Timeline for Local global principle for a system of polynomial equations
Current License: CC BY-SA 4.0
9 events
when toggle format | what | by | license | comment | |
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Aug 27, 2021 at 15:41 | comment | added | roydiptajit | Thanks @Gro-Tsen | |
Aug 27, 2021 at 15:40 | comment | added | roydiptajit | Thanks. @WillSawin | |
Aug 27, 2021 at 14:17 | comment | added | Will Sawin | In fact as soon as there are two equations of degree 2, the Hasse principle can fail. (With one, it's of course OK.) Given any elliptic curve E over Q whose Tate-Shafarevich group has 4-torsion, we can produce such a variety by taking a 4-torsion element, looking at the corresponding $E$-torsor, which fails the Hasse principle, and embedding it into $\mathbb P^3$ by a degree $4$ line bundle, where it will be the intersection of two quadrics. | |
Aug 27, 2021 at 10:57 | comment | added | Gro-Tsen | Restricting to homogeneous polynomials of degree $2$ (quadrics) doesn't help, because, as the answers to this question explain, every projective variety can be written (after a suitable embedding) as an intersection of quadrics. So the answer to your question is no, there is no local-global principle. (But I don't think the question is stupid and you shouldn't have been downvoted.) | |
Aug 27, 2021 at 10:52 | comment | added | roydiptajit | Please look at my updated question. In case of homogenous polynomials of degree 2 is the answer still no? | |
Aug 27, 2021 at 10:51 | history | edited | roydiptajit | CC BY-SA 4.0 |
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Aug 27, 2021 at 10:35 | review | Close votes | |||
Sep 1, 2021 at 3:08 | |||||
Aug 27, 2021 at 10:15 | comment | added | abx | No. Look at Hasse principle. | |
Aug 27, 2021 at 9:54 | history | asked | roydiptajit | CC BY-SA 4.0 |