DEFORMATION OF SCHEMES DEFINED BY VANISHING OF PFAFFIANS

1. INTRODUCTION.

Let k be a field and Y an algebraic scheme over k • By a _9-efo:gnation of Y we mean a flat morphism q:V ~ W of algebraic schemes V and W such that Y is isomorphic to the fiber of q at some rational point of W. The deforma- .. · tion is called ~~~E=~~ha~ if the fiber of q at each point in an open dense subset of W is non-singular.

We shall in the following work deal with the problem of constructing deformations of an affine scheme defined by vanishing of Pfaffians of an alternating matrix (see section 2 for definitions).

In section 3 we show that the generic schemes P28 (X) (the scheme of all alternating mXm-matrices whose Pfaffians of order 2s vanish) have properties like generic determinan- tal schemes: they are reduced and irreducible and P2s_2 (X) is the singular locus of P2s(X). It is proved by Room (see 7, 10.4.t1r, p. 200) that the scheme P2s(X) has dimension [m(m-1)-(m-2s+1)(m-2s+2)]/2. We give another proof of this dimension formula.

In section 4 we use results proved by D. Laksov in 5 to construct defamations of schemes defined by vanishing of Pfaffians. Especially, we show that a .·Gorenstein point Y in A3 ₌ Spec(k[x1,x2 ,x3]) ^{has a} non-~ingular deformation.

To obtain this we use a structure theorem for Gorenstein ideals I of height three in a regular local ring R • D.A. Buchsbaum and D. Eisenbud have in 1 showed that such ideals can be generated by Pfaffians of order 2n of an alternating (2n+1) X (2n+1)-matrix with entries in R •

From this we obtain an element d in k[x~,x

2

^,x

3

^] such that Y is the closed subscheme of Spec(k[x1,x2,x3]d) (k[x1,x2,x3Jd is the quotient ring of k[x 1,x2

,x

3] by the multiplicative set {1,d,d2 , •.•

1)

where the Pfaffians of an alternating matrix with entries in k[x1,x2 ,x3]d vanish. Uning this matrix we construct a deformation q:V ~ W of Y such that the fiber of q at all points in an open dense subset of W consists of distrinct points each of multiplicity one.

A. Iarrobino and J. Emsalem have in 4 proved that certain types of local Gorenstein algebras of length n in three variables have no deformations to Spec (k[x]/(xn)) • Together with our result on non-singular deformations of

Gorenstein points in &~ ?: , this gives the existence of points in A3 which have non-singular deformations although

they have no deformations to Spec(k[x]/(~)) • 2. BA. SIC PROPERTI~S _Q~FFIANS.

Let R be a commutative ring. A square matrix with entries in R is called alternating if it is skew symme- tric and if all its diagonal elements are zero.

Let M be an alternating nXn-matrix with entries in R • If n is an odd number, then det (M) -- 0 • For n even, det(M)

=

(Pf(M)) 2 , where Pf(M) is a polynomial function of the entries ⁱⁿ M (see 1, Lemma 2.3 or 2, p. 82-84 or 6, Theorem 7, p. 373). The polynomial Pf(I-1:) is called the Pfaffi.?-A of N

Denote by obtained from

M . . the alternating (n-2)x (n-2)-matrix

~,J

IJI by deleting the ith and jth row and the ith and jth column. Then for any 1 ~ i ~ n , the Pfaffian

of M can be computed by the formula

Pf(M)

=

^(2.1)

where m. . is the

l,J ( . . )th l,J ^{entry of M .} (see 1, Lemma 2.4).

If we delete the same n-2s rows and columns from

r1 ,

we get an alternating 2s X 2s-ma trix. The Pfaff ian of this matrix is called ¹¹a Pfaffian of M of order 2s .. u Deno·[;e by Pf28^(M) the ideal in R generated by all Pfaffians of M of order 2s. By virtue of the formula (2.1~ we get that

where 1 ~ s

5

n/2 -1 •

If It(M) is the ideal in R generated by all minors of M of order t , then for each s with 1 < s ~ n/2 we have that

I2^{~ (M)}

S

_{Pf 28} (1\1) :::, Rad ( I 28 ^{(H) )}

(2.2)

(2.3) (2.4) where Rad(I 2s(M)) denotes the redical of the ideal I^2s(M)

(see 1, Corollary 2.6).

