If a normal quartic surface admits a singular point that is not a rational double point, then the surface is determined by the triplet consisting of the minimal desingularization , the pullback of a general hyperplane section, and a nonzero effective anti-canonical divisor of . Geometric constructions of all the possible triplets are given.
Yuji ISHII. Noboru NAKAYAMA. "Classification of normal quartic surfaces with irrational singularities." J. Math. Soc. Japan 56 (3) 941 - 965, July, 2004. https://doi.org/10.2969/jmsj/1191334093