Algebraic subvarieties of projective spaces

7 Algebraic subvarieties of projective spaces

0 ,...,x n ] be a set, all whose elements are homogeneous polynomials. Then it defines a subset Z(S) n ⊂P , which consists

7.1 n Closed subsets Z(S) ⊂P Let S ⊂ C[x

of all points ha 0 ,...,a n i such that f(a 0 ,...,a n ) = 0 for all f ∈ S. This condition is well-defined, as

f (λa d

0 , . . . , λa n )=λ f (a 0 ,...,a n )

for any homogeneous polynomial of degree d. Under the quotient map

q:C n − {0} → P

n+1

the inverse image of Z(S) n+1 ⊂P is the subset Z(S) ∩(C −{0}), where Z(S) ⊂ C denotes the Zariski closed subset defined by S. Hence by definition of quotient

n+1

topology, Z(S) is Zariski closed in P n . Similarly, Z(S) is Euclidean closed in P .

7.2 n+1 Exercise Let X ⊂C be a closed subvariety, which is conical (that is, if v ∈

X then λv ∈ X for all λ ∈ C). Then show that there exists a set S ⊂ C[x 0 ,...,x n ] of homogeneous polynomials, such that X = Z(S). (Hint: If f vanishes over X, show that each homogeneous component of f also vanishes over X).

7.3 Homogeneous ideals An ideal I ⊂ C[x 0 ,...,x n ] is called homogeneous if for any f ∈ I, all the homogeneous components of f are in I. Deduce from the above exercise that a subset X n ⊂P is Zariski closed if and only if there exists a

homogeneous ideal I ⊂ C[x 0 ,...,x n ] such that X = Z(I).

7.4 n Projective variety A projective variety is a Zariski closed subset of P .

7.5 Homogeneous Nullstellensatz Let I ⊂ C[x 0 ,...,x n ] be a homogeneous ideal. Let f ∈ C[x 0 ,...,x n ] be a homogeneous polynomial of degree √ ≥ 1. Show that Z(I) ⊂ Z(f) if and only if f ∈ I.

7.6 Exercise: When is Z(I) empty Let I ⊂ S = C[x 0 ,...,x n ] be a homo- √ geneous ideal. Show that Z(I) is empty if and only if S + ⊂

I, where S + ⊂

C [x 0 ,...,x n ] is the ideal (x 0 ,...,x n ).

7.7 Theorem There is an inclusion-reversing bijective correspondence between the set HRI of all homogeneous radical ideals in S = C[x 0 ,...,x n ] which do not contain the ideal S n

+ , and the set of all non-empty closed subvarieties of P , given by I 7→ Z(I), with inverse given by

X 7→ I X = { f ∈ C[x 0 ,...,x n ] | f(a 0 ,...,a n ) = 0 for all ha 0 ,...,a n i∈X} Under this correspondence, the set of all maximal elements of HRI is in bijection

with the set of all points of P n . The set of all prime ideals in HRI is in bijection with the set of all irreducible closed subvarieties in P n .

7.8 n Affine cone b X over a projective variety X ⊂P Given any closed subset

X n ⊂P defined by a homogeneous radical ideal I ⊂ C[x

0 ,...,x n ], the closed subset

b X = Z(I) n+1 ⊂C is called the affine cone over X. The ring C[x 0 ,...,x n ]/I = O( b X) is called the homogeneous coordinate ring of X n ⊂P .

7.9 n Exercise: Noetherianness Show that P under Zariski topology is a noethe- rian topological space, that is, any decreasing sequence of closed subsets is finite.

Deduce that any locally closed subvariety of P n is noetherian. In particular, any locally closed subvariety X n ⊂P is quasi-compact, that is, every Zariski open cover

of X has a finite subcover. Moreover, X can be uniquely written as a minimal finite union of irreducible closed subsets of X.

7.10 Principal open subsets Let f ∈ C[x 0 ,...,x n ] be a homogeneous polyno- mial. The Zariski open subset U n

f =P − Z(f) is called the principal open subset defined by f . Show that principal open subsets form a basis of open sets for the Zariski topology on P n .

7.11 Regular functions on an open subvariety A function φ : V → C on an open subvariety V n ⊂P is called a regular function if given any principal open

f ⊂ V , the restriction φ| U f is of the form g/f where g ∈ C[x 1 ,...,x n ] is a homogeneous polynomial and r is a non-negative integer such that deg(g) = r deg(f ) (so that the rational function g/f r is homogeneous of degree 0).

