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Math. Z. 201,139-144 (1989) Mathematische zeitschrift

9 Springer-Verlag 1989

On the p-constant Stratum and the V-filtration: an Example

Jan Stevens Mathematisches Seminar der Universitfit Hamburg, BundesstraBe 55, D-2000 Hamburg 13, Federal Republic of Germany

We give an example of a #-constant deformation of an isolated singularity for which the V-filtration on the Jacobi algebra jumps. Using one of Luengo's exam- ples of a singular #-constant stratum we obtain non-smoothness of Z, as corol- lary.

Introduction

The spectrum of an isolated complex hypersurface singularity turns out to be an important invariant in deformation theory. In/~-constant deformations the spectrum is constant and it gives an upper bound for the dimension of the /~-constant stratum Z. in Q1" = 112 {x}/(Sif). This bound is relatively easy to com- pute, but it is not sharp. A better estimate is obtained from another related filtration on QI, determined by the operation of Qy on ~"+ 1/dfA fP by multipli- cation [V-C].

In this note we give an example, where non-smoothness of the #-constant stratum [L 2] can be detected from this multiplication filtration: in one of Luen- go's examples this filtration jumps along Z.. For a one-parameter deformation f~, whose general point is a smooth point of Zu, we show that the linear span of the tangent cone of r.(ft) is the weight > 1 subspace of the filtration. We only use that the stratum described by Luengo is contained in Z,, so as corollary we obtain a proof that Z.(fo) is singular without using Perron's result [P].

We first review the relation between the V-filtration and Z.. A closer analysis of Luengo's example prepares on our computations. These are made with the Newton diagram: we replace the equation by a stably equivalent non-degenerate one.

A general reference for the concepts of singularity theory used in this note is the second volume of the book by Arnol'd, Gusein-Zade and Varchenko [A-V-G].

1. The V-filtration

Let f: (~" + l, 0) -~ (C, O) be a function with an isolated singularity. Let QY =f~e++llo/dfAf2" and Qs=(E{Xo .... ,x.}/(aif), the base of the miniversal

140 J. Stevens

unfolding off. On QS we have the V-filtration [S-St]; the spectrum o f f consists of numbers a~il~ with multiplicity dim Gr}Q I. The choice of a volume form transfers the V-filtration to Qs. If f is non-degenerate for its Newton diagram, then one may compute the V-filtration from the Newton filtration via the inclu- sion (2" + 1 ~ (E{x0, ..., x,} dx o/x ... /x dx,/xo. . . . .x, [V-Kh] (this normalization of the spectrum differs 1 from the one used in [A-V-G]),

One defines a filtration on End QI by: ~0e V ~ End QI, if (o (WQI )c V~+~Q I for all-t,. This filtration induces one on Qr via the inclusion Qsc End QY (let g~ Qs operate on Q r by multiplication).

Theorem [V-C]. Let K ~ Qf be the tangent cone to the #-constant stratum ~ in Q y . Then:

K ~ Q In V I End QI ~ Vl +p Ql '

where fl is the smallest spectrum number.

2. Super-isolated singularities

Definition [L2]. A hypersurface singularity (X, 0) c(lE "+ 1, 0) is called super-iso- lated, if X is resolved by blowing up the origin in C "+ 1.

Lemma [L2]. Let f=fd+fd+l+- ' , with fk homogeneous of degree k. Then f defines a super-isolated singularity, if and only if V(fa+ 1)r~ Sing V(fd) =0 in IP =.

