Congruent zeta functions. No.3

Yoshifumi Tsuchimoto

DEFINITION 3.1   Let $R$ be a ring. A polynomial $f(X_0,X_1,\dots,X_n)\in R[X_0,X_1,\dots, X_n]$ is said to be homogenius of degree $d$ if an equality

$$f(\lambda X_0,\lambda X_1,\dots, \lambda X_n) = \lambda^d f(X_0,X_1,\dots,X_n)$$

holds as a polynomial in $n+2$ variables $X_0,X_1,X_2,\dots, X_n, \lambda$.

For any homogeneous polynomial $F(X_0,X_1,\dots,X_n)$, we may obtain its inhomogenization as follows:

$$f(x_1,x_2,\dots,x_n)= F(1,X_1,\dots,X_n).$$

Conversely, for any inhomogeneous polynomial $f(x_1,\dots,x_n)$ of degree $d$, we may obtain its homogenization as follows:

$$F(X_1,X_2,\dots,X_n)= f(X_1/X_0,\dots,X_n/X_0)X_0^d.$$

DEFINITION 3.2   Let $k$ be a field.

1. We put
   $$\P ^n(k)=(k^{n+1}\setminus \{0\}) /k^\times$$
   and call it (the set of $k$-valued points of) the projective space. The class of an element $(x_0,x_1,\dots,x_n)$ in $\P ^n(k)$ is denoted by $[x_0:x_1:\dots:x_n]$.
2. Let $f_1,f_2,\dots, f_l \in k[X_0,\dots, X_n]$ be homogenious polynomials. Then we set
   $$V_h(f_1,\dots,f_l)= \{ [x_0:x_1:x_2:\dots x_n] ; f_j (x_0,x_1,x_2,\dots,x_n)=0 \qquad(j=1,2,3,\dots,l) \}.$$
   and call it (the set of $k$-valued point of) the projective variety defined by $\{f_1,f_2,\dots,f_l\}$.

(Note that the condition $f_j(x)=0$ does not depend on the choice of the representative $x\in k^{n+1}$ of $[x]\in \P ^n(k)$.)

LEMMA 3.3   We have the following picture of $\P ^2$.

1. $$\P ^2=\mathbb{A}^2\coprod \P ^1.$$
   That means, $\P ^2$ is divided into two pieces $\{Z\neq 0\}=\complement V_h(Z)$ a nd $V_h(Z)$.

2. $$\P ^2=\mathbb{A}^2\cup \mathbb{A}^2 \cup \mathbb{A}^2.$$
   That means, $\P ^2$ is covered by three “open sets” $\{Z\neq 0\}, \{Y\neq 0\}, \{X \neq 0\}$. Each of them is isomorphic to the plane (that is, the affine space of dimension 2).