A NOTE ON HILBERT SCHEMES OF NODAL CURVES. Ziv Ran

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1 A NOTE ON HILBERT SCHEMES OF NODAL CURVES Ziv Ran Abstract. We study the Hilbert schee and punctual Hilbert schee of a nodal curve, and the relative Hilbert schee of a faily of curves acquiring a node. The results are then extended to flag Hilbert schees, paraetrizing chains of subschees. We find, notably, that if the total space X of a faily X/B is sooth (over an algebraically closed field k), then the relative Hilbert schee Hilb (X/B) is sooth over k and the flag Hilbert schees are noral and locally coplete intersection, but generally singular. The Hilbert schee paraetrizes ideal sheaves or subschees of Projective Space or ore generally, or a fixed schee X. Perhaps the siplest case is where X is a sooth curve, for then the Hilbert schee Hilb (X) paraetrizing length- subschees of X coincides with the syetric product Sy (X). Our purpose in this note is to study what is in a sense the next siplest case, where X is essentially a curve with ordinary nodes, with planar equation forally equivalent to xy = 0. Besides Hilb (X) itself, there are (at least) 2 other types of Hilbert schee of natural interest here: the punctual one Hilb 0 (X), paraetrizing length- subschees supported at a node; and the relative one Hilb ( X/B), paraetrizing length- subschees in the fibres of the faily X/B that is the ap A 2 A 1 given by xy = t (both Hilb (X) and Hilb ( X/B) ay conveniently be viewed as gers, or foral schees, along Hilb 0 (X). All these will be deterined below. We find that Hilb 0 (X) is a connected chain of ( 1) nonsingular rational curves eeting transeversely; Hilb (X) is a connected chain of (+1) nonsingular -diensional gers, whose first and last ebers are supported on points and each other eber is supported on a coponent of Hilb 0 (X), and only consecutive ebers intersect (and those transversely); finally, Hilb ( X/B) is a sooth ( + 1)-fold. To elucidate the relations between the Hilb for different, with an eye to enuerative applications based on recursion on, we will also study soe flag Hilbert schees Hilb., paraetrizing chains of ideals whose colengths for a given sequence. = ( 1 > 2 >...). For soe purposes, these are easier to work with than ordinary Hilbert schees, due to the natural aps between the Hilb. for different (.). For exaple, for. = (, 1), we find that the punctual Hilbert schee Hilb 0.(X) is a chain of 2 3 copies of P 1 that alternate between those coing fro Hilb 0 (X) and fro Hilb 0 1(X). The relative Hilbert schee Hilb. ( X/B) is still sooth. Things begin to change though with. = (, 1, 2). Here Hilb 0.(X) is still a chain of 2 3 coponents, but now only the 2 external ones on each side are P 1 s and the rest are P 1 P 1. The relative 1991 Matheatics Subject Classification. 14C05, 14H10. Research Partially supported by NSA Grant MDA ; reproduction and distribution of reprints by US governent peritted. 1

