Zero-cycles on Del Pezzo surfaces

In 1974, D. Coray showed that on a smooth cubic surface with a closed point of degree prime to 3 there exists such a point of degree 1, 4 or 10. We show how a combination of generization, specialisation, Bertini theorems and large fields avoids considerations of special cases in his argument. For del Pezzo surfaces of degree 2, we give an analogue of Coray's result. For smooth cubic surfaces with a rational point, we show that any zero-cycle of degree at least 10 is rationally equivalent to an effective cycle. For smooth cubic surfaces without a rational point, we relate the question whether there exists a degree 3 point which is not on a line to the question whether rational points are dense on a del Pezzo surface of degree 1.