3.4 Survey of Prohibitions on the Complex Schemes Which Come From Topology

Recall that $l$ denotes the total number of ovals on the curve. The following theorem is a reformulation of 3.2.E.

3.4.A. (See 2.6.A)   A curve with empty real point set is of type II.

3.4.B (See 2.6.C)   If the curve is of type I, then

$$l\equiv \left[\frac m2\right]\mod 2.$$

3.4.C. Rokhlin Complex Orientation Formula (see 2.7.C))   Let $A$ be a nonsingular curve of type I and degree $m$. Then

$$\int (i_{\mathbb{R}A}(x))^2 d\chi(x)=\frac{m^2}4$$

Extremal Properties of Harnack's Inequality

3.4.D. (Cf. 2.6.B)   Any M-curve is of type I.

3.4.E. Kharlamov-Marin Congruence.   Any $(M-2)$-curve of even degree $m=2k$ with

$$p-n\equiv k^2+4\mod 8$$

is of type I.

Extremal Properties of the Refined Arnold Inequalities

3.4.F   If $m\equiv 0\mod 4$ and $p^-+p^0=\frac{(m-2)(m-4)}8+1$, then the curve is of type I.

3.4.G   If $m\equiv 0\mod 4$ and $n^-+n^0=\frac{(m-2)(m-4)}8$, then the curve is of type I.

Extremal Properties of the Viro-Zvonilov Inequality

3.4.H   Under the hypothesis of 3.3.P, the curve is of type I.

Congruences

3.4.I. Nikulin-Fiedler Congruence.   If $m\equiv 0\mod 4$, the curve is of type I, and every even oval has even characteristic, then $p-n\equiv 0\mod 8$.

The next two congruences are included violating a general promise given at the beginning of the previous section. There I promised exclude prohibitions which follow from other prohibitions given here. The following two congruences are consequences of Rokhlin's formula 3.4.C. The first of them was discovered long before 3.4.C. The second was overlooked by Rokhlin in [Rok-74], where he even mistakenly proved that such a result cannot exist. Namely, Rokhlin proved that the complex orientation formula does not imply any result which would not follow from the prohibitions known by that time and could be formulated solely in terms of the real scheme. Slepian congruence 3.4.K for M-curves is the only counter-example to this Rokhlin's statement. Slepian was Rokhlin's student, he discovered a gap in Rokhlin's arguments and deduced 3.4.K.

3.4.J. Arnold Congruence (see 2.7.D))   If $m$ is even and the curve is of type I, then

$$p-n\equiv \frac{m^2}4\mod 4.$$

3.4.K. Slepian Congruence.   If $m$ is even, the curve is of type I, and every odd oval has even characteristic, then

$$p-n\equiv \frac{m^2}4\mod 8.$$

Rokhlin Inequalities

Denote by $\pi$ and $\nu$ the number of even and odd nonempty ovals, respectively, bounding from the outside those components of the complement of the curve which have the property that each of the ovals bounding them from the inside envelops an odd number of other ovals.

3.4.L   If the curve is of type I and $m\equiv 0\mod 4$, then

$$4\nu+p-n\le \frac{(m-2)(m-4)}2+4.$$

3.4.M   If the curve is of type I and $m\equiv 2\mod 4$, then

$$4\pi+n-p\le \frac{(m-2)(m-4)}2+3.$$