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Zero-cycles on del Pezzo surfaces. Variations on a theme by Daniel Coray. (Zéro-cycles sur les surfaces de Del Pezzo (Variations sur un thème de Daniel Coray).) (French. English summary) Zbl 1479.14031

Summary: 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 D. Coray in his thesis [Arithmetic on cubic surfaces. Cambridge: Trinity College (PhD Thesis) (1974)]. Combining Bertini theorems and large fields, we introduce some flexibility 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 [Compos. Math. 39, 301–332 (1979; Zbl 0386.14003)]). 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.

MSC:

14G05 Rational points
14J26 Rational and ruled surfaces
14C25 Algebraic cycles

Citations:

Zbl 0386.14003
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References:

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