3.1 Flexible Curves

In Section 1 all prohibitions were deduced from the Bézout Theorem. In Section 2 many proofs were purely topological. A straightforward analysis shows that the proofs of all prohibitions are based on a small number of basic properties of the complexification of a nonsingular plane projective algebraic curve. It is not difficult to list all these properties of such a curve $A$:

1. Bézout's theorem;
2. $\mathbb{C}A$ realizes the class $m[\mathbb{C}P^1]\in H_2(\mathbb{C}P^2)$;
3. $\mathbb{C}A$ is homeomorphic to a sphere with $(m-1)(m-2)/2$ handles;
4. $conj(\mathbb{C}A)=conj$;
5. the tangent plane to $\mathbb{C}A$ at a point $x\in\mathbb{R}A$ is the complexification of the tangent line of $\mathbb{R}A$ at $x$.

The last four are rough topological properties. Bézout's theorem occupies a special position. If we assume that some surface smoothly embedded into $\mathbb{C}P^2$ intersects the complex point set of any algebraic curve as, according to Bézout's theorem, the complex point set of an algebraic curve, then this surface is the complex point set of an algebraic curve. Thus the Bézout theorem is completely responsible for the whole set of properties of algebraic curves. On the other hand, its usage in obtaining prohibitions involves a construction of auxiliary curves, which may be very subtle.

Therefore, along with algebraic curves, it is useful to consider objects which imitate them topologically.

An oriented smooth closed connected two-dimensional submanifold $S$ of the complex projective plane $\mathbb{C}P^2$ is called a flexible curve of degree $m$ if:

(i)

$S$ realizes $m[\mathbb{C}P^1]\in H_2(\mathbb{C}P^2)$;

(ii)

the genus of $S$ is equal to $(m-1)(m-2)/2$;

(iii)

$conj(S)=S$;

(iv)

the field of planes tangent to $S$ on $S\cap \mathbb{R}P^2$ can be deformed in the class of planes invariant under $conj$ into the field of (complex) lines in $\mathbb{C}P^2$ which are tangent to $S\cap \mathbb{R}P^2$.

A flexible curve $S$ intersects $\mathbb{R}P^2$ in a smooth one-dimensional submanifold, which is called the real part of $S$ and denoted by $\mathbb{R}S$. Obviously, the set of complex points of a nonsingular algebraic curve of degree $m$ is a flexible curve of degree $m$. Everything said in Section 2.1 about algebraic curves and their (real and complex) schemes carries over without any changes to the case of flexible curves. We say that a prohibition on the schemes of curves of degree $m$ comes from topology if it can be proved for the schemes of flexible curves of degree $m$. The known classification of schemes of degree $\le 6$ can be obtained using only the prohibitions that come from topology. In other words, for $m\le 6$ all prohibitions come from topology.