Singular and Non-singular Varieties

10 Singular and Non-singular Varieties

Regular local rings: definition

10.1 Proposition If A is a noetherian local ring with maximal ideal m and residue field k = A/m, then m/m 2 as a k-vector space has dimension ≥ the Krull dimension

of A.

dim 2

k (m/m ) ≥ dim(A)

10.2 Definition: Regular local ring A noetherian local ring with maximal ideal m and residue field k = A/m is called regular if equality holds: dim 2

k (m/m )= dim(A).

Non-singularity: intrinsic definition

10.3 Non-singular point on a subvariety A point P on a variety X is said to be non-singular point of X (or a variety X is said to be non-singular at a point P ∈ X) if the local ring O X,P is a regular local ring. Otherwise P is said to be a singular point of X (or X is said to be singular at P ).

10.4 Non-singular variety A variety X is said to be non-singular if each point P ∈ X is a non-singular point of X.

10.5 Lemma Any regular local ring is an integral domain.

10.6 Exercise Let X be a variety, let X 1 ⊂ X and X 2 ⊂ X be two different irreducible components of X, and let P ∈X 1 ∩X 2 . Then show using the above lemma that X is singular at P . In particular, if X is a non-singular variety, then X is the disjoint union of all its irreducible components.

Embedded non-singularity: Jacobian criterion

10.7 n Lemma Let X ⊂C be a locally closed subvariety, and let P ∈ X. Let m P ⊂ O 2

X,P denote the maximal ideal of the local ring of X at P , and let dim C (m P /m P ) denote the dimension of the C-vector space m 2

P /m P . Then the following holds. Given any principal open subset U n

h ⊂C such that P ∈U h and X ∩U h is closed in U h , together with any finite set of generators f 1 ,...,f m of the radical ideal I X∩U h ⊂ O(U h ) = C[x 1 ,...,x n , 1/h], we have

rank(∂f 2

i /∂x j )(P ) + dim C (m P /m P )=n

where rank(∂f i /∂x j )(P ) is the rank of the m × n Jacobian matrix (∂f i /∂x j )(P ).

X) = n. By 9.25, the vector spaces T 2

Proof By 9.26, we have rank(∂f i /∂x j )(P ) + dim C (T P

X and m/m are dual of each other so have the same dimension.

10.8 n Theorem For any point P on a locally closed subvariety X ⊂C , the following conditions are equivalent. (1) X is non-singular at P , that is, the local ring O X,P is regular. (2) The algebraic Zariski tangent vector space T P

X has dimension equal to the local dimension dim( O X,P ) of X at P .

(3) Given any principal open subset U ′

h ⊂C such that P ∈U h and X =X ∩U h is closed in U h , together with any finite set of generators f 1 ,...,f m of the radical ideal I X ′ ⊂ O(U h ) = C[x 1 ,...,x n , 1/h], the following holds: the rank of the m ×n Jacobian matrix (∂f i /∂x j )(P ) equals n − d(P ) where d(P ) = dim(O X,P ) denotes the local dimension of X at P . Proof As principal open sets form a basis of open sets in C n , and as X is locally

closed in C n , there exists a principal open subset U

h ⊂C such that P ∈U h and

X ∩U h is closed in U h . For any finite set of generators f 1 ,...,f m of the radical ideal

I X ′ ⊂ O(U h ) = C[x 1 ,...,x n , 1/h], by the above lemma we have

rank(∂f 2

i /∂x j )(P ) + dim C (m P /m P )=n

P /m P ) = dim( O X,P ) = d(P ) the local dimension of X at P ) if and only if rank(∂f i /∂x j )(P ) = n − d(P ). This proves the theorem.

Therefore, X is non-singular at X (that is, dim 2 C (m

Non-singular locus is open and non-empty.

10.9 Theorem For any non-empty variety X, there exists a nonempty open sub- variety U ⊂ X such that a point P lies in U if and only if X is non-singular at P.

