Zn) for all λ ∈ C

1. Algebraic Varieties in Pⁿ.

Let (ζ₀, · · · , ζ_n) denote the standard coordinate system on Cⁿ⁺¹and (ζ₀ : · · · : ζ_n) be the corresponding homogeneous coordinate system on Pⁿ.

Definition 1.1. A polynomial f (Z₀, · · · , Z_n) in S = C[Z0, · · · , Z_n] is called homogeneous of degree d ≥ 0 if f (λZ0, · · · , Zn) = λ^dP (Z0, · · · , Zn) for all λ ∈ C. The set of all homogeneous polynomials of degree d together with 0 is denoted by S_d.

Then we have a direct sum decomposition of S :

S =M

d≥0

S_d

such that Si· S_j ⊂ S_i+j for any i, j.

Given a homogeneous polynomial f in S, we say that a point P (a₀ : · · · : a_n) ∈ Pⁿ is a zero of f if f (a0, · · · , an) = 0 for any representative (a0, · · · , an) of P (a0 : · · · : an). It is easy to check that the notion of zero of a homogeneous polynomial in Pⁿ is well-defined.

When P is a zero of f, we denote f (P ) = 0. The set of zeros of f, is denoted by V (f ). In general, let T be a family of homogeneous polynomials over C. We denote V (T ) the set of all common zeros of polynomials in T, i.e. V (T ) is the set of all P ∈ Pⁿ so that f (P ) = 0 for all f ∈ T.

Definition 1.2. We say that Y ⊂ Pⁿ is an algebraic set if there exists a family T of homogeneous polynomials so that Y = V (Y ). We say that a subset U of Pⁿ is a Zariski open set of Pⁿ if Pⁿ\ U is an algebraic set.

Theorem 1.1. The Zariski open sets in Pⁿ forms a topology. We call this topology the Zariski topology on Pⁿ.

Proof. To do this, it is equivalent to show that (1) ∅ and Pⁿ are algebraic.

(2) Any intersection of algebraic sets is also algebraic.

(3) Any finite union of algebraic sets is algebraic.

Since V (1) = ∅ and V (0) = Pⁿ, (1) is obvious. To prove (2), we only need to show V (S

iT_i) =T

iV (T_i) for any family {T_i} of sets of homogeneous polynomials. Again, this is obvious.

Let T₁, · · · , T_mbe sets of homogeneous polynomials. Let T be the set {f₁· · · f_m: f_i∈ T_i}.

Then elements of T are homogeneous polynomials. If P ∈ S

iV (Ti), then P ∈ V (Tj) for some j. Hence f_j(p) = 0 for all f_j ∈ T_j. Then P is a zero of all f in S, i.e. P ∈ V (T ).

Conversely, if P ∈ V (T ), then f (P ) = 0 implies that P ∈ V (Ti) for some i. Hence P ∈ S

iV (Ti). We conclude thatS

iV (Ti) = V (T ). We completes the proof.  An ideal a in S is said to be a homogeneous ideal if

a=M

d

(a ∩ S_d).

If a is a homogeneous ideal in S, we define

V (a) = V (T ), where T is the set of homogeneous polynomials in a.

Definition 1.3. We say that X ⊂ Pⁿ is a projective variety if X is irreducible algebraic set. A quasi projective variety is an open subset of a projective variety.

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Let Y be any subset of Pⁿ. The homogeneous ideal I(Y ) of Y is the ideal generated by f ∈ S such that f a homogeneous polynomial and f (P ) = 0. The homogeneous coordinate ring of Y is defined to be the quotient ring S(Y ) = S/I(Y ).

Let U_i = {(ζ₀ : · · · : ζ_n) ∈ Pⁿ: ζ_i 6= 0}. Then U_i is the complement of the zero set V (ζ_i) of ζi for 0 ≤ i ≤ n and hence Ui is Zariski-open. We find that {Ui : 0 ≤ i ≤ n} forms an open covering of Pⁿ. We also consider the map

ϕ_i : U_i → Cⁿ

defined by ϕ_i(ζ₀ : · · · : ζ_n) = (ζ₀/ζ_i, · · · , ζ_n/ζ₀). Then ϕ_i is a bijection for all 0 ≤ i ≤ n. Let us claim that ϕi are homeomorphism in the Zariski-topology on Pⁿ and Cⁿ.

