Papers › Hypersurfaces passing through the Galois orbit of a point
Hypersurfaces passing through the Galois orbit of a point
Shamil Asgarli, Jonathan Love, Chi Hoi Yip
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Asgarli, Ghioca, and Reichstein proved that if K is a field with |K|>2, then for any positive integers d and n, and separable field extension L/K with degree m=n+dd, there exists a point P∈ℙⁿ(L) which does not lie on any degree d hypersurface defined over K. They asked whether the result holds when |K| = 2. We answer their question in the affirmative by combining various ideas from arithmetic geometry. More generally, we show that for each positive integer r and separable field extension L/K with degree r, there exists a point P ∈ℙⁿ(L) such that the vector space of degree d forms over K that vanish at P has the expected dimension. We also discuss applications to linear systems of hypersurfaces with special properties.
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