7.12 Exercise Show that if (U f i ) i=1,...,m is a principal open cover of an open subvariety V n ⊂P , and if φ : V → C is a function such that each restriction φ|

fi

is of the above form g i /f i where g i ∈ C[x 1 ,...,x n ] is a homogeneous polynomial and r i is a non-negative integer such that deg(g i )=r i deg(f i ), then φ is regular.

7.13 Exercise Show that every regular function is continuous in Zariski topology. Is the converse true?

7.14 Exercise Show that every regular function is holomorphic. Is the converse true?

7.15 n Exercise Show that all regular functions on an open subvariety V ⊂P form a C-algebra (which we denote by O(V )) under point-wise operations.

be the principal open set U x i . Show that its ring of regular functions is the polynomial ring

7.16 n Exercise For any 0 ≤ i ≤ n, let U

More generally, for any principal open U n

f ⊂P , show that

O(U f ) = C[x 0 ,...,x n , 1/f ] 0 ⊂ C[x 0 ,...,x n , 1/f ] the homogeneous component of degree zero in C[x 0 ,...,x n , 1/f ].

be a locally closed subvariety. A function φ : X → C is called a regular function if for

7.17 n Regular functions on a locally closed subvariety Let X ⊂P

each P n ∈ X, there exists a Zariski open neighbourhood U ⊂ P together with a regular function on ψ : U → C such that φ| X∩U =ψ | X∩U .

7.18 Exercise Show that the regular functions on any locally closed subvariety

X n ⊂C form a C-algebra under point-wise operations. (We denote this algebra by O(X)) over C.

7.19 n Exercise Show that O(P ) = C, that is, the only regular functions on all of P n are the constant functions.

7.20 n Exercise More generally, for any connected closed subvariety X ⊂P , show that the only regular functions are constants, that is, O(X) = C.

The category of all quasi-affine or quasi-projective varieties We now define a category

V of varieties. The objects of this category are all quasi-

affine or quasi-projective varieties (locally closed subsets of C n or P ). Given two varieties X and Y (objects of

V) a morphism f : X → Y in the category V is by definition a continuous map f from X to Y in the Zariski topology such that for any open subset V ⊂ Y and any regular function φ ∈ O(V ), the composite map φ

− 1 − ◦f:f 1 (V ) → C is a regular function on f (V ). These will be called regular morphisms.

− 7.21 1 Pull-back homomorphism f : O(V ) → O(f (V ) If f : X → Y is

a morphism in the category

V and if V ⊂ Y is any open subset, then we get a

− -algebra homomorphism f 1 : O(V ) → O(f (V )) under which φ 7→ φ ◦ f.

7.22 n Exercise Show that the map ψ

i :C →P defined earlier by putting

i :C →P : (z 0 ,...,z i−1 ,z i+1 ,...,z n ) 7→ hz 0 ,...,z i−1 , 1, z i+1 ,...,z n i

is an isomorphism in the category n V of C with the open subvariety U x i ⊂P . (Hint: Show that ψ i is a homeomorphism which has the property that for any open

− subset V 1 ⊂U

i and function φ : V → C, the composite map φ ◦ ψ i :ψ i (V ) → C is − a regular function on ψ 1

i (V ) if and only if φ : V → C is regular.

7.23 Exercise: Composition of regular morphisms Let X, Y and Z be objects of

V and let f : X → Y and g : Y → Z be regular morphisms. Then show that the composite g ◦ f : X → Z is again a regular morphism. Moreover, show that

the composite f # ◦g of the pull-back homomorphisms f : O(Y ) → O(X) and

g # : O(Z) → O(Y ) equals the pull-back homomorphism (g ◦ f) for the composite,

that is, f # ◦g = (g ◦ f) : O(Z) → O(X).

7.24 n Exercise If a locally closed subvariety X ⊂P can be expressed as an

f where Z ⊂P is Zariski closed and U f =P − Z(f) is a principal open defined by some homogeneous polynomial f ∈ C[x 0 ,...,x n ] of degree ≥ 1, then show that X is affine, that is, X is isomorphic to a closed subvariety of some C m .

intersection Z n ∩U