It is instructive to compute the Milnor number in the more general situation f=fd+fe+k+ .-- with V(fd+k)c~SingV(fe)=O; then V(fd) has only isolated

singularities, say Pl .... , p=. We have #(f) = (d - I)" + 1 + k ~ #(V(fd), P3 (cf. Iom- i= l

din's formula [I]). This formula follows from a formula for the characteristic polynomial Pl(t) of the monodromy of f (whose proof will be published else- where): let Pd(t) denote the characteristic polynomial of an isolated homogeneous singularity of degree d; let pik(t)=det(H~-tI), where Hi is the monodromy of the singularity (V(fa), P3. Then:

P~(t) ~ (td+"). PC(t) = (t d _ 1)kS.," I~ P'

i

Returning to super-isolated singularities we see that an equimultiple deformation F(x, t) is g-constant, if and only if Fe(x, t) defines a/~-constant family of projective hypersurfaces. Let Nu, d be the #-constant stratum in the base Sa of the miniversal equimultiple deformation with section [T]. For super-isolated singularities the base Se maps injectively into the base of the versal deformation, if and only if the polynomials O~fd are linearly independent. Luengo's observation is that the equations for Zu. e involve only the degree d part:

Theorem. The #-constant stratum in S d is smooth, if and only if the #-constant stratum in the versat projective deformation of V(fa) is smooth.

The p-constant Stratum and the V-filtration 141

Luengo proves that a g-constant deformation of a super-isolated surface singularity is equimultiple, so 2;, is the stratum obtained from the #-constant deformations of V(f~). His result depends on the work of Perron [p].l

3. A Family of Plane Curves

Let 2;,=S r ...... r~ be the system of plane projective curves of degree d with k simple singularities of type T~ (A, D or E). This system forms an algebraic subset of IW ~d+ 3)/z [W], [G-K].

The infinitesimal description is as follows: let C be a reduced curve in S,

define the sheaf N '=ker N~ p~, where N is the normal bundle of C.

The Zariski tangent space to 2; is H~ ') and the obstructions lie in Ht(N ') [WI. One has the exact sequence:

O--) H~ H~ Q T~p ---r HI(N')~O.

Consider the curve C, given by f=x9+y(xy3+z4) 2 I L l ] . It has an A35 singularity in (0:1:0). We describe the map/_/0 (N) -o T 1. From the exact sequence O~H~176176 we see that the monomials of degree 9 span H~ while the equation f gives one relation. To compute T 1 we use affine coordinates (x, z). Then:

(f, Oxf, 0z f ) = (x 9 -1- (x -~- z4) 2 , 9 x 8 + 2 (x + z4), 8 z 3 (x + z4)).

This ideal contains x 9 and xSz3; as a basis for T 1 we take xiz j, i

142 J. Stevens

So h ~ (N') = 20 and H 1 (N') ~- C. c l(x 7 z3). The polynomials x 9, y axf z ~xf and ms (x + z 4) with x lm5 or z 3 [ms, where deg m5 < 5, span H ~ (N').

The system Z.-=Z~ 35 has a smooth 16-dimensional singular locus, consisting of curves of the form ncp]+l 9, where 1 is a hyperflex of cp4 and n a line; a non-zero element of Hl(N')V=Hom(N',~o) is given by ms (Pg~---~rnsl, mll 8~--~ -mln(p4.

The deformation f+2(X+Zg)(e60XS+~slxgz+e42x3zZ+e33xZz3 ) can be lifted to a trivial deformation of the singular point, by completing the square, but the coefficient %o ~33 + %a 842 of X 7 z 3 iS the obstruction to rewrite the result- ing equation mode 3 as a polynomial of degree 9; in fact, e6o e33+es~ e4z is the obstruction for a general deformation. We have recovered Luengo's result that Z has a cone-like hypersurface singularity [-L 1]. Luengo's methods enable him to find such nice examples.

We also need a smooth point of S; we take

f =x 9 +y(xy 3 +z4) 2 + txS(xy 3 +z4).

We can use the same basis for T 1 as before, but now H~ T 1 is surjective for t+0:

(*) (x + z 4) x2z 3 txT z

So S is smooth in the points f , t ~: 0.

4. The Example

We will compute Qsr~ v 1 End QY for

Je t = X 9 ~- y(x y 3 + z4) 2 -~- t x 5 (x y 3 -~- 7. 4) -~- ylO,

using Newton diagram methods and knowledge of the spectrum of our function. Write q~ = xy 3 + z 4. Then f is stably equivalent to the nondegenerate function

f +(u+ y(v -q~) - t xS)(v+q~)=yl~ + x9 +uv+uxy3 +uz4 + yv2- tvxS. Fig. 2 shows (the projection on a hyperplane of) the Newton diagram re-

stricted to z = 0.

