1.10 Hilbert Curves

In 1891 Hilbert [Hil-91] seems to have been the first to clearly state the isotopy classification problem for nonsingular curves. As we saw, the isotopy types of Harnack M-curves are very special. Hilbert suggested that from the topological viewpoints M-curves are the most interesting. This Hilbert's guess was strongly confirmed by the whole subsequent development of the field.

There is a big gap between property 1.9.A of Harnack M-curves and the corresponding prohibition in 1.3.C. Hilbert [Hil-91] showed that this gap is explained by the peculiarities of the construction and not by the intrinsic properties of M-curves. He proposed a new method of constructing M-curves which was close to Harnack's method, but which gives M-curves with nests of any depth allowed by Theorem 1.3.C. In his method the role a line plays in Harnack's method is played instead by a nonsingular conic, and a line or a conic is used for the starting curve. Figures 10-11 show how to construct M-curves by Hilbert's method.

In Table 3 we list the isotopy types of M-curves of degree $\le 8$ which are obtained by Hilbert's construction.

Table 3:

$m$	Isotopy types of the Hilbert M-curves of degree $m$
4	$\langle 1\rangle$
6	$\langle 9\amalg 1\langle 1\rangle\rangle \qquad\qquad\qquad\langle 1\amalg 1\langle 9\rangle\rangle$
8	$\langle 5\amalg 1\langle 14\amalg 1\langle 1\rangle\rangle\rangle\ \langle 17... ...amalg 1\langle1\rangle\rangle\rangle\ \langle 18\amalg 1\langle3\rangle\rangle$
	$\langle 1\amalg 1 \langle 2\amalg 1\langle17\rangle\rangle\rangle\ \langle 1\amalg 1\langle 14\amalg 1\langle5\rangle\rangle\rangle$
5	$\langle J\amalg 6\rangle$
7	$\langle J\amalg 15\rangle \quad \langle J\amalg 12\amalg1\langle 2\rangle\rang... ...\langle 12\rangle\rangle\quad \langle J\amalg 1\amalg 1\langle 13\rangle\rangle$

The first difficult special problems that Hilbert met were related with curves of degree 6. Hilbert succeeded to construct M-curves of degree $\ge6$ with mutual position of components different from the scheme $\langle 9\amalg 1\langle 1\rangle\rangle$ realized by Harnack. However he realized only one new real scheme of degree 6, namely $\langle1\amalg1\langle9\rangle\rangle$. Hilbert conjectured that these are the only real schemes realizable by M-curves of degree 6 and for a long time affirmed that he had a (long) proof of this conjecture. Even being false (it was disproved by D. A. Gudkov in 1969, who constructed a curve with the scheme $\langle 5\amalg 1\langle5\rangle\rangle$) this conjecture caught the things that became in 30-th and 70-th the core of the theory.

In fact, Hilbert invented a method which allows to answer to all questions on topology of curves of degree 6. It involves a detailed analysis of singular curves which could be obtained from a given nonsingular one. The method required complicated fragments of singularity theory, which had not been elaborated at the time of Hilbert. Completely this project was realized only in the sixties by D. A. Gudkov. It was Gudkov who obtained a complete table of real schemes of curves of degree 6.

Coming back to Hilbert, we have to mention his famous problem list [Hil-01]. He included into the list, as a part of the sixteenth problem, a general question on topology of real algebraic varieties and more special questions like the problem on mutual position of components of a plane curve of degree 6.

The most mysterious in this problem seems to be its number. The number sixteen plays a very special role in topology of real algebraic varieties. It is difficult to believe that Hilbert was aware of that. It became clear only in the beginning of seventies (see Rokhlin's paper ``Congruences modulo sixteen in the sixteenth Hilbert's problem'' [Rok-72]). Nonetheless, sixteen was the number assigned by Hilbert to the problem.