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Could anybody help me with examples of surfaces $X\subset\mathbb P^3$ (projective, over $\mathbb C$) having many isolated singularities of the type $A_1$ ($x^2+y^2+z^2=0$) or $A_2$ ($x^2+y^2+z^3=0$) and no other singular points? «Many» means «as close to the known upper bound (in terms of $\deg X$) as possible».

Thank you in advance,

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For nodal surfaces, maybe this site would help – BS. Aug 16 '13 at 15:34
@BS: Thanks a lot! – Serge Lvovski Aug 16 '13 at 15:50
What is the known upper bound? – Mariano Suárez-Alvarez Aug 16 '13 at 19:59
There is a nice paper by Bruce and Wall that might be of interest to you, in which they classify all possible singularities which may occur for cubic surfaces. – Daniel Loughran Aug 16 '13 at 20:47
Many thanks, Daniel! – Serge Lvovski Aug 16 '13 at 22:28

For the case of surfaces of degree $\le 6$, you may consult Catanese, F.; Ceresa, G.: J. Pure Appl. Algebra 23 (1982), and, if you read Italian,

Ezio Stagnaro: Rend. Sem. Mat. Univ. Padova 59 (1978), 179–198 (1979).

Octic surfaces with many nodes were constructed by Marco Kühnel: Geom. Dedicata 109 (2004), 189–195.

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There is an example of an octic with 84 A_2 singularities (known upper bound is 98) in arXiv:1108.1820, section 9.

Also, check out this paper

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