Classification of singularities of plane curves of fixed degree (reference request)

Question

We know the answers to some questions like What is the maximal number of singularities of (reduced) plane curves of degree $d$? for general $d$ (in this case $\tfrac{1}{2}d(d-1)$, obtained by $d$ lines in general position). On the other hand, more subtle questions like Given $d$, what is the largest $n$ such that some plane curve of degree $d$ has a singular point of type $A_n$ can be really hard. Nonetheless, in small degrees, complete lists can be given and in very small degrees, the lists are actually rather short and obvious. I would really appreciate if someone could give an up to date list of the major results in the field.

Since this is probably too much asked for, here are some more, rather concrete questions:

1. I am aware of Chung-Man Hui's '79 thesis Plane quartic curves classifying singularities of plane curves of degree four. Is there something similar for quintics or even sextics?
2. Wall's book Singular Points of Plane Curves (most notably sections 7.5-7.7) contains relevant information. Is the information I can find there (more or less) complete and up to date?
3. Back to one of the questions above, out of curiosity, for which degrees do we know the maximal $n$ such that some curve of the given degree has an $A_n$-singularity? We have: $$\begin{array}{lccccc} d & 2 & 3 & 4 & 5 & 6 & \cdots \\ \text{max. }A_n\text{ on a }C_d & 1 & 3^{1)} & 7^{2)} & 12^{3)} & 19^{3)} & ? \end{array}$$ 1) On an irreducible cubic, there is at worst an $A_2$. 2) On an irreducible quartic, there is at worst an $A_6$. 3) Added after JNS' answer; here, the maximal $A_n$ is attained by irreducible curves.

Answer 1

Here is what is known for quintics and sextics. Starting from $d=7$, I do not know the answer to your question.

• Quintics: The maximal $A_n$ on a $C_5$ is $n=12$. For example, the curve given by the zero set of $$ (y^2-xz)^2(\frac{1}{4}x+y+z)-x^2(y^2-xz)(x+2y)+x^5 $$ has an $A_{12}$-singularity at $(0:0:1)$, c.f. Wall, C. (1996). Highly singular quintic curves. Mathematical Proceedings of the Cambridge Philosophical Society, 119(2), 257-277. He gives a classification of quintics with ''large'' Milnor number.
• Sextics: The maximal $A_n$ on a $C_6$ is $n=19$. This was proved by Yang, who gave a complete classification of sextics with simple singularities. See: Yang, Jin-Gen. Sextic curves with simple singularities. Tohoku Math. J. (2) 48 (1996), no. 2, 203--227.

As a side note, you have observed that the maximal $A_n$-singularities for $d=2,3,4$ occur only for reducible curves. However, for $d=5,6$ the maximal singularities are achieved only by irreducible curves.

Answer 2

About question 1, Namba's book Geometry of projective algebraic curves contains a classification of irreducible quintics. The list is rather long and unpleasant, and I think that getting a list of all reducible quintics would be a bit painful (but somewhat feasible). Rational cuspidal curves of degrees up to 7 are also classified (see Torgun Karoline Moe's master thesis and here). In general, the number of possible configurations of singularities explodes, and things become non-trivial even for line arrangements.

About question 3, abx mentioned Guseĭn-Zade and Nekhoroshev's paper. There's a more recent paper by Orevkov, Some examples of real algebraic and real pseudoholomorphic curves, where he slightly improve on previous results.