I was told singular quartic algebraic surfaces in $\mathbb{P}^3_{\mathbb{C}}$ have been completely classified and their singularities have been described. Can anyone provide me with a resource where this classification is described, in particular the possible combination of singularities on such surfaces?

A search provided http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1370.pdf, but i am wondering if there's something in one of the published books? I did not find anything in Reid - chapters on surfaces, and Beauville's book on surfaces treats only smooth surfaces.


  • $\begingroup$ What kind of classification are you looking for? Just in terms of singularities? $\endgroup$ – J.C. Ottem May 3 '13 at 16:29
  • $\begingroup$ @Ottem, yes maybe i should have said this more clearly in the first place. My question should actually be: "What are the possible combinations of singularities on quartics in $\mathbb{P}^4$? Furthermore, for each such possible combination, can we say anything else about the surface?" $\endgroup$ – Joachim May 3 '13 at 17:12
  • $\begingroup$ In fact, maybe i should say something about the application i have in mind. I have a quartic surface $V$ and a plane $W$ in $\mathbb{P}^3$, and i know $V$ is not singular along a curve in $V \cap W$. I would like to determine the possible singularities that $V$ can have along the curve $V \cap W$. $\endgroup$ – Joachim May 3 '13 at 17:24

Look at the classical book by C. M. Jessop Quartic surfaces with singular points (1916).

The paper by C. Segre Etude des différentes surfaces du 4-eme ordre à conique double ou cuspidale, Math. Ann. 24 (1884), no. 3, 313–444 is also useful.

At any rate, I do not know if there exists a really complete classification.


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