# Resolutions of singularities.

Yoshifumi Tsuchimoto

DEFINITION 02.1   Let be a field. For any , we put

and call it (the set of -valued point of) the affine variety defined by .

DEFINITION 02.2   Let be a ring. A polynomial is said to be homogenius of degree if an equality

holds as a polynomial in variables .

DEFINITION 02.3   Let be a field.
1. We put

and call it (the set of -valued points of) the projective space. The class of an element in is denoted by .
2. Let be homogenious polynomials. Then we put

and call it (the set of -valued point of) the projective variety defined by .
(Note that the condition does not depend on the choice of the representative of .)

2014-04-17