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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 $4$, and x, y, z are real.

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?

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 $4$.

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 $4$, and x, y, z are real.

Does anyone know the classification of fourth order surfaces? By "fourth order surfaces", I mean, a surface defined by fourth order equation like ax^4+by^4+cz^4+dx^3y+...+a1x+a2y+a3z+d=0. Many thanks!

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 $4$.