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Resolutions of singularities.

Yoshifumi Tsuchimoto

\fbox{02. Affine varieties, projective spaces and projective varieties.}

DEFINITION 02.1   Let $ \mathbbm{k}$ be a field. For any $ f_1,\dots f_m \in \mathbbm{k}[X_1,\dots, X_n]$ , we put

$\displaystyle V(f_1,\dots ,f_m)(\mathbbm{k})
=\{ x \in \mathbbm{k}^n; f_1(x)=0,\dots ,f_m(x)=0\}

and call it (the set of $ \mathbbm{k}$ -valued point of) the affine variety defined by $ \{f_1,f_2,\dots,f_m\}$ .

DEFINITION 02.2   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)

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

DEFINITION 02.3   Let $ \mathbbm{k}$ be a field.
  1. We put

    $\displaystyle \P ^n(\mathbbm{k})=(\mathbbm{k}^{n+1}\setminus \{0\}) /\mathbbm{k}^\times

    and call it (the set of $ \mathbbm{k}$ -valued points of) the projective space. The class of an element $ (x_0,x_1,\dots,x_n)$ in $ \P ^n(\mathbbm{k})$ is denoted by $ [x_0:x_1:\dots:x_n]$ .
  2. Let $ f_1,f_2,\dots, f_l \in \mathbbm{k}[X_0,\dots, X_n]$ be homogenious polynomials. Then we put

    % latex2html id marker 612
$\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 $ \mathbbm{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 \mathbbm{k}^{n+1}$ of $ [x]\in \P ^n(\mathbbm{k})$ .)