2 2 Z. RAN Hilbert schee Hilb. ( X/B) is no longer sooth, but it is still noral with locally coplete intersection singularities. A siilar picture eerges for longer chains,and in particular for the full flag case fhilb = Hilb, 1,...,1. In [R], which uses soe of these results, we will develop an alternative and ore geoetric approach to these objects. In particular, we will identify fhilb ( X/B) with a certain space W ( X/B) constructed as an explicit blow-up of the relative Cartesian product X /B. The proof of this identification uses soe of the results here. We will then apply the spaces W ( X/B) to soe enuerative probles. In these applications, the relatively siple relationship between fhilb and fhilb 1 is critical. We work over a fixed algebraically closed field k. We denote by R localizarion of the ring k[x, y]/(xy) and at its axial ideal (x, y). A typical eleent of R can be written in the for u(a + i 1 b i x i + c i y i ) where u is a unit and the su is finite. The foral copletion ˆR = k[[x, y]]/(xy) = {a + i 1 b i x i + c i y i } (su not necessarily finite) is isoorphic to the foral copletion of the local ring at any 1-diensional ordinary node. We seek first to deterine the punctual Hilbert schee Hilb 0 (R) of colength- ideals in R which, as is well known, is naturally isoorphic to Hilb 0 ( ˆR). At this point, we do not seek to define or copute a natural schee structure on Hilb 0 (R) (so calling it the punctual Hilbert schee is soething of a isnoer) this will be done later (see Corollary 8). For now we siply view Hilb 0 (R) as an algebraic set endowed with a flat faily of ideals that yields a bijective correspondence between the (closed, k valued) points of Hilb 0 (R) and the colength- ideals of R. Theore 1. (i) Every ideal I < R of colength is of one of the following, said to be of type (c i ), (q i ), respectively: I i (a) = (y i + ax i ), 0 a k, i = 1,..., 1; Q i = (x i+1, y i ), i = 1,...,. (ii) The closure Ci in the Hilbert schee of the set of ideals of type (c i ) is isoorphic to P 1 and consists of the ideals of types (c i ) or (q i ) or (q i+1 ). In fact, we have li a 0 I i (a) = Q i, li a I i (a) = Q i+1. (iii) The punctual Hilbert schee Hilb 0 (R), as algebraic set, is a rational chain (1) C 1 Q 2 C 2 Q 1 C 1;

3 HILBERT SCHEME OF NODAL CURVES 3 it has ordinary nodes at Q 2,..., Q 1 and is sooth elsewhere. proof. Recall that eleents z, z R are said to be associate if z = uz for soe unit u. Note that any nonzero nonunit z R is associate to a uniquely deterined eleent of the for x α or y β or x α + ay β, a 0, α, β > 0, in which case we will say that z is of type (α, 0) or (0, β) or (α, β), respectively. Note also that for any ideal I of colength we have x, y I. Now given I of colength, pick z I of inial type (α, β), with respect to the natural partial ordering on types. Suppose to begin with that α, β > 0. Then note that x α+1, y β+1 I, and consequently (α, β) is unique: indeed if (α, β ) is also inial then we ay assue α > α, hence x α I, hence y β I, contradicting iniality. Hence (α, β) is unique and it is then easy to see that the eleent z = x α + ay β I is unique as well, so clearly z generates I and I is of type (c β ). Thus we ay assue that any inial eleent of I is of type (α, 0) or (0, β). Since x, y I, I clearly contains inial eleents of type (α, 0) and (0, β), and then it is easy to see that I is of type (qβ ). This proves assertion (i). Since Ii (a) contains yi+1, x i+1, assertion (ii) is easy. As for (iii), let C = Ci i be an abstract nodal curve as in (1). It follows fro (ii) that each Ci carries a flat faily of ideals, i.e. adits a natural orphis to Hilb (R) which is clearly injective on k-valued as well as k[ɛ]-valued points (copare the coputations in the proof of Theore 2 below). Since these orphiss agree on the intersections Ci Ci+1 = Q i+1, they yield a orphis, again clearly injective on k and k[ɛ] valued points, fro C to Hilb (R), which identifies C with Hilb 0 (R) as an algebraic set. Next we deterine the structure of the full Hilbert schee of R and ˆR: Theore 2. The Hilbert schee Hilb (R) (resp. Hilb ( ˆR) is a chain D 0 D 1 D 1 D where each Di is a sooth and diensional ger (resp. foral schee) supported on Ci for i = 1,..., 1 or Q i for i = 0, ; for i = 1,..., 1, Di eets its neighbors Di±1 transversely in diension 1 and eets no other D i. The generic point of Di corresponds to subschee of Spec(R) coprised of i points on the x-axis and i points on the y-axis. proof. Clearly Hilb (R) (resp. Hilb ( ˆR)) is a ger (resp. foral schee) supported on Hilb 0 (R), so this is a a atter of deterining the schee structure of Hilb (R) and Hilb ( ˆR) at each point of Hilb 0 (R), which ay be done forally by testing on Artin local algebras. Again, we shall do so at Q i, i > 1 as the cases of Ii (a) and Q 1 are siilar and sipler. Given S artinian local augented, a flat S-deforation of I = Q i is given by an ideal I S = (f, g),