Let Y be a Zariski closed subset of Cⁿ. We want to show that ϕ⁻¹_i (Y ) is Zariski closed in Pⁿ.

Let T be a set of polynomials in C[X1, · · · , X_n]. Given a polynomial f (X₁, · · · , X_n) of degree d, we define a homogeneous polynomial ϕ^∗f of degree d in variables Z0, · · · , Zn by

(ϕ^∗_if )(Z₀, · · · , Z_n) = Z_i^df (Z₀/Z_i, · · · , Z_n/Z_i).

We set ϕ^∗T = {ϕ^∗f : f ∈ T }.

Lemma 1.1. Ui∩ V (ϕ^∗_iT ) = ϕ⁻¹_i V (T ) for any set of polynomials T in C[X1, · · · , Xn].

Proof. Let P ∈ ϕ⁻¹_i (Y ) and f ∈ T. We have ϕ^∗_if (P ) = 0. Hence P ∈ V (ϕ^∗_iT ). (By definition P ∈ U_i.) Conversely, if P ∈ V (ϕ^∗_iT ) ∩ U_i, then for all f ∈ T, f (ϕ_i(P )) = 0. Hence P ∈

ϕ⁻¹_i V (T ). We prove the statement. 

If Y is a Zariski closed set in Cⁿ, Y = V (T ) for some T ⊂ C[X1, · · · , X_n]. Hence ϕ⁻¹_i Y = Ui∩ V (ϕ^∗_iT ) is closed in Ui. We see that ϕi is continuous in the Zariski topology.

Let f (Z₀, · · · , Z_n) be a homogeneous polynomial of degree d. We set f_ϕi(X₁, · · · , X_n) = f (X₁, · · · , X_i−1, 1, X_i, · · · , X_n).

For any family T of homogeneous polynomials in C[Z0, · · · , Zn], we set Tϕi = {fϕi : f ∈ T }.

Lemma 1.2. ϕ_i(U_i ∩ V (T )) = V (T_ϕi) for any set T of homogeneous polynomials in C[Z0, · · · , Zn].

Proof. Let Q ∈ ϕ_i(U_i ∩ V (T )). Then we can find P ∈ V (T ) ∩ U_i so that ϕ_i(P ) = Q. (If Q = (a1, · · · , an) ∈ Cⁿ, P = [a1, · · · , ai−1, 1, ai, · · · , an].) Since P ∈ V (T ), f (P ) = 0 for all f ∈ T. We see that f_ϕi(Q) = f (P ) = 0. Hence Q ∈ V (T_ϕi).

Let Q ∈ V (T_ϕi). Then f_ϕi(Q) = 0 for all f ∈ T. Hence f (ϕ⁻¹_i (Q)) = f_ϕi(Q) = 0. We see that ϕ⁻¹_i (Q) ∈ U_i∩ V (T ) and hence Q ∈ ϕ_i(U_i∩ V (T )). We complete the proof.  Hence if Y is a Zariski closed set in Ui, then there exists a family T of homogeneous polynomials in S so that Y = U_i ∩ V (T ). Then ϕ(Y ) = V (T_ϕ) in Cⁿ and hence ϕ_i is a closed mapping in the Zariski topology.

Proposition 1.1. ϕ_i : U_i → Cⁿ are homeomorphisms in the Zariski topology.

Corollary 1.1. If Y is a projective (resp, quasi-projective) variety, then Y is covered by open sets Y ∩Uifor 0 ≤ i ≤ n which are homeomorphic to affine (reps. quasi-affine) varieties via ϕi defined above.

Proof. Since {Ui} is an open covering of Pⁿ, {Ui∩ Y : 0 ≤ i ≤ n} forms an open covering for Y. If Y is a projective variety, it is Zariski closed. Hence for 0 ≤ i ≤ n, U_i∩ Y is Zariski closed in Ui. Since ϕi is continuous in the Zariski-topology, ϕi(Ui∩ Y ) is Zariski closed in

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Cⁿ. The irreducibility of Ui∩ Y implies the irreducibility of ϕ_i(Ui∩ Y ). Hence ϕ_i(Ui∩ Y ) is an affine variety in Cⁿ for each 0 ≤ i ≤ n. Since ϕ_i is a homeomorphism, U_i ∩ Y is homeomorphic to the affine variety ϕi(Ui∩ Y ) in Cⁿ.