2.8 Complex Schemes of Degree $\le 5$

As it was promised in Section 2.5, we can prove now that only schemes realized in Figures 17, 18 and 19 are realizable by curves of degree 3, 4 and 5, respectively. For reader's convinience, I present here a list of all these complex schemes in Table 5.

Table 5:

$m$

Complex schemes of nonsingular plane curves of degree $m$

1

$\langle J\rangle^1_I$

2

$\langle1\rangle^2_I$
	$\langle0\rangle^2_{II}$

3

$\langle J\amalg 1^-\rangle^3_I$
	$\langle J\rangle^3_{II}$

4

$\langle 4\rangle^4_{I}$
	$\langle 3\rangle^4_{II}$
$\langle 1\langle 1^-\rangle\rangle^4_{I}$	$\langle2\rangle^4_{II}$
	$\langle 1\rangle^4_{II}$
	$\langle0\rangle^4_{II}$

5

$J\amalg 3^+\amalg3^-\rangle_{I}^5$
	$\langle J\amalg5\rangle^5_{II}$
$\langle J\amalg1^+\amalg3^-\rangle_{I}^5$	$\langle J\amalg 4\rangle^5_{II}$
	$\langle J\amalg 3\rangle^5_{II}$
$\langle J\amalg1^-\langle 1^-\rangle\rangle^5_{I}$	$\langle J\amalg 2\rangle^5_{II}$
	$\langle J\amalg 1\rangle^5_{II}$
	$\langle J\rangle^5_{II}$

Degree 3. By Harnack's inequality, the number of components is at most 2. By 1.3.B a curve of degree 3 is one-sided, thereby the number of components is at least 1. In the case of 1 component the real scheme is $\langle J\rangle$, and the type is II by Klein's congruence 2.6.C. In the case of 2 components the type is I by 2.6.B. The real scheme is $\langle J\amalg 1\rangle$. Thus we have 2 possible complex schemes: $\langle J\amalg 1^-\rangle_I^3$ (realized above) and $\langle J \amalg 1^+ \rangle_I^3$. For the first one $\int (i_{\mathbb{R}A}(x))^2 d\chi(x)=9/4$ and for the second $\int (i_{\mathbb{R}A}(x))^2 d\chi(x)=1/4$. Since the right hand side of the complex orientation formula is $m^2/4$ and $m=3$, only the first possibility is realizable.$\qedsymbol$

Degree 4. By Harnack's inequality the number of components is at most 4. We know (see 1.4) that only real schemes $\langle 0\rangle$, $\langle 1\rangle$, $\langle 2\rangle$, $\langle1\langle1 \rangle\rangle$, $\langle3 \rangle$ and $\langle 4\rangle$ are realized by nonsingular algebraic curves of degree 4. From Klein's congruence 2.6.C it follows that the schemes $\langle 1\rangle$ and $\langle3 \rangle$ are of type II. The scheme $\langle 0\rangle$ is of type II by 2.6.A. By 2.6.B $\langle 4\rangle$ is of type I.

The scheme $\langle 2\rangle$ is of type II, since it admits no orientation satisfying the complex orientation formula. In fact, for any orientation $\int(i_{\mathbb{R}A}(x))^2 d\chi(x)=2$ while the right hand side is $m^2/4=4$.

By 2.6.D the scheme $\langle1\langle1 \rangle\rangle$ is of type I. A calculation similar to the calculation above on the scheme $\langle 2\rangle$, shows that only one of the two semiorientations of the scheme $\langle1\langle1 \rangle\rangle$ satisfies the complex orientation formula. Namely, $\langle1\langle1^-\rangle\rangle$. It was realized in Figure 18.

Degree 5. By Harnack's inequality the number of components is at most 7. We know (see 1.4) that only real schemes $\langle J\rangle$, $\langle J\amalg 1\rangle$, $\langle J\amalg 2\rangle$, $\langle J\amalg1\langle1\rangle\rangle$, $\langle J\amalg 3\rangle$, $\langle J\amalg 4\rangle$, $\langle J\amalg 5\rangle$, $\langle J\amalg 6\rangle$ are realized by nonsingular algebraic curves of degree 5. From Klein's congruence 2.6.C it follows that the schemes $\langle J\amalg 1\rangle$, $\langle J\amalg 3\rangle$, $\langle J\amalg 5\rangle$ are of type II. By 2.7.F $\langle J\rangle$ and $\langle J\amalg 2\rangle$ are of type II.

By 2.6.B $\langle J\amalg 6\rangle$ is of type I. The complex orientation formula gives the value of $\Lambda _-$ (cf. Proof of 2.7.F): $\Lambda _-=\frac12k(k+1)=3$. This determines the complex scheme. It is $\langle J\amalg3^-\amalg3^+\rangle_I^5$.

By 2.6.D $\langle J\amalg1\langle1\rangle\rangle$ is of type I. The complex orientation formula allows only the semiorientation with $\Lambda _-=2$. Cf. Figure 19.

The real scheme $\langle J\amalg 4\rangle$ is of indefinite type, as follows from the construction shown in Figure 19. In the case of type I only one semiorientation is allowed by the the complex orientation formula. It is $\langle J\amalg3^-\amalg1^+\rangle_I^5$.