4 4 Z. RAN (4) f = x +1 i + f 1 (x) + f 2 (y), g = y i + g 1 (x) + g 2 (y), where f i, g j have coefficients in S, and such that R S /I S is S free of rank, in which case it is clear by Nakayaa s Lea that 1, x,..., x i, y,..., y i 1 is a free basis for R S /I S. It is easy to see that we ay assue f 1, g 1 are in fact polynoials of degree i and f 2, g 2 are of degree < i and f 2, g 2 have no constant ter. Let s write (5) f 1 (x) = i 0 i 1 a j x j, f 2 (y) = b j y j, 1 (6) g 1 (x) = i 0 i 1 c j x j, g 2 (y) = d j y j. 1 Now obviously yf b i 1 g 0 xg c i f od I S. Writing these eleents out in ters of 1, x,..., x i, y,..., y i 1 yields relations aong 1, x,..., x i, y,..., y i 1. Since the latter eleents for an S-free basis of R S /I S, those relations ust be trivial. In other words, we have exact equalities rather than congruences: yf b i 1 g = 0 = xg c i f. Writing out this equality ter by ter yields the following identities b j = b i 1 d j+1, j = 1,..., i 2, b i 1 d 1 = a 0, (7) b i 1 c j = 0, j = 0,..., i, c j = c i a j+1, j = 0,..., i 1, c i a 0 = 0, c i b j = 0, j = 1,..., i 1. (If i = 1 only lines 4 and 5 of display (7) are operational.) Conversely, suppose the relations (7) are satisfied, or equivalently (8) yf b i 1 g = 0 = xg c i f.

5 HILBERT SCHEME OF NODAL CURVES 5 By Nakayaa s Lea, 1, x,..., x i, y,..., y i 1 generate R S /I S, hence to show (4) defines a flat faily it suffices to show these eleents adit no nontrivial S-relations od I S. To this end,, suppose (9) u i (x) + v i 1 (y) = A(x, y)f + B(x, y)g where u, v, A, B are all polynoials in the indicated variables and of the indicated degees (if any) with coefficients in S and v has no constant ter; in fact it clear a priori that then A, B ust have coefficients in S. Then the relations (8) allow us to rewrite (9) as (9 ) u i (x) + v i 1 (y) = A (x)f + B (y)g, with A (x) = a k xk S [[x]], B (x) = b k yk S [[y]]. Coparing coefficients of x i+1,..., y i,... in (9 ) we get relations a 0 = a 1a i +..., b 0 = b 1b i (10)... a k = a k+1a i +..., b k = b k+1b i Starting fro the fact that a j, a k S, j, k, we infer fro these first that, in fact, a k 2 S, k; plugging the latter fact back into the relations (10) we then infer a k 3 S, k, and so on. Since S is artinian it follows that a k = 0, k and likewise for b k. Thus A = B = u i = v i 1 = 0, hence there are no nontrivial relations, as claied. Thus the Hilbert schee is ebedded in the space of the variables i.e. A +1, and defined by the relation a 1,..., a i, d 1,..., d i 1, b i 1, c i, (11) b i 1 c i = 0. Thus it is a union of 2 sooth diensional coponents eeting transversely in a sooth ( 1) diensional subvariety. The generic point on the coponent where b i 1 = 0 (resp. c i = 0) is clearly an ideal generated by g (resp. f), which has the properties as claied. In the case I = I i (a) = (yi + ax i ), a siilar analysis shows that an S- deforation of I is given by the principal ideal I S = (y i + ãx i + f 1 (x) + f 2 (y)) where ã S, ã a od S, i 1 i 1 f 1 (x) = a j x j, f 2 (y) = b j y j, 0 1

