Zéro-cycles sur les surfaces de del Pezzo (Variations sur un thème de Daniel Coray)

摘要

Let X be a smooth, projective, geometrically rational surface over a field of characteristic zero. To any such surface one associates two integers N.X/ and M.X/ which are simple functions of the square of the canonical class. We prove: (a) If the gcd of the degrees of closed points on X is 1, then there exist closed points on X the degrees of which are coprime to one another as a whole and are less than or equal to N.X/ . (b) If X has a rational point, then any zero-cycle on X of degree at least equal to M.X/ is rationally equivalent to an effective cycle. Effective zero-cycles of degree less than or equal to M.X/ generate the Chow group of X . Result (a) extends a theorem on cubic surfaces obtained by Daniel Coray in his thesis (1974). Combining Bertini theorems and large fields, we introduce some fiexibility in his method. The results (a) and (b) then follow from a case by case analysis of the various birational equivalence classes of geometrically rational surfaces: del Pezzo surfaces and conic bundle surfaces (the latter type had been handled with D. Coray in 1979). In a last section, 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.