4.1 Curves with Maximal Nest Revised

To begin with, I present another proof of Theorem 2.6.D. It gives slightly more: not only that a curve with maximal nest has type I, but that its complex orientation is unique. This is not difficult to obtain from the complex orientation formula. The real cause for including this proof is that it is the simplest application of the technique, which will work in this section in more complicated situations. Another reason: I like it.

4.1.A   If a nonsingular real plane projective curve $A$ of degree $m$ has a nest of ovals of depth $[m/2]$ then $A$ is of type I and all ovals (except for the exterior one, which is not provided with a sign in the case of even $m$) are negative.

Recall that by Corollary 1.3.C of the Bézout theorem a nest of a curve of degree $m$ has depth at most $m/2$, and if a curve of degree $m$ has a nest of depth $[m/2]$, then it does not have any ovals not in the nest. Thus the real scheme of a curve of 4.1.A is $\langle 1\langle 1\dots \langle1 \rangle \dots \rangle\rangle$, if $m$ is even, and $\langle J\langle1\langle\dots1\langle1\rangle\dots\rangle\rangle$ if $m$ is odd. Theorem 4.1.A says that the complex scheme in this case is defined by the real one and it is

$$\langle 1\langle 1^-\dots \langle1^- \rangle\dots\rangle\rangle_I^m$$

for even $m$ and

$$\langle J\langle1^-\langle\dots1^-\langle1^-\rangle\dots\rangle\rangle_I^m$$

if $m$ is odd.

Proof. [Proof of 4.1.A] Take a point $P$ inside the smallest oval in the nest. Project the complexification $\mathbb{C}A$ of the curve $A$ from $P$ to a real projective line $\mathbb{C}L$ not containing $P$. The preimage of $\mathbb{R}L$ under the projection is $\mathbb{R}A$. Indeed, the preimage of a point $x\in \mathbb{R}L$ is the intersection of $\mathbb{C}A$ with the line connecting $P$ with $x$. But since $P$ is inside all ovals of the nest, any real line passing through it intersects $\mathbb{C}A$ only in real points.

The real part $\mathbb{R}L$ of $L$ divides $\mathbb{C}L$ into two halves. The preimage of $\mathbb{R}L$ divides $\mathbb{C}A$ into the preimages of the halves of $\mathbb{R}L$. Thus $\mathbb{R}A$ divides $\mathbb{C}A$.

The projection $\mathbb{C}A\to\mathbb{C}L$ is a holomorphic map. In particular, it is a branched covering of positive degree. Its restriction to a half of $\mathbb{C}A$ is a branched covering of a half of $\mathbb{C}L$. Therefore the restriction of the projection to $\mathbb{R}A$ preserves local orientations defined by the complex orientations which come from the halves of $\mathbb{C}A$ and $\mathbb{C}L$. $\qedsymbol$