×

Counter-examples to the Ragsdale conjecture. (Contre-exemples à la conjecture de Ragsdale.) (French. Abridged English version) Zbl 0787.14040

The subject is related to the first part of Hilbert’s 16th problem, the classification of oval arrangements of real plane non-singular algebraic curves. One of the most important questions has been Ragsdale’s conjecture on curves of even degree \(2k\): \[ p\leq (3k^ 2 - 2k + 2)/2, \quad n \leq (3k^ 2 - 3k)/2, \] where \(p\) (resp. \(n\)) is the number of ovals, lying inside even (resp. odd) number of other ovals. O. Ya. Viro [Sov. Math. Dokl. 22, 566-570 (1980; Zbl 0422.14032)] constructed curves with \(n = (3k^ 2 - 3k+2)/2\) and proposed a corrected Ragsdale’s conjecture \[ \max\{p,n\}\leq (3k^ 2 - 3k+2)/2, \] and its generalization to non-singular complex simply connected surfaces \(X\) with an antiholomorphic involution \(\text{conj}\): \(\dim H_ 1(\text{Fix (conj)},\mathbb{Z}/2\mathbb{Z})\leq h^{1,1}(X)\). This note presents examples of curves with either \(p\) or \(n\) equal to \(13k^ 2/8 + O(k)\), which disproves the corrected Ragsdale conjecture (and thereby the Viro conjecture). The construction is based on Viro’s method of glueing polynomials, and has a purely combinatorial nature.

MSC:

14P25 Topology of real algebraic varieties
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H45 Special algebraic curves and curves of low genus
PDFBibTeX XMLCite