6 6 Z. RAN a j, b j S, and via (ã, a 0,..., a i 1, b 1,..., b i 1 ) we ay identify Hilbert schee locally with A. Reark 2.1. We note that in ters of the above coordinates the subset Hilb 0 (R) Hilb (R) is defined by i.e. by the conditions a 1 =... = a i = d 1 =... = d i 1 = 0, f(x, 0) = x i+1, g(0, y) = y i. We shall see below that this, in fact, defined the natural schee structure on Hilb 0 (R). Next we consider the relative local situation, i.e. that of a ger of a (1- paraeter) faily of curves with sooth total space specializing to a node. Thus set R = k[x, y] (x,y), B = k[t] (t), and view R as a B odule via xy = t. As is well known, this is the versal deforation of the node singularity xy = 0, so any faily of nodal curves is locally a pullback of this. Theore 3. The relative Hilbert schee Hilb ( R/B) is forally sooth, forally ( + 1) diensional over k. proof. The relative Hilbert schee paraetrizes length- schees contained in fibres of Spec(R) Spec(B). This eans ideals I S < R S of colength containing xy s for soe s S, such that R S /I S is S free. The analysis of these is virtually identical to that contained in the proof of Theore 2, except that the relation b i 1 c i = 0 gets replaced by (*) b i 1 c i = s and lines 3,5,6 of display (7) are replaced, respectively, by b i 1 c j = sa j+1, j = 0,..., i 1 c i a 0 = sd 1 c i b j = sd j+1, 1 j i 2, relations which already follow fro the other relations (in lines 1,2,4 of display (7)) cobined with (*). Thus, the relative Hilbert schee is the subschee of the affine space of the variables a 1,..., a i, d 1,..., d i 1, b i 1, c i, t defined by the relation (12) b i 1 c i = t, hence is sooth as claied. Reark 3.1. After this was written, the author was infored by Prof. I. Sith of soe related work by hiself and Prof. S. Donaldson [DS, S] which considers the relative Hilbert schee of a pencil of nodal curves on a sooth surface fro a

7 HILBERT SCHEME OF NODAL CURVES 7 rather different viewpoint, valid in the syplectic category over C; in particular, they prove in this context an analogue of Theore 3 (soothness of the total space of the relative Hilbert schee). Construction 3.2. An explicit construction of Hilb (R), globally along Hilb 0 (R), can be given as follows (also see [R2] for further developents). Let C 1,..., C 1 be copies of P 1, with hoogenous coordinates u i, v i on the i-th copy. Let C C 1... C 1 A 1 be the subschee defined by v 1 u 2 = tu 1 v 2,..., v 2 u 1 = tu 2 v 1. Thus C is a reduced coplete intersection of divisors of type (1, 1, 0,..., 0), (0, 1, 1, 0,..., 0),..., (0,..., 0, 1, 1) and it is easy to check that the fibre of C over 0 A 1 is C 0 = [1, 0]... [1, 0] C i [0, 1]... [0, 1] i and that in a neighborhood of C 0, C is sooth and C0 is its unique singular fibre over A 1. We ay identify C 0 is an obvious way with Hilb 0 (R). Next consider an affine space A 2 with coordinates a 0,..., a 1, d 0,..., d 1 and let H C A 2 be the subschee defined by a 0 u 1 = tv 1, d 0 v 1 = tu 1 a 1 u 1 = d 1 v 1,..., a 1 u 1 = d 1 v 1. Set L i = p C i O(1). Then consider the subschee of H A 1 Spec( R) defined by the equations F 0 := x + a 1 x a 1 x + a 0 Γ(O H R)... F 1 := u 1 x 1 + u 1 a 1 x u 1 a 2 x + u 1 a 1 + v 1 y Γ(L 1 R) F i := u i x i +u i a 1 x i u i a i+1 x+u i a i +v i d i+1 y+...+v i d 1 y i 1 +v i y i... Γ(L i R) F := d 0 + d 1 y d 1 y 1 + y Γ(O H R). In view of the above results, it is easy to check that the ideal sheaf I generated by F 0,..., F defines a subschee of H Spec(R) that is flat over H and that ay serve to identify H with Hilb( R)/B. Locally at a point of type c i (resp. Q i ), I is generated by F i (resp. F i 1, F i ). Now let E be the universal bundle on H, whose fibre at a point corresponding to a length- schee z is Γ(O z ), and let V be the trivial rank (2 + 1) bundle on the sybols 1, x,..., x, y,..., y. Since 1, x,..., x, y,..., y generate Γ(O z ) for all z Hilb ( R), we have a surjection V E. Then F 0,..., F together yield a ap 1 F : O H 1 L 1 i O H V

