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Does anyone know the classification of fourth order surfaces? By "fourth order surface" I mean a surface defined by an equation of the form $$f(x, \, y, \, z)=0,$$ where $f$ is a polynomial of degree $4$.

Edit. The two answers below (J. Silverman and F. Polizzi) concern the case of complex surfaces. Are there known results in the case of real surfaces?

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What do you mean by "classification"? It's better to work with a homogeneous polynomial, in which case the polynomial defines a quartic surface in $\mathbb P^3$. If the surface is non-singular, then it is a K3 surface, and "most" such K3 surfaces have automorphism group $\text{Aut}(X)=\{1\}$ and Neron-Severi group $\text{NS}(X)=\mathbb Z$. (Most means off of a countable union of proper subvarieties of the associated moduli space.) If you search on the terms "quartic surface" and "K3 surface", you'll find lots of information on the web.

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Regarding singular quartic surfaces in $\mathbb C \mathbb{P}^3$, the classical reference is Jessop's book Quartic surfaces with singular points (1916).

An electronic copy of the book is freely available for legal download here.

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The classification of real non-singular quartic surfaces in $\mathbb{RP}^3$ (up to various possible equivalence relations like homeomorphism, isotopy,...) is discussed in

  • A. Degtyarev, V. Kharlamov: Topological properties of real algebraic varieties: Rokhlin's way. Uspekhi Mat. Nauk 55 (2000), no. 4(334), 129-212; translation in Russian Math. Surveys 55 (2000), no. 4, 735–814

See also Sections 3.5.3 and 3.5.4 in the arXiv-version of the paper (as well as the references given there).

Some statements concerning the structure and classification of singular real quartic surfaces are given in:

  • A. Degtyarev and I. Itenberg: On real determinantal quartics. Proceedings of the Gökova Geometry-Topology Conference 2010, 110–128, Int. Press, Somerville, MA, 2011.
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