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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