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On the Conjecture for the Rational Points on a Cubic Surface

✍ Scribed by Mordell, L. J.


Book ID
120098372
Publisher
Oxford University Press
Year
1965
Tongue
English
Weight
222 KB
Volume
s1-40
Category
Article
ISSN
0024-6107

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We show that the number of nontrivial rational points of height at most B, which lie on the cubic surface x 1 x 2 x 3 = x 4 (x 1 + x 2 + x 3 ) 2 , has order of magnitude B(log B) 6 . This agrees with Manin's conjecture.

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In this paper we find an algorithm which computes the Hilbert function of schemes Z of ''fat points'' in ‫ސ‬ 3 whose support lies on a rational normal cubic curve C. The algorithm shows that the maximality of the Hilbert function in degree Ε½ t is related to the existence of fixed curves either C its