BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Jean-Louis Colliot-Thelene (Université Paris-Sud) DTSTART:20200922T140000Z DTEND:20200922T150000Z DTSTAMP:20260423T035959Z UID:ZAG/51 DESCRIPTION:Title: Zer o-cycles on del Pezzo surfaces\nby Jean-Louis Colliot-Thelene (Univers ité Paris-Sud) as part of ZAG (Zoom Algebraic Geometry) seminar\n\n\nAbst ract\nLet k be an arbitary field of characteristic zero and X be a smooth\ , projective\, geometrically rational surface. Birational classification o f such surfaces (over k) is due to Enriques\, Manin\, Iskovskikh\, Mori. W e are interested in zero-cycles on such surfaces. In 1974\, Daniel Coray s howed that on a smooth cubic surface X with a closed point of degree prim e 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 a voids considerations of special cases in Coray's argument. For smooth cubi c 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 deg ree 1. This completes the proof that for any geometrically rational surfac e 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 le ast N is rationally equivalent to an effective zero-cycle. For smooth cubi c surfaces X without a rational point\, we relate the question whether the re 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.\n LOCATION:https://researchseminars.org/talk/ZAG/51/ END:VEVENT END:VCALENDAR