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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.
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    $\begingroup$ The paper On singularities of type $A_k$ on simple curves of fixed degree, by S.M. Guseĭn-Zade and N.N. Nekhoroshev, Funct. Anal. Appl. 34 (2000), no. 3, 214–215, gives an upper bound for the maximum $A_k$ on a curve of degree $d$, approximately $k<\dfrac{3d^2}{4} $, and constructs some examples with $k>\dfrac{d^2}{2} $. $\endgroup$ – abx Mar 23 '18 at 10:19
  • $\begingroup$ Thank you, @abx, this is indeed very interesting. It also gives a reference to Greuel, Lossen and Shutsin, Plane curves of minimal degree with prescribed singularities, Invent. Math. 133 (1998), no. 3, 539–580, which attempts the question from the opposite side, asking for asymptotic bounds on the degree given a finite list of singularities. From these two articles, it seems that asymptotically, there is more we can say that for small numbers. $\endgroup$ – Ben Mar 26 '18 at 8:46
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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.

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