Is there a known classification of singular cubic surfaces over finite fields?
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$\begingroup$ Can you please clarify the question? What you mean by "classification"? Do you only care about large finite fields or do you really care about small finite fields? Do you e.g. want to know whether every possible ADE singularity for cubic surfaces over $\mathbb{C}$ can occur for some cubic surfaces over $\mathbb{F}_2$? As finite fields are not algebraically closed there are also subtitles regarding whether the singularities are defined over the base field or over a finite field extension and conjugate under the Galois action. $\endgroup$– Daniel LoughranCommented Nov 21, 2017 at 17:51
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$\begingroup$ Hirschfeld, in his book titled "Finite projective spaces of three dimensions" classified all the quadric surfaces in $\mathbb{F}_q$. I am looking for similar results for cubic surfaces. I do care about every finite fields, one could ignore characteristic 2 though. It is fine to consider the singularities even over algebraic closure of $\mathbb{F}_q$. $\endgroup$– M. D.Commented Nov 21, 2017 at 22:34
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$\begingroup$ It might help to recall Hirschfeld's classification to make precise what kind of "classification" you are looking for. $\endgroup$– Daniel LoughranCommented Nov 22, 2017 at 9:46
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$\begingroup$ I do not have a soft copy for Hirschfeld's book, but the following paper (ac.els-cdn.com/S1071579706000414/…) mentions it in page 621, Table 1. $\endgroup$– M. D.Commented Nov 22, 2017 at 10:30
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$\begingroup$ So you are interested in a classification up to the action of $\mathrm{PGL}_4(\mathbb{F}_q)$? Do you want specific list of possible equations over each $\mathbb{F}_q$? Or do you just want to know all possible singularity types? The case of quadrics is quite different from the case of cubic surfaces; there are $6$ isomorphism classes of quadric surfaces over each $\mathbb{F}_q$, but the number of isomorphism classes of cubic surfaces over $\mathbb{F}_q$ grows as $q \to \infty$; this is why I'm not sure what you want by classification, as it the number of possibilities is growing. $\endgroup$– Daniel LoughranCommented Nov 22, 2017 at 11:56
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