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Upper bounds on the number of lines on a surface Elkies, Noam D.


We prove that for q>1 a smooth surface of degree q+1 over any field has at most (q+1)(q^3+1) lines, and thus that when q is a prime power the "Hermitian" (a.k.a. diagonal) surface of that degree over a field of q^2 elements has the maximal number of lines for its degree. This was previously known only for q=3 (Rams-Schuett 2015) and of course q=2. This is one of several cases where we obtain a sharp bound; other examples are the 126 tritangents of the Fermat sextic in characteristic 5 (and likewise for other "Hermitian" curves in odd characteristic), and the 891 planes on the diagonal cubic fourfold in characteristic 2.

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