Let k be a field and let x. . , 1 < i < j < m

l ' J =., ~~ be

m(m-1)/2 algebraically independent elements over k (m a number > 2). If we put

i > j , the mX m-matrix

PROPOSITION 3.1.

x . .

=

0

l,J

X

=

(x. . ) l,J

and x . .

=

-x . , .

l , J J. ~

is alternating.

is for

Let Q be a minimal prime ideal in the polynomial ring

!~

P

=

k[x 1 , 2, •••• ,xm- 1,m] containing the ideal Pf28(X). Then . the height of Q is (m-2s+2)(m-2S+1)/2 •

Proof: We use induction on m • For S=1 the state- ment in the proposition is obvious.

Suppose m ~ 4 and s >

-

2 •

Since Pf 2s(X) is a homogeneous ideal we have that Rad (Pf 2s(X))

S

(x 1 2, •••• ^,~_1 m) • But the closed subset

, '

V(Pf 2s(X)) of the m(m-1)/2-dimensional affine space contains points· not in V(x_{1 ,}2 , •••• ^J~_1 ^,m), e.g. point

(1,0, •••• ,0) • Hence Rad(Pf2s(X))

*

(x1, 2 , •••• ^,~_1 ^{,m) and} we may suppose that x 1 2 is not in Q .

Considered as a ^matri~

'

with elements in the localized ring Px 1 2 we can operate on the rows and columns of

,

^X until X has the form

0 1 0

.

•

.

⁰

-1 0 0

. . .

⁰

0 0

X' =

xn

0 0

where X" is an alternating (m-2) x (m-2)-matrix with entries x . . 1,J + c. . , l,J 3 :S i < j < m and c. . l,J consists of sums of elements from the first two rows of X •

Clearly the ideals Pf (X')

2s and Pf (X^{11 )}

2.s-2 in ~1 2

are equal. Using the formula (2.4) of section 2 v1e get

'

that

Rad (Pf 2s(X)) ⁼ Rad(I 2s(X)) Rad (I2s(X')) ⁼ Rad(Pf2s_ 2 (X^{11 ))}

But, considered as ideals in Px 1 2 , Rad(I2s(X)) is equal to Rad (I 2s(X)')) • Consequently the ideal

' QPx

1 2 will be a minimal prime ideal containing Pf (X^{11 )}

'

2s-2 Thus, by induction, the height of

hence also the height of (m-2s+1)/2 , as required.

PROPOSITION 3.2.

Q in

QP.x1 2

,

in P) is equal to

in

Q.E.D.

The affine scheme P2s (X)

=

Spec(k[x1, 2 , ••.•• ,xm_ 1 ,mJ/

Pf2 (X» 2:,5 2s ^~ m , has the following properties:

(A) P2s(X) is a reduced and irreducible subscheme of codimension (m-2s+2)(m-2s+1)/2 in the affine m(m-1)/2-simensional space of all alternating

mX m-matrices.

(B) The scheme P28 _2^(X) is the singular locus of the scheme P_{28 (X) •}

Proof: The codimension formula of (A) follows at once from Proposition 3.1. Moreover if S=1 both (A) and (B) is obvious. Suppose s > 2 and let b 1 , •••• ,b₁ denote the Pfaffians of X of order 2s-2 Let P2 (X)b· , 1 s l ^< i < 1 , be the affine open subscheme of P26 (X) defined by

LEMMA 3.3.

(i) P2s(X)bi is regular and irreducible.

