Zero-cycles on del Pezzo surfaces

Abstract: Let k be an arbitary field of characteristic zero and X be a smooth, projective, geometrically rational surface. Birational classification of such surfaces (over k) is due to Enriques, Manin, Iskovskikh, Mori. We are interested in zero-cycles on such surfaces. In 1974, Daniel Coray showed that on a smooth cubic surface X with a closed point of degree prime to 3 there exists a closed point of degree 1, 4 or 10. Whether 4 and 10 may be omitted is still an open question. We first show how a combination of generisation, specialisation, Bertini theorems and "large" fields avoids considerations of special cases in Coray's argument. For smooth cubic surfaces X with a rational point, we show that any zero-cycle of degree at least 10 is rationally equivalent to an effective cycle. We establish analogues of these results for del Pezzo surfaces X of degree 2 and of degree 1. This completes the proof that for any geometrically rational surface X with a rational point, there exists an integer N which depends only on the geometry of the surface, such that any zero-cycle of degree at least N is rationally equivalent to an effective zero-cycle. For smooth cubic surfaces X 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.

algebraic geometry

Audience: researchers in the topic

ZAG (Zoom Algebraic Geometry) seminar

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