Mutual arrangement of the ovals of nonsingular real plane algebraic curves: classification of the realizable real schemes (the plane-curve part of part I of Hilbert's sixteenth problem)

The source paper (D. Hilbert, 'Ueber die reellen Züge algebraischer Curven', Math. Ann. 38 (1891) 115-138) introduced the question of the mutual arrangement of the connected components of a nonsingular real plane algebraic curve - in particular of an M-curve, which attains Harnack's maximal number $(n-1)(n-2)/2+1$ of components at degree $n$ - and conjectured that an M-curve of degree 6 cannot have all eleven ovals mutually external. The question became the curve part of part I of Hilbert's sixteenth problem (1900). As the later literature establishes, the classification was completed only for degree 6 by Gudkov (1969-70, with the congruence machinery corrected and completed by Arnol'd and Rokhlin, 1971-73) and, through Viro's patchworking (a gluing construction of real curves with prescribed topology) and subsequent work, for degrees up to 7; for degree 8 and above no complete classification exists, and for degree 9 even the M-curve case is described as 'wide open'. The problem remains open.

Background

A real plane algebraic curve of degree $n$ is the zero locus in the real projective plane $\mathbb{RP}^2$ of a nonzero real homogeneous polynomial of degree $n$; the curve is nonsingular when the polynomial and its partial derivatives have no common zero (so the real part is a smooth compact one-dimensional manifold). A connected component homeomorphic to a circle is called an oval; when $n$ is odd the curve has exactly one additional one-sided component (a pseudoline), and all other components are ovals. Harnack (1876) proved that a nonsingular real plane curve of degree $n$ has at most $(n-1)(n-2)/2+1$ connected components, and he constructed curves attaining this bound for every $n$; such curves are called M-curves (Maximal number of components). Harnack's construction realizes only special arrangements of the components, and the natural question - raised for the first time in Hilbert's 1891 paper, as recorded in Gudkov's 1974 survey of the problem - is which mutual arrangements actually occur: ovals may be nested (one lying in the interior disk of another), and the whole configuration up to homeomorphism of $\mathbb{RP}^2$ is called the real scheme (or isotopy type) of the curve. An oval lying inside an odd number of other ovals is called odd, and otherwise even; for a curve of even degree one writes $P$ and $L$ for the numbers of even and odd ovals. The classification problem asks, for each degree $n$, for the exact list of realizable real schemes. The 1891 paper contributed a gluing construction with a small parameter (an early precursor of patchworking) that produced M-curves of degree 6 with arrangements unreachable by Harnack's method, and it conjectured that an M-sextic cannot have type 11, that is, eleven ovals all outside one another. In his 1900 lecture, published in the standard 1902 English translation, Hilbert posed this classification as part I of problem 16 of his famous list, together with the analogous question for real algebraic surfaces in $\mathbb{RP}^3$; the plane-curve part is the problem considered here. The problem has two dual sources of difficulty. Constructions (proving a scheme realizable) are typically built by gluing methods: the small-parameter technique of the 1891 paper and its modern descendant, Viro's patchworking, which assembles real curves with prescribed topology from pieces of toric varieties (algebraic varieties containing a dense algebraic torus on which the geometry is combinatorial). Obstructions (proving a logically possible scheme unrealizable) use arithmetic restrictions such as the Gudkov-Arnol'd-Rokhlin congruences on $P-L$, and the complex orientation method, developed by Rokhlin in the 1970s (his 1978 survey is the standard reference), which studies the action of complex conjugation on the complex points of the curve and refines the real scheme by a complex scheme; for M-curves of even degree the central congruence is $P-L \equiv (n/2)^2 \pmod 8$.

Problem Statement

For degree $n = 8$ - the smallest degree for which no complete classification is known, and a case of the plane-curve part of part I of Hilbert's sixteenth problem posed in the 1891 source paper - determine exactly which real schemes are realized by nonsingular real plane algebraic curves of degree $8$ in $\mathbb{RP}^2$: produce the complete list of realizable mutual arrangements of the components (the nesting patterns of at most $22$ ovals, with emphasis on M-curves attaining this maximal number of $22$ ovals), together with (i) an explicit realization or existence proof for each listed scheme and (ii) a proof of non-realizability for every remaining logically possible scheme.

References

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