(ii) b 1, ••• ,b1 can be arranged in a sequence such tbat for each 2 < k < 1 , P2 (X)bk

n

P2s(X)bt is non- empty for at least one t , 1 < t < k •

(lli) The union of the schemes P2s(X)bi is dense in P2s(X) • Proof of Leiil£la ~-~~ Let b be the Pfaffian of order

2s-2 obtained by deleting the first m-2s+2 rows and the first m-2s+2 columns from X • Using the formula for

expansion of Pfaffians along a row (see{2.1) in section 2) we get that (m-2s+2){m-2s+1)/2 of the generators of Pf 2s(X)

can be written

where

bx . . +A . .

l,J l,J ^{1 <} _~· ⁱ < · < _J m-2s+2 A . .

l,J is a polynomial in the variables v ~ m-2s+3 •

of order 2s

Indeed, bx. . + A. . is the Pfaffian of X l,J l,J

obtained by deleting the first m-2s+2 rows expect the ith and jth row and the first m-2. +2 columns

t th .;th and

excep . e .... .th J

Let I be the ideal in k[x 1 2, ••• ,xm_ 1 m] generated

' '

by bx. . + A. . 1 < i < j < m-2s+2 • The scheme l,J l,J

I

pb =Spec (k[x1,2'"""'xm-1,m]b/Ik[x1,2'"""'xm-1,m]b) isomorphic to Spec (k[x1 2, ••• ,x 1 ]b/J where J _{, m- ,m} ideal in k[x 1 2, ••• ,xm_ 1 m]b generated by

' '

x . . l,J

I

'

is is the

1 < i < j ~ m-2s+2 • This gives that Pb is a regular and

irreducible affine scheme of dimension im(m-1)-(m-2s+2)(m-2s+1)]/2.

But is a closed subscheme of of the same

I I

simension as Pb • Hence Pb and P2s(X)b are equal, and P2s(X) 0 is regular and irreducible.

To prove t i i ) we must a:.crange ^b1, ••• , b1 in a sequence such that for every k, 2 ~ k ~ 1 , bkbt is not in

Rad{Pf28 {X)) for at least one t , 1 < t < k . But, suppose bk and bt are Pfaffians of two submatrices of X of size 2s-2 which has (2s-3)(2s-4)/2 common entries.

Then any Pfaffian of X of order 2s consists of sums of monomials such that each term in this sum contains a variable which is not in bk and bt • Hence bkbt is not in

Rad (Pf 28 (X)) ^~

On the other hand we can list b1 ^~ ••• ,b₁ in a sequence such that for each k > 2 , the matrix defining bk has (2s-3)(2s-4)/2 common entries with at least one of the

matrices defining b 1, ••• ,bk_ 1 This gives a proof of (ii).

Let Q be a minimal prime ideal in P2s(X) From Propostion 3.1. we conclude that Q is not in P₂s_ 2 (X) But the complement of P2s_ 2 (X) in P2s(X) is equal to

l

U P2 (X)b- , so this m1ion is dense in ^P~·s(X) Thus the last

i=1 s ^l ~

part of the lemma is shown.

We now complete the proof of Propostion 3.2.

Let S be the singular locus of P₂s(X) and denote by f_{1 , •••} ,fr the Pfaffians of X of order 2s. Using the expansion formula for Pfaffians (see (2.1) of section get that all entries in the Jacobian matrix (911__)

ox

_u,v

Pfaffians of X of order 28-2 or zero. It follows

2) we are at once that P28_2 (X) :: S • But the complement of P2s_2 (X) in

l

P2s(X) is equal to the union ^i~

1

P28 (X)bi and each of the schemes P2s(X)bi are regular (see (i) of Lemma 3.3).

Therefore S

=

P2s_2 (X) and (B) is proved.

Let R be a noetherian ring and look at the following conditions about R for k

=

0,1,2 •••• ~

(Sk) it holds that depth pE Spec(R) •

(R ) > inf(k,ht(p))

p - for all

(Rk) if p E Spec (R) and ht(p) < k , then Rp is

regular.

