Let F be a field of positive characteristic $p$ and let $d,n$ be two positive integers.
Can we explicitly write down an equation defining a smooth hypersurface $X_d⊂\mathbb P^n_F$ of degree d ?
This is easy if $p$ does not divide $d$: the Fermat equation $∑x^d_i=0$ will do.
But what if $p$ does divide $d$ ?

  • 1
    $\begingroup$ The table in section 3 of Poonen's www-math.mit.edu/~poonen/papers/projaut.pdf does this. $\endgroup$
    – SashaP
    Dec 29, 2020 at 19:06
  • 3
    $\begingroup$ (Katz also faced this problem in web.math.princeton.edu/~nmk/baby16.pdf —- see Section 6.) $\endgroup$
    – alpoge
    Dec 29, 2020 at 21:19
  • $\begingroup$ @SashaP : thank you for your comment $\endgroup$
    – lefuneste
    Dec 29, 2020 at 21:31
  • $\begingroup$ In the case $p\vert d$ the equation mentioned by @alpoge is due to Ofer Gabber and is $x_0^d+\sum _{i=0}^{n-1}x_ix_{i+1}^{d-1}=0$. This is much better than my answer, which I'm thus deleting. $\endgroup$
    – lefuneste
    Dec 30, 2020 at 0:22
  • $\begingroup$ @alpoge Ofer Gaber's example in your reference is much better than mine. Could you please explain in an answer, (which can then be upvoted) , since I have deleted my own answer . $\endgroup$
    – lefuneste
    Dec 30, 2020 at 0:26


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