MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question

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.

share|cite|improve this answer

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.

share|cite|improve this answer

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.
share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.