It is proved in EGA (see 3, Proposition 5.8.5, p. 108) that R is reduced if and only if (R_{0 )} and (S 1) are satisfied.

Put R ⁼ k[ x 1 , 2 , ••• ,

:xw__

1,m ]/Pf 2_s (X) and take a prime ideal Q in R with ht(Q) ~ 1 • Then by Proposition 3.1.

Q is not in P28_2 (X) and it follows from statement (B) of the proposition that RQ is regular. Hence both (R_{0 )} and (S 1) holds for R and we have shown that P2s(X) is reduced.

It remains to prove that P28 (X) is irreduc1ble.

Suppose P2 (X) _{S .} ^{= ZA} _I U

z

2 and suppose we have ^prove~

that P2 (X)b. s ^l

= z-1

^I il P? (X),_,. , 1 ^{'-B ,.)l} <_ i <_ k-1 , 2 < ^~ k < ^~-~ 1 •

We have that P28 (X)bk

=

_{[P2 8}(X)bk n

z

₁

J

^U _[P28(X)bk il

z

_{2 ] .}

But P2s(X)bk is irreducible (see Lemma 3.3, (i)) and there- fore equal to P2 s(X)bkn

z

1 or P28 (X)bk ^il

z

2 • Using that p2s(X)bk intersects one of the schemes p2:s(X)bi

'

^{1 <} i ::; k-1

(see Lemma 3.3, (ii)) and that P2 s(X)bk is non-singular

(see Lemma 3.3, (i)) we conclude that P2 s(X)bk = P2 s(X)bk n 21 Thus l.J1 ^'7 contains the union of the schemes p2 (X)b·

l

1 < i < 1

'

and since this union is dense in P2 s(x) (see Lemma 3.3, (iii)) we have that P2 s(X) is irreducible.

