Resolutions of singularities.

Yoshifumi Tsuchimoto

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

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

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

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

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

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

1. We put
   $$\P ^n(\mathbbm{k})=(\mathbbm{k}^{n+1}\setminus \{0\}) /\mathbbm{k}^\times$$
   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
   $$V_h(f_1,\dots,f_l)= \{ [x_0:x_1:x_2:\dots x_n] ; f_j (x_0,x_1,x_2,\dots,x_n)=0 \qquad(j=1,2,3,\dots,l) \}.$$
   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 $[x]\in \P ^n(\mathbbm{k})$ .)