Congruent zeta functions. No.3

Yoshifumi Tsuchimoto

\fbox{projective space and projective varieties.}

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

$\displaystyle 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:

$\displaystyle 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:

$\displaystyle 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

    $\displaystyle \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

    % latex2html id marker 689
$\displaystyle 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. $\displaystyle \P ^2=\mathbb{A}^2\coprod \P ^1.
$

    That means, $ \P ^2$ is divided into two pieces % latex2html id marker 712
$ \{Z\neq 0\}=\complement V_h(Z)$ a nd $ V_h(Z)$.
  2. $\displaystyle \P ^2=\mathbb{A}^2\cup \mathbb{A}^2 \cup \mathbb{A}^2.
$

    That means, $ \P ^2$ is covered by three “open sets” % latex2html id marker 720
$ \{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).