3.5 Ideas of Some Proofs

Theorems formulated in 3.3 and 3.4 are very different in their profundity. The simplest of them were considered in Subsection 3.2.

Congruences

There are two different approaches to proving congruences. The first is due basically to Arnold [Arn-71] and Rokhlin [Rok-72]. It is based on consideration of the intersection form of two-fold covering $Y$ of $\mathbb{C}P^2$ branched over the complex point set of the curve. The complex conjugation involution $conj:\mathbb{C}P^2\to\mathbb{C}P^2$ is lifted to $Y$ in two different ways, and the liftings induce involutions in $H_2(Y)$, which are isometries of the intersection form. One has to take an appropriate eigenspace of one of the liftings and consider the restriction of the intersection form to the eigenspace. The signature of this restriction can be calculated in terms of $p-n$. On the other hand, it is involved into some congruences of purely arithmetic nature relating it with the discriminant of the form and the value of the form on some of characteristic vectors. The latters can be calculated sometimes in terms of degree and the difference between the number of ovals and the genus of curve. Realizations of this scheme can be found in [Arn-71] for 3.4.J, [Rok-72] for 3.3.C, [Kha-73] and [GK-73] for 3.3.D, [Nik-83] for 3.3.F, 3.3.G, 3.4.I and a weakened form of 3.3.E. In survey [Wil-78] this method was used for proving 3.3.C, 3.3.D and 3.4.J.

The second approach is due to Marin [Mar-80]. It is based on application of the Rokhlin-Guillou-Marin congruence modulo 16 on characteristic surface in a 4-manifold, see [GM-77]. It is applied either to the surface in the quotient space $\mathbb{C}P^2/conj$ (diffeomorphic to $S^4$) made of the image of the flexible curve $S$ and a half of $\mathbb{R}P^2$ bounded by $\mathbb{R}S$ (as it is the case for proofs of 3.3.C, 3.3.D and 3.4.E in [Mar-80]), or to the surface in $\mathbb{C}P^2$ made of a half of $S$ and a half of $\mathbb{R}P^2$ (as it is the case for proofs of 3.3.E, 3.4.I and special cases of 3.3.F and 3.3.G in [Fie-83]).

The first approach was applied also in high-dimensional situations. The second approach worked better than the second one for curves on surfaces distinct from projective plane, see [Mik-94]. Both were used for singular curves [KV-88].

Inequalities

Inequalities 3.3.H, 3.3.I, 3.3.J, 3.3.K, 3.4.J and 3.4.K are proved along the same scheme, originated by Arnold [Arn-71]. One constructs an auxiliary manifold, which is the two-fold covering of $\mathbb{C}P^2$ branched over $S$ in the case of 3.3.H, 3.3.I, 3.3.J and 3.3.K and the two-fold covering of $\mathbb{C}P^2/conj$ branched over the union of $S/conj$ and a half of $\mathbb{R}P^2$ in the case of 3.4.J and 3.4.K. Then preimages of some of the components of $\mathbb{R}P^2\smallsetminus \mathbb{R}S$ gives rise to cycles in this manifold. Those cycles define homology classes with special properties formulated in terms of their behavior with respect to the intersection form and the complex conjugation involutions. On the other hand, the numbers of homology classes with these properties are estimated. See [Arn-71], [Gud-74], [Wil-78] and [Rok-80].