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The aim of this note is to construct a two-dimensional family of smooth quartic surfaces in P3 containing 16 mutually disjoint conics and altogether exactly 432 conics. For an integer d ≥ 1 denote by N4(d) the maximal number of smooth rational curves of degree d that can lie on a smooth quartic surface in P3. Schur [9] gave an example of a smooth quartic surface with 64 lines and Segre [10] showed that there are no smooth quartics containing more than 64 lines, so

Quartic surfaces with 16 skew conics.