Timeline for lines on quintic del Pezzo 3-fold of degree 5
Current License: CC BY-SA 4.0
11 events
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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 |