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Let $X\subset \mathbb{P}^3$ be an irreducible quartic surface, defined over an algebraically closed field $k$. Suppose that $X$ is rational (i.e. birational to $\mathbb{P}^2$). Is is true that $X$ has one of the following type of singularities?

a) a point $q\in X$ of multiplicity $4$ (hence $X$ is a cone over a rational quartic curve).

b) a point $q\in X$ of multiplicity $3$ (hence the projection away from $q$ gives a birational map $X\dashrightarrow \mathbb{P}^2$).

c) a curve of singular points.

d) a double point $q\in X$ and a double line $l$ in the exceptional divisor $E_q\simeq \mathbb{P}^2$ of $q$ edit: new case added

e) a double point $q\in X$, a double point $q_1$ in the exceptional divisor $E_q\simeq \mathbb{P}^2$ of $q$ and a double line $l$ in the exceptional divisor $E_{q_1}\simeq \mathbb{P}^2$ of $q_1$ edit: new case added

It seems from the classification of Noether indicated by abx in

M. Noether, "Ueber die rationalen Flächen vierter Ordnung" (Mathematische Annalen, 1189, page 546-571)

that these are the only possible cases. Is there a modern proof of this ?

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    $\begingroup$ No, there are three other types — they are described in detail in Jessop's book Quartic surfaces with singular points, Chap. VIII (but this goes back to M. Noether). For the first type the surface has a tacnode, the two other types are more complicated. $\endgroup$
    – abx
    Commented Dec 29, 2018 at 20:45
  • $\begingroup$ @abx, Thanks for the reference. I've looked at the book of Jessop and at the article of Noether in Math. Annalen. The three types are described in terms of the singularities of the sextic which is the branched curve of the double cover induced by the projection away from a double point (1: contains a double line, 2: has a point of multiplicity 4, 3: has two point of multiplicity 3, infinitely near to each other). $\endgroup$
    – Jérémy Blanc
    Commented Jan 1, 2019 at 11:33
  • $\begingroup$ In terms of singularities, the first one has one double pt and one double line, and the two others have one double pt, then one infinitely near double point and then one double line. It is not completely clear to me that they really prove that no other example is possible (and that the examples are rational, but this is not very hard to check). The proof relies on an article of Clebsch on pencil of genus 2 curves and on the associated branched sextic. $\endgroup$
    – Jérémy Blanc
    Commented Jan 1, 2019 at 11:35

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