800 conics on a smooth quartic surface (Q2136135)

Counting rational curves on \(K3\) surfaces is classically a hard problem. While bounds of the maximal number of lines on a polarised \(K3\) surface of degree \(2n\) are well-known, much less is known for higher degree rational curves, also for conics. In the present paper it is shown the existence of a smooth quartic surface in \(\mathbb P^3\) with 800 irreducible conics, improving the previously known examples of a quartic \(K3\) containing 352 or 432 conics, while the best upper bound for conics on a quartic \(K3\) surface is 5016. It must be remarked that while the \(K3\) surface is implicitly constructed in the paper under revision, its equation was later found by \textit{X. Roulleau} and equations of all the conics were found by \textit{B. Naskręcki} in [``Explicit equations of 800 conics on a Barth-Bauer quartic'', in press]. The main theorem is proven by using the extended binary Golay code to produce a specific lattice containing 800 vectors of square -2, called conics, and a vector \(h\) of square 4 such that \(h\cdot c = 2\) for each conic \(c\). The existence of the quartic \(K3\) surface with the desired properties follows then by the surjectivity of the period map and results of \textit{B. Saint-Donat} [Am. J. Math. 96, 602--639 (1974; Zbl 0301.14011)].