a workpiece by laser emission relies upon local cooling of the workpiece at the point at which cutting starts and ends. Optionally, local cooling can also be provided at a cross point where two cutting lines intersect. Relying upon the difference in thermal stress between the laser beam incidence point and the surrounding cooled portion, an initial crack can be generated in a specified direction, whereby the initial crack is propagated to form the desired cutting line. In one embodiment, the cooling is provided by a low temperature solid such as a Peltier cooling plate and the temperature of the workpiece near a point of incidence of the laser beam is determined. The determination can be made by measuring the temperature of a plasma generated near the point of incidence, or by directly measuring the temperature of the portion of the workpiece near the point of incidence.