Q.E.D.

~~~~-~.:.1·

If m

=

2s+1 the scheme P2

:s<x)

is Cohen-J'f.JB.ca.nlay, i.e.

the :ring k[ x 1 2 , ••• , x2

s

2 . + 1 ]/Pf 2 'S(X) is Cohen-Macanlay

, '

^~

(see 1, Proposition 6.1).

For other values of s (except the tri""tial cases s = 1

0

or 2s=m) it is not known if P2s(X) is Cohen-Macanlay or not.

4.

CONSTR1[Q~~F9RMATlONS OF ~I001ES*_DEFINED BY

JANISHING 0~ PFAF¥I&NS.

Let Z = Spec(A) be an affine open subset of the p- deimensional affine space ~p

=

Spec(k[Z 1, ••• ,zp]) • Put .!A.q = Spec(k[Y1, ••• ,Yq]) and let ^{f:Z ~.J:Aq} be a morphism

of affine schemes. Denote by the image of ^y,

J by the homomorphism k[Y 1, ••• ,Yq] ^{-7 A} corresponding to the

morphism f • J:l1oreover, denote by ^G = Spec ^{(k[ U} 1 1, ^U 1 2 , ••• ,

' '

up,q'v 1, ... ,vq]) the affine space of (p+1)xq-matrices and by e the rational point of G corresponding to the matrix with all entries equal to zero.

by the

Define a homomorphism of rings

$:k[Y 1, ••• ,YqJ ^-7 A[u1, 1,u1, 2 , ••• ,up,q'v1, ••• ,vq]

p

$(Y.) = E U . . Z. + V. + fJ.(Z) • Let F:GX Z ~A\.q J i=1 l,J l J

morphism of affine schemes corresponding to $ • Let ~ = D₀

S

_D1 ^~ ••• ~ Dc = D be a sequence of irreducible subschemes of .J:Aq = M and suppose D is

be

Cohen-!l[acaulay. J:l1oreover, assume that _{D. 1}

l - is the singular locus of Di , i=1, ••• ,c •

Denote by V the open subscheme of the scheme F- 1(D) = (GX

z)y

where the morphism

qD : J!,--1 (D) ~ G

induced by the projection of GX Z onto the first factor, is flat (see 3, rv3, (11.1.1)).

For each rational point g of the scheme G we denote by _fg the restriction of the morphism F to the scheme (gx Z)

'"::; z •

Note that by the associativity formula, the fiber q"D1 (g) = g xG(GX Z)xMD is isomorphic to the inverse image f- 1 (D) g = {gx Z)xMD of D by fg •

D. Laksov has in 5 proved that qD and fg have the following properties (see

5,

Theorem 2 of section

3

and the proposition of section

4):

E~Q~Q§!~!Q~-±~1~ (D. Laksov)

If f-_e ¹(D) is a subscheme of Z of pure codimension codim (D'M) , then the following conditions hold:

(a) The fiber· q~(e) is contained in V.

(b)

(i) (ii)

There exists an open dense subset

u

of G such

that for each point g of

u

the following assertions holds:

The fiber qD g - f g D ^{-1( ) rv -1( )} is contained in

v

_•

Each scheme in the sequence

••• c f-1 (D )

=

f-g·J (D)

- g c

is of pure codimension codim (D. ,M) in Z

l

• .C'

l.L codim (D. ,M) _{' l} is greater than dim M) •

(empty

(ni)

f~

¹

(Di_ 1 ⁾

is the singular locus of the scheme

f-_g 1 (D.) for i

=

1, .•• ,c.

l

We are interested in the following special case~ Let

Y be a closed subscheme of pure codimension three in Z

=

Spec(A) defined by vanishing of Pfaffians of ^orde:t.~ 2n of an alter-·

nating (2n+1) X (2n+1 )-matrix A •

F

=

(f. . )

l , J with entries in

Let M=Spec(k[x1 2, ••• ,x2n 2n+ 1]) be the affine

' '

n(2n+1 )-dimensional space of alternating (2n+1) x (2n+1 )- matrices. Denote by P28 the scheme of all alternating

(2n+1)X (2n+1)-matrices whoSe Pfaffians of order 2s vanish

0 < s < n •

In section 3 we have proved the following:

Y>

=

^p o ~- ^p 2 <::. • • • • S ^p 2n

is a sequence of irreducible subschemes of M and P2 _2 is the singular locus of P28 , s=1, ••• ,n (see Proposition 3.2). Moreover, P2n

=

P is Cohen-Macanlay (see Remark 3.4).

Now, define a homomorphism of rings

by sendign x. . to f. . , 1 < i < j < 2n+ 1 • Then Y

l,J l,J -

is the scheme theoretic inverse image of P by the morphism of affine schemes

corresponding to ~ •

Remember that codim(P,M) is three, and since V is supposed to have pure codimension three in Z we can use Propostion 4.1 to obtain the following result:

THEOREM 4 .. 2 •

---

Let Z

=

^Spec(A) be an affine open subset of the p-

dimensional affine space Ap , p ~ 3 • Suppose Y is the closed subscheme of Z where the Pfaffians of order 2n of an alternating (2n+1)X (2n+1) -matrix F with entries in A vanish. Moreover, suppose Y has pure codimension three in Z •

Then there exists a flat morphism

q:V-+1:!

from an algebraic scheme V to a regular, irreducible

algebraic scheme W and an open dense subset U of W such that:

(a) There exists a rational point e in vi such that

the scheme Y is isomorphic to the fiber of q at e • (b) For each rational point g of U there exists an

alternating (2n+1)X {2n+1} matrix F(g) with

(i)

entries in A with the following properties:

q-1(g) The fiber

8losed subscheme of

is isomorhic to P2n(F(g)) (the Z where the Pfaffians of F(g) of order 2n vanish).

(ii) Each scheme P2s(F(g)) in the sequence

is empty or of pure codimension (2n-28+3)(n-s+1) in Z •

(ili) P28 _2(F(g)) is the singular locus of the scheme P28 (F(g)) , 1 < s < n •

2Q~Q~~!!!_1!~·

Let Y be a Gorenstein point in & 3 , i.e.

Y ⁼ Spec(k[x1,x2,x3]/I) where k[x1,x2,x3]/I is a local Gorenstein ring of dimension zero. Then Y has non-singular deformations.

g1:oof of~the~~o~Ja~~ We will show that there exists an element d in k[x 1,x2,x3] such that k[x1,x2,x3]/I is ssomorphic to k[x 1,x2 ,x3 ]d/Ik[x1,x2 ,x3Jd and such that the ideal Ik[x 1,x2 ,x3]d is generated by Pfaffians of an alter- nating matrix with entries in k[x 1,x2,x3]d •

First, localizing in the maximal ideal Q containing I , we can write k[x 1,x2 ,x3]/I as a quotient of the local ring k[x1,x2,x3

]Q

by the ideal Ik[x1,x2 ,x3

]Q •

^Vie then use the Pfaffian structure of Gorenstein ideals of height three in regular local rings (see 1, Theorem 2.1)~ If R is a regular local ring and J is a Gorenstein ideal in R of height three (i4e. R/J is a Gorenstein ring of demension

dimR-3) then there exists an alternating (2n+1) x (2n+1 )- matrix N with entries in R such that J is equal to

generated by the Pfaffians of order 2n of an alternating, matrix F' with entries in kfx1,x2 ,x3

]Q .

If we multiply

each entry in F' by the procuct of the denominators of the entries in F' we get an altenating matrix F with entries in k[x 1,x2 ,x3] such that Pf2n(F')

=

Pf 2n(F)k[x1,x2 ,x3

]Q.

Since Ik[x1,x2 ,x3

]Q =

Pf211 (F)k[x1,x2 ,x3

]Q

we can find an element d in k(x 1,x2 ,x3] , d not in Q , such that

Ik[x1 ,x2 ,x3]d

=

Pf2n(F)k[x1,x2 ,x3 ]d •

By Theorem 4.2 with Z

=

Spec(k[x1,x2 ,x3Jd) we can con- struct a deformation

where the fiber of q at all points g in an open dense subset of W has a stratification

such that each member in this stratification is the singular locus of the preceding. Moreover either P2s(g) has

codimension (2n-2S+3)(n-s+1) in Z or P28(g) is empty.

But since Z has dimension three P2n_ 2 (g) is empty and hence P2n(g) is non-singular.

Q.E.D.

