1.8 End of the Proof of Theorem 1.6.A

In Section 1.6 it was shown that for any $m$ there exist nonsingular curves of degree $m$ with the minimum number $(1-(-1)^m)/2$ or with the maximum number $(m^2-3m+4)/2$ of components. Nonsingular curves which are isotopic to one another form an open set in the space $\mathbb{R}C_m$ of real projective plane curves of degree $m$ (see Section 1.7). Hence, there exists a real pencil of curves of degree $m$ which connects a curve with minimum number of components to a curve with maximum number of components and which intersects the set of real singular curves only in its principal part and only transversally. As we move along this pencil from the curve with minimum number of components to the curve with maximum number of components, the curve only undergoes Morse modifications, each of which changes the number of components by at most 1. Consequently, this pencil includes nonsingular curves with an arbitrary intermediate number of components.$\qedsymbol$