Timeline for Is each rationally chain connected surface rational?

6 events

when toggle format	what		by	license	comment
Sep 4, 2014 at 6:10	vote	accept	Mikhail Skopenkov
Sep 3, 2014 at 21:55	comment	added	Roberto Pignatelli		I edited my post explaining this point, I hope is ok now. Really, i thik you could avoid solving the indeterminacy, as this is just a finite set (see Beauville's proof of Proposition III.20). Anyway, I solved them to be safe: then you pull-back holomorphic forms, and this gives the injective map we need.
Sep 3, 2014 at 21:50	history	edited	Roberto Pignatelli	CC BY-SA 3.0	Explained the implication "uniruled implies vanishing plurigenera"
Sep 3, 2014 at 17:31	comment	added	Mikhail Skopenkov		Thank you very much, this completes the proof. Indeed, once all plurigenera vanish, the surface is ruled by the Enriques theorem [2,Theorem VI.17]. However, the assertion ``if a surface S is uniruled, all plurigenera vanish'' requires some work for the proof. E.g., we cannot just take a pullback of a pluricanonical section because the dominant map $X\times \mathbb{P}^1\to S$ may not be defined everywhere. Thus we need to start with eliminating indeterminacy using [2,Theorem II.7] and so on. That is why a reference to the assertions is preferable.
Sep 3, 2014 at 15:20	history	edited	Roberto Pignatelli	CC BY-SA 3.0	misprints corrected, removed some useless sentence
Sep 3, 2014 at 13:07	history	answered	Roberto Pignatelli	CC BY-SA 3.0