Proof (1) Openness of the set U of nonsingular points: Let P ∈ U. As by assump- tion X is non-singular at P , by an above exercise (which uses the fact that regular local rings are domains), it follows that P lies on exactly one irreducible component of X. Replacing X by the complement in X of all other irreducible components of

X, we can assume without loss of generality that X is irreducible. As the question is local on X, we may assume that X is a closed subvariety of C n . For any finite

set of generators f 1 ,...,f m of the radical ideal I = I X , by the above theorem at the point P , we have equality (∂f i /∂x j )(P ) equals n − d(P ) where d(P ) = dim(O X,P ) denotes the local dimension of X at P . As X is irreducible, dim( O X,P ) = dim(X), hence we get an equality

rank(∂f i /∂x j )(P ) = n − dim(X)

However, the rank of a matrix of continuous functions is lower-semicontinuous (as if a certain minor has determinant non-zero at a point, it remains non-zero in a However, the rank of a matrix of continuous functions is lower-semicontinuous (as if a certain minor has determinant non-zero at a point, it remains non-zero in a

rank(∂f i /∂x j )(Q) ≥ n − dim(X)

On the other hand, we know that rank(∂f 2

i /∂x j )(Q) = n − dim(m Q /m Q ) ≤ n − dim(O X,Q )=n − dim(X) Therefore we have the equality rank(∂f i /∂x j )(Q) = n − dim(O X,Q ) at all points of

V . Hence by the above theorem X is non-singular at all points of V , showing that

V ⊂ U. Therefore U is open as claimed. (2) U is non-empty when X is non-empty: Without loss of generality, we can assume

that X is a closed irreducible subvariety of C n . Let I ⊂ C[x

1 ,...,x n ] be the corre- sponding radical ideal, which is therefore a prime ideal, and O(X) is the C-algebra

C [x 1 ,...,x n ]/I which is a domain of finite-type over C. Let y 1 ,...,y d ∈ O(X) be

a maximal set of algebraically independent elements over C (where d is the dimen- sion of X). The field of fractions K(X) of O(X) is therefore a finite extension the rational function field C(y 1 ,...,y d ). As C is characteristic zero, this is a separable extension. (More generally, when working over an algebraically closed base field k of finite characteristic instead of C, we use the theorem on the existence of a sep-

arating transcendence basis y 1 ,...,y d ). Hence by the primitive element theorem, there exists an element α ∈ K(X) such that

K(X) = C(y 1 ,...,y d )[α]

The element α satisfies a monic irreducible polynomial g(t) ∈ C(y 1 ,...,y d )[t] so that

K(X) = C(y 1 ,...,y d )[t]/(g) with α the class of t.

The coefficients of g(t) are rational functions in y 1 ,...,y d . If b ∈ C[y 1 ,...,y d ] is the L.C.M. of their denominators (determined up to a non-zero constant by unique factorisation on polynomials), then we obtain an irreducible polynomial

f (t) = bg(t) ∈ C[y 1 ,...,y d , t]

of degree ≥ 1 in t, such that K(X) is isomorphic to C(y 1 ,...,y d )[t]/(f ). Let C d+1 have linear coordinates denoted by y

1 ,...,y d , t. Consider the closed sub- variety

Y = Z(f (t)) d+1 ⊂C

defined by the vanishing of f . We claim that X has a non-empty open subvariety

V which is isomorphic to a non-empty open subvariety W of Y . Let the chosen primitive element α ∈ K(X) be written as a quotient α = u/v where u, v ∈ O(X), v 6= 0. Then the vanishing of v defines a proper closed subvariety Z(v) ⊂ X, which

has a non-empty open complement V = X − Z(v). Then we have y 1 ,...,y d ,α ∈ O(V ). Hence we have a morphism

θ = (y d+1

1 ,...,y d , α) : V →C

As the image of θ lies in Y d+1 ⊂C , this defines a morphism V → Y which we again denote by θ.

Note that x 1 ,...,x n ∈ O(X) ⊂ K(X) = C(y 1 ,...,y d )[α]. Hence each x i can be expressed as a ratio of polynomials a i (y 1 ,...,y d , α)/b i (y 1 ,...,y d ), where the de- nominators are not identically equal to zero. As t does not occur in b i , note that

f (t) does not divide b i . Hence the vanishing each b i defines a proper closed subva- riety Z(b i ) ⊂ Y . As Y is irreducible, Y is not a union of any collection of finitely many proper closed subvarieties, hence the open complement W = Y −∪ i Z(b i ) is

non-empty. Then x 1 ,...,x n ∈ O(W ), hence we get a morphism

η = (x n

1 ,...,x n ):W →C

As the image of η lies in X n ⊂C , this defines a morphism W → X which we again denote by η.

