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We know every cubic surface in $\mathbb{P}^3$ is obtained by blowing up $\mathbb{P}^2$ at 6 points in general position. Hence they are all birational to $\mathbb{P}^2$.

My question is: Do we have more refined ways to classify these cubics? I heard we can do something with configurations?

Thanks!

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  • $\begingroup$ Look at [Hartshorne, V.4]. $\endgroup$ – Sándor Kovács Mar 7 '16 at 20:35
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    $\begingroup$ Not all cubic surfaces are rational, even over complex numbers. $\endgroup$ – Sasha Mar 7 '16 at 20:49
  • $\begingroup$ @Sasha Sorry I should have said I meant smooth cubic surfaces over complex numbers. $\endgroup$ – Xuqiang QIN Mar 7 '16 at 20:52
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Well, you can classify smooth cubic surfaces up to isomorphism instead. This gives rise a 4-dimensional moduli space.

This can be seen by noting that $\mathrm{PGL}_3$ acts transitively on collections of $4$ points in $\mathbb{P}^2$ in general position, so one may assume that the first $4$ blown-up points are $[0:0:1], [0:1:0], [1:0:0]$ and $[1:1:1]$. There are $2$ parameters for each of the remaining two points, hence one has a $4$-dimensional moduli space.

Doing this a bit more rigorously shows that the moduli stack of smooth cubic surfaces is a smooth $4$-dimensional Deligne-Mumford stack. It will not be a scheme as there are cubic surfaces with non-trivial automorphisms. People therefore often consider marked cubic surfaces: these are cubic surfaces together with a choice of numbering of the lines. The automorphism group of a marked cubic surface is trivial, and there is a fine moduli space of marked cubic surfaces which is a $4$-dimensional scheme. See for example:

A.-S. Elsenhans and J. Jahnel: Moduli spaces and the inverse Galois problem for cubic surfaces.

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  • $\begingroup$ Hacking, Keel and Tevelev arxiv.org/abs/math/0702505 have a nice construction of a compactification of the moduli space of marked cubic surfaces, if that is of interest. $\endgroup$ – David E Speyer Mar 7 '16 at 21:38

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