X 9

/

vx '/ ,'/, \ \

,' I I [ , . .

v2y ylO

Fig. 2

The #-constant Stratum and the V-filtration 143

The top-dimensional faces are: I. x + y+ z + 5u+4v=9, containing UZ4, uxy3, uv, V2y, X 9 (and vx 5)

II. 40x + 36y+ 37z + 212u+162v=360 III. 12x+8y+9z+44u+36v=80. The partial derivatives are: 9xS+uy3-5tvx 4, lOy9+3uxyZ+v2, 4uz 3,

v + x y3+ z 4, u + 2y v - tx s. One can take the same basis for Qs as for f (x , y, z), but one has the possibility to write expressions m5 q~ as monomial m5 v.

We now describe the spectrum off(x, y, z). In what follows spectrum numbers k/360 will be denoted by their numerator k, while the other ones are given as full fraction. One obtains Sp f by removing from the spectrum of a homoge- neous polynomial of degree 9 (e.g. Xg+y9+zg) 35 arithmetic progressions of length 9 with steps 1/9 and replacing them by progressions of length 10 and step 1/10, the first being 5/9, 6/9, 7/9, 8/9, 9/9, 10/9, 11/9, 12/9, 13/9, replaced by 199, 235, 271, 307, 343, 379, 415, 451, 487, 523. In general, for fg+y k one finds arithmetic progressions ~ + ilk, i = 0, ..., k - 1 with ~ = (19 k + 9)/36 k, 20 k~ 36k, (21k + 27)/36k . . . . , (53k + 27)/36k, i.e. (mk +9r)/36k, re+r-0(4) , 0__

144 J. Stevens

Every non-trivial linear combination is also not in V I End Q f~ so TSL = Q:o :', V 1 End Q:o

For t4=0 one has dim vt+#QyJTSL=6. We find, besides the same mono- mials as before, an extra one: vx2z 3 (cf. 3 (*)). In this case 9xS+uy 3 - -5 tvx 4 and t v x 4 y, t v x 4 z ~ T S L. The computations above give five independent elements of VX+:QA/Q::~ V 1 End Q A. Because vx2yza=txrz3 /2 in Q:,, we find that e([vx2z3] 9 [y]) = 878 < 880, Therefore TSL = Q:, c~ v 1 End QA.

Corollary 1. The dimension of Q: n V 1 End QY jumps along the #-constant stratum.

Corollary 2 [L2], The #-constant stratum o f f o is not smooth.

Proof. For t # 0 the tangent cone K, to S, (i.e. the set of tangent lines to curves on S. , a space of the same dimension as Z,) is a linear space of the same dimension as the smooth space SL, SO the germ (SL,ft) is an irreducible compo- nent of (S,,ft). Therefore the singular germ (Sr.,fo) is an irreducible component of (S,,fo).

References

[A-V-G]

[G-K]

[L1]

[L23 [P]

[S-St]

[St]

ET]

[V-Kh]

IV-C]

[w]

Arnol'd, V.L, Varchenko, A.N, Gusein-Zade, S.M.: Osobennosti differentsiruemykh oto- brazhenii. Monodromiya i asimptotiki integralov. Moskwa, Nauka, 1974. English transla- tion: Singularities of Differentiable Maps, Volume IL To appear Greuel, G.-M., Karras, U.: Families of varieties with prescribed singularities, Preprint Kaiserslautern 1987 Iomdin, I.N.: Complex surfaces with a one-dimensional set of singularities. (Russian) Sibirsk. Mat. ~, 15, 1061-1082 (1974), MR 56 4~ 5931 Luengo, I.: On the existence of complete families of projective plane curves which are obstructed. J. Lond. Math. Soe. 36, 33-43 (1987) Luengo, I.: The p-constant stratum is not smooth. Invent. Math. 90, 139-152 (1987) Perron, B.: ~> implique