It follows from their definitions that η ◦ θ = id V and θ ◦ η = id W . We will now show that for the subvariety Y d+1 ⊂C , the set of non-singular points is nonempty. As Y is closed of dimension d, with its radical ideal equal to the principal ideal (f ), by the above theorem it is non-singular at all points P ∈ Y where the

1 × (d + 1)-matrix (∂f/∂y 1 , . . . , ∂f /∂y d , ∂f /∂t)(P ) is not zero. As t occurs in f with degree ≥ 1, the derivative ∂f/∂t is not the zero polynomial. (Here we used that C has characteristic zero. However, it is possible to appropriately modify the argument so as to apply to any perfect field of finite characteristic). The t-degree of ∂f /∂t is less that that of f , so f cannot divide it, so ∂f /∂t is not identically zero on Y . Hence Y has a non-empty open subset N of non-singular points.

By irreducibility of Y , it follows that N ∩ W is non-empty. Using the isomorphism θ:V → W , it follows that V (and hence X) has at least one non-singular point.

This completes the proof of the theorem.

10.10 Note Inside the part (2) of the above proof is hidden the basic concept of birationality. It will be made explicit later in some other series of lectures in this Instructional School.

Some holomorphic facts

10.11 n Exercise If V is a non-empty connected open subset of C in the Euclidean topology, then the ring H(V ) of all holomorphic functions on V is a domain.

10.12 n Proposition Let X ⊂C

be Zariski closed, and let f 1 ,...,f m generate

I X ⊂ C[x 1 ,...,x n ], the ideal consisting of all polynomial functions vanishing on

be a Euclidean-open subset and let g : V → C be a holomorphic function such that g vanishes on X ∩ V . Then there exist holomorphic functions

X. Let V n ⊂C X. Let V n ⊂C

J X∩V = H(V )I X

Proof (Sketch) Take any P = (a 1 ,...,a n ) ∈ X ∩ V . The ideal I is radical, hence its extension in the completion b O C n ,P = C[[x 1 −a 1 ,...,x n −a n ]] is again a radical ideal. Note that O C n ,P ⊂H C n ,P ⊂b O C n ,P , so I X generates a radical ideal in H C n ,P . By analytic nullstellensatz, if g is a germ of a holomorphic function around P such that g vanishes on X, then g lies in the radical of the analytic ideal generated by

I X . From this the result follows.

10.13 GAGA The above proposition also follows by GAGA, but the above proof is more elementary.

10.14 n Exercise Let X ⊂C and Y ⊂C be irreducible closed subvarieties, and let f : X → Y be a regular morphism. Suppose that the induced C-algebra homo-

morphism f # : O(Y ) → O(X) makes O(X) a finite algebra over O(Y ). Then show that (i) the map f is closed in Zariski topology, and (ii) the map f is proper hence

closed in Euclidean topology (hint: show that the inverse image of any bounded set is bounded).

10.15 n Proposition If X ⊂C is an irreducible closed subvariety, then any proper Zariski-closed subset of X is no-where dense in the Euclidean topology. Conse-

quently, any non-empty Zariski-open subset is Euclidean-dense, and any non-empty Euclidean-open subset is Zariski-dense in X. Proof (Sketch) When X = C n , we leave this as an exercise. In the general case, use Noether normalisation to find a finite projection X m →C , and then use the inverse function theorem to show that outside a Zariski-closed subset it is a finite covering projection, to prove the result for X.

Comparison theorem: Nonsingularity and submanifolds

10.16 n Theorem Let X ⊂C be a locally closed subvariety, and let P ∈ X. Then the following statements are equivalent. (1) The variety X is non-singular at the point P . (2) There exists a Euclidean open neighbourhood W of P in C n such that X ∩W is a closed connected holomorphic submanifold of W . (3) There exists a Euclidean open neighbourhood W of P in C n such that X ∩W

is a closed connected real ∞ C -submanifold of W .

(4) There exists a Euclidean open neighbourhood W of P in C n such that X ∩W is a closed connected real differential submanifold of W of class 1 C .

Moreover, when the above equivalent conditions are fulfilled, the dimension of X ∩W as a topological manifold is equal to twice the the local dimension of X at P as an algebraic variety.

Proof (1) ⇒ (2) Suppose X is non-singular at P . Then there exists a principal open neighbourhood U n

h of P in C such that X ∩U h is closed in U h , irreducible and non-singular. If X ∩U h defined by the radical ideal I ⊂ O(U h ), then for any set of generators f 1 ,...,f m of the radical ideal and any point Q ∈X∩U h , the rank of the m × n Jacobian matrix (∂f i /∂x j )(Q) equals n − d where d = dim(X ∩ U h ).

