Giacomo Maletto: Hilbert’s 16th problem for arrangements of a quartic curve and three lines

Abstract.

Motivated by Hilbert’s 16th problem for real planar curves and a related ongoing classification of arrangements of three quadrics in real space, we study the classification of the topological types of arrangements formed by a real quartic curve f(x, y, z) = 0 together with three lines in general position that intersect the quartic transversely - thought of, after a projective change of coordinates, as the coordinate axes xyz = 0.

We can encode the topology of an algebraic curve and the way it interacts with the axes purely as combinatorial data. This data admits several equivalent representations, by means of Dyck words and trees, graphs, or cell complexes. Using these, we investigate which configurations can actually be realized by algebraic arrangements, providing both necessary and sufficient criteria for realizability.

As sufficient condition, we employ Viro’s patchworking method to construct explicit realizable cases; additional realizable configurations are obtained from these by translations. As necessary condition, we introduce a combinatorial analogue of Bézout’s theorem: a configuration is said to \textit{satisfy Bézout at order $n$} if, roughly speaking, the addition of any $n$ new lines forces no violation of Bézout’s theorem.

By comparing the configurations that satisfy Bézout at order 1 with those obtained through patchworking, we achieve a complete classification of all arrangements in which the quartic has exactly one oval, and we derive both upper and lower bounds for the number of configurations of the remaining topological types.