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I have a (geometrically irreducible) cubic surface defined over a finite field $F_q$ with three non-$F_q$-rational singularities (defined over the cubic extension of $F_q$). Counting the number of $...

Probably not research level, feel free to downvote. We got construction of bounded degree projective curves with no points over finite fields. This construction generalizes to higher dimension. One of ...