Let (after possible re-indexing) the first r = n − d rows and columns of the matrix (∂f i /∂x j )(P ) be linearly independent. The non-vanishing of the determinant of the

r ×r-minor (∂f i /∂x j ) 1≤ i≤r,1≤j≤r defines a Zariski-open neighbourhood N of P in U h . Let φ : N r →C

be the map (f 1 ,...,f r ). Hence applying the implicit function theorem to the map φ, we see that the vanishing of φ defines a closed holomorphic submanifold Y of N of dimension n −r = d. As df r+1 , . . . , df m are linear combinations

of df 1 , . . . , df r on Y , it follows by integration that f r+1 ,...,f m are locally constant on Y . Hence a Euclidean open neighbourhood of P in Y lies in X. This shows that

X is a holomorphic manifold of dimension d. (2) ⇒ (3) ⇒ (4) : Obvious. (Note: Actually, we have not introduced the notion of a

C 1 -submanifold in these lectures so far: the reader may either just ignore statement (4) and go from (3) back to (1) using the below argument for (4) implies (1), or

otherwise the reader can learn the definition of 1 C -submanifolds elsewhere.) (4) n ⇒ (2) Suppose that there exists a Euclidean open neighbourhood W of P in C

such that X 1 ∩ W is a closed connected C -submanifold of W of dimension r. As non-singular points are dense in the Euclidean topology, the real tangent space

X to X at P is the limit of tangent spaces at a sequence of regular points tending to P . Hence T P

X is a complex-linear subspace, say of complex dimension d (so r must be even, equal to 2d). Then there exists a coordinate plane E of dimension d in

X on E is an isomorphism. Hence in a Euclidean neighbourhood of π(P ) 1 ∈ E, there exist n − d complex C functions h

C n such that the projection π of T P

1 ,...,h n−d whose graph is the intersection of X with a cubical neighbourhood of P inside W . Again, as regular points are dense, it follows that the functions h 1 ,...,h n−d satisfy the Cauchy-Riemann equations. Hence X ∩ W is in fact a holomorphic submanifold of W .

(2) n ⇒ (1) Let W be a Euclidean open neighbourhood of P in C such that X ∩W is a closed connected holomorphic submanifold of W of complex dimension d. Let U h be

h is closed. We first show that P lies on a unique irreducible component of X ′ . Let P

a principal open neighbourhood of X in C ′ such that X =X ∩U

∈D⊂X ′ where D is Euclidean open in X ′ and is holomorphically isomorphic to a polydisk. If X

1 and

2 are two irreducible components of X passing through P , then by a Proposition above (which says that any non-empty Euclidean open subset is Zariski-dense in an irreducible variety) it follows that D ∩X i is Zariski-dense in X i for i = 1, 2. Hence

from the fact that H(D) is a domain, we see that X 1 =X 2 .

Hence we can now assume that X ′ =X ∩U

h is closed in U h and irreducible.

Let d ′ denote the dimension of X as an algebraic variety, which by irreducibil- ity equals dim( ′ O

X,Q ) for all Q ∈X . There exists a holomorphic coordinate chart (U ; z ′

1 ,...,z n ) in U h such that X ∩ U is defined by vanishing of z d+1 ,...,z n . The (n − d) × n Jacobian matrix (∂z i /∂x j ) where d + 1 ≤ i ≤ n and 1 ≤ j ≤ n has rank n ′ − d. If f

1 ,...,f m generate the radical ideal of X in O(U h ), then by a proposition above, the functions z d+1 ,...,z n are holomorphic linear combinations of f 1 ,...,f m around P . Clearly, f 1 ,...,f m are holomorphic linear combinations of z d+1 ,...,z n . Hence we get

rank(∂f i /∂x j )(P ) = n −d

As any non-empty Zariski-open is Euclidean-dense, there exists some Q ∈ B such

that X ′ is non-singular at Q. Then rank(∂f

i /∂x j )(Q) = n −d where d = dim(X ). This shows d ′ = d. Hence the equality rank(∂f

i /∂x j )(P ) = n − d now shows that

X ′ is non-singular at P . This completes the proof of the theorem.

¤ The above proofs of Proposition 10.12 and Theorem 10.16 are taken from page 13 of

John Milnor: Singular points of complex hypersurfaces, Princeton University Press, 1968. I thank Gurjar for showing me this reference.