~~~-1!.1·

Iarrobino and Ernsalem ask in 4 if a point Y in 1A r which has non-singular deformations, has a deformation to Spec(k(x]/(xn)) too, i.e. a deformation q ~ V ~ W where the fiber of q at every point in an open dense subset of W is isomorphic to Spec (k[x]/ (xn)) •

But there exists a Gorenstein point in .M.3 which has no deformations to Spec(k[x]/(xn)) (see 4, Theorem 3.35).

Thus, byvirtueof Corollary 4.3 there is not, in general, a positive answer to the question.

R E F E R E N C E S :

1. D.A. BUCHSBAUM AND Do EISE~illUD, Algebra structures for finite free resolutions, and some structure theorem for ideals of codimension 3, unpublished.

2. N. BOURBAKI, ¹¹Algebre^{11 ,} Hermann, Paris, 1958.

3. A. GROTHENDIECK, Elements de geometrie algebrique, Fubl.

Math. de I.H.E.S., no. 20, 24, 28, 32 (1964, 1965, 1966, 1967).

4. A. IARROBINO AND J. EHSALElv1, Finite algebras having small tangent space; some zerodimensional generic singularities.

unpublished.

5. D. LA.KSOV, Deformation of determinantal varieties, Compo- si tio : 1>1ath. ( 1975), 273-292.

6.. S. LA.NG, "Algebra 11 , Addison-Wesley, Massachusetts, California. London, Sydney, Manila, 1971.

7. T.G. HOON, ¹¹The geometry of determinantal loci, ¹¹ Cambridge University Press, Cambridge, 1938.