RESOLUTIONS OF SINGULARITIES.

YOSHIFUMI TSUCHIMOTO

02. Affine varieties, projective spaces and projective varieties.

Definition 2.1. Let 𝕜 be a field. For any f_{1}, … f_{m} ∈ 𝕜[X_{1}, …, X_{n}], we put

V (f_{1}, …, f_{m})(𝕜) = {x ∈ 𝕜^{n}; f_{1}(x) = 0, …, f_{m}(x) = 0}

and call it (the set of 𝕜-valued point of) the affine variety defined by {f_{1}, f_{2}, …, f_{m}}.

Definition 2.2. Let R be a ring. A polynomial f(X_{0}, X_{1}, …, X_{n}) ∈ R[X_{0}, X_{1}, …, X_{n}] is said to be homogenius of degree d if an equality

f(λX_{0}, λX_{1}, …, λX_{n}) = λ^{d}f(X_{0}, X_{1}, …, X_{n})

holds as a polynomial in n + 2 variables X_{0}, X_{1}, X_{2}, …, X_{n}, λ.

Definition 2.3. Let 𝕜 be a field.

(1) We put

ℙ^{n}(𝕜) = (𝕜^{n+1} \ {0})/𝕜^{×}

and call it (the set of 𝕜-valued points of) the projective space. The class of an element (x_{0}, x_{1}, …, x_{n}) in ℙ^{n}(𝕜) is denoted by [x_{0} : x_{1} : ⋯: x_{n}].

(2) Let f_{1}, f_{2}, …, f_{l} ∈ 𝕜[X_{0}, …, X_{n}] be homogenious polynomials. Then we put

V_{h}(f_{1}, …, f_{l}) = {[x_{0} : x_{1} : x_{2} : … x_{n}]; f_{j}(x_{0}, x_{1}, x_{2}, …, x_{n}) = 0 (j = 1, 2, 3, …, l)}.

and call it (the set of 𝕜-valued point of) the projective variety defined by {f_{1}, f_{2}, …, f_{l}}.

(Note that the condition f_{j}(x) = 0 does not depend on the choice of the representative x ∈ 𝕜^{n+1} of [x] ∈ ℙ^{n}(𝕜).)