Algebraic Geometry

Definition 1.1 Affine space of dimension n over k is defined
An(k) = {(a1, a2, . . . , an) : ai ∈ k}.

Definition 1.2 A morphism of affine spaces
φ : An(k) → Am(k)

is a map given by a polynomial rule
(x1, x2, . . . , xn) → (φ1(x1, . . . , xn), . . . , φm(x1, . . . , xn)),
with the φi ∈ k[x1, . . . , xn].

Definition 1.7 A hypersurface of degree d is the locus
V( f ) := {(a1, . . . , am) ∈ Am(k) : f (a1, . . . , am) = 0} ⊂ Am(k),
where f is a polynomial of degree d.
A regular parametrization of a hypersurface V( f ) ⊂ Am(C) is a morphism
φ : An(C) → Am(C)
such that
1. the image of φ is contained in the hypersurface, i.e., f ◦ φ = 0;
2. the image of φ is not contained in any other hypersurface, i.e., for any h ∈
C[y1, . . . , ym] with h ◦ φ = 0 we have f |h.

Let Pn,d ⊂ k[x1, . . . , xn] denote the vector subspace of polynomials of degree ≤ d.

The monomials , α1 +· · ·+αn ≤ d form a basis for Pn,d , so we have (see Exercise 1.4)
dim Pn,d =choose n from n+d

Problem 1.13 (Simple Interpolation Problem) Given distinct points
p1, . . . , pN ∈ An(k)
what is the dimension of the vector space Id (p1, . . . , pN ) of polynomials of degree
≤ d vanishing at each of the points?

Definition 1.14 Given S ⊂ An(k), the number of conditions imposed by S on
polynomials of degree ≤ d is defined
Cd (S) := dim Pn,d − dim I_d (S).
S is said to impose independent conditions on Pn,d if
Cd (S) = |S|.
It fails to impose independent conditions otherwise.

 quadrics (d = 2)

Proposition 1.15 Let S ⊂ An(k) and consider an invertible affine-linear transformation
φ : An(k) → An(k). Then Cd (S) = Cd (φ(S)) for each d.