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Aug 21, 2022 at 9:46 comment added isomorphismes for Hilbert schemes, see also Eisenbud & Harris, "Schemes: the language of modern algebraic geometry" (1992) and Nakajima, "Lectures on Hilbert schemes of points of surfaces" (1991).
Mar 9, 2020 at 16:16 comment added abx A general hyperplane section is a quintic del Pezzo surface. It is well known that it contains 10 lines.
Mar 9, 2020 at 16:05 comment added anon @abx Ok, but I'm not quite sure what you mean by ''a general hyperplane section contains some lines''. I would do it by constructing an incidence correspondence and show that the subvariety in Grass(2,6) parametrising lines on V(5) has dimension 2 (which is consistent with the fact that the Hilbert scheme is P2). Plus my argument in the question has shown that there are finitely many lines passing through a given point. So this solves this question in an elementary way.
Mar 9, 2020 at 14:12 vote accept CommunityBot
Mar 9, 2020 at 10:39 comment added abx Yes. For instance a general hyperplane section contains some lines.
Mar 9, 2020 at 9:05 comment added anon @abx Do you know the lines are infinite without knowing the Hilbert scheme is P2?
Mar 9, 2020 at 5:25 comment added Sasha The Hilbert scheme is the natural way to parameterize subschemes of a fixed scheme (e.g., of the quintic del Pezzo 3-fold) of given type (e.g., of lines). The references are Grothendieck's FGA (original), or "FGA explained" (recent). The Hilbert scheme of lines on the quintic del Pezzo 3-fold is isomorphic to P2.
Mar 9, 2020 at 5:24 comment added abx What do you mean? It is infinite of course, read Sasha's answer.
Mar 8, 2020 at 23:47 comment added anon Is there a way to know the number of lines on V(5)?
Mar 8, 2020 at 23:46 comment added anon I don't know what a Hilbert scheme is. Would you mind giving me some references?
Mar 8, 2020 at 20:56 history answered Sasha CC BY-SA 4.0