8 8 Z. RAN so we get an exact sequence 1 0 O H L 1 i O H V E 0. 1 In particular, it follows that c 1 (E) = L L 1. Note that A a 0,...,a 1, A d 0,...,d 1 ay be identified, respectively, with Sy A 1 x, Sy A 1 y and accordingly it will be convenient to use the notation σ i = a i, τ i = d i, these being respectively the i-th eleentary syetric functions in the roots of F 0, F. I clai, in fact, that the projection c : H A 2+1 ay be identified with the cycle ap H Sy (Spec( R)/B) : this is because the natural ap Φ : Sy (Spec( R)/B) A σ A τ B is an ebedding. Φ is an ebedding because its restrictions on the generic fibre and the special fibre over B are ebeddings. Moreover, it is not hard to see that the iage of Φ (which coincides with that of c) is schee-theoretically defined by the equations σ i τ j = tσ i 1 τ j 1, i, j = 1,...,, i + j > where we set σ 0 = τ 0 = 1. Setting S =i(φ), we have that H is a Weil divisor in S C. The local analysis iediately yields soe conclusions for the Hilbert schee of a nodal curve: Corollary 4. Let C 0 be a curve with only k nodes as singularities and c irreducible coponents. Then (i) Hilb (C 0 ) is reduced and has precisely ( ) +c 1 coponents, the general eleent of each of which corresponds to a reduced subschee of the sooth part of C 0 ; (ii) let I be a point of Hilb (C 0 ) having colength i at the i-th node of C 0 ; then locally at I,Hilb (C 0 ) is a cartesian product of k factors, each of which is a 2- coponent noral crossing of diension i, i = 1,..., k, or a point if i = 0, ties a sooth factor. (iii) the fibre of the cycle ap cyc : Hilb (C 0 ) Sy (C 0 ) (cf. [A]) over a cycle having ultiplicity i at the ith node is a product of 1- diensional rational chains of length i 1. proof. It is clear fro the explicit analysis in the proof of Theore 2 that any subschee of C 0 defors to a reduced subschee supported on the sooth part. Such subschees are paraetrized by an open dense subset of the syetric product Sy (C 0 ). This clearly yields (i) and (ii), while (iii) follows fro the fact that the fibres of cyc are products of punctual Hilbert schees. Next we extend (ost of) the above results to the case of flag Hilbert schees (see [Se] for a general discussion of those). By definition, for any decreasing sequence of positive integers. = ( 1 >... > k ),

9 HILBERT SCHEME OF NODAL CURVES 9 the flag Hilbert schee Hilb. (R) paraetrizes nested chains of ideals (13) I 1 S <... < I k S < R S such that R S /I j S is S-free of rank j, j = 1,..., k (i.e. such that each I j S is an S point of Hilb j (R)); siilarly for punctual and relative Hilbert schees. Thus a flag Hilbert schee is by definition a subschee of a product (or, in the relative case, a fibre product) of its constituent ordinary Hilbert schees and as such coes equipped with forgetful projection orphiss to those constituents; oreover the defining equations for Hilb. (R) in k Hilb j (R) are just the conditions that the inclusions (13) hold, and these equations involve only pairs of successive factors Hilb j (R), Hilb j+1 (R), j = 1,..., k 1. In the case of full flags, i.e. the case. = ( > 1 >... > 1), we will denote Hilb. (R) by fhilb (R). We begin by analyzing the case of pairs of ideals of relative colength 1: Theore 5. (i)the only colength-( 1) ideal containing Ii (a) is Q 1 i ; the only colength-( 1) ideals containing Q i are the I 1 i 1 (a) for a 0 and their liits Q 1 i and Q 1 i 1. (ii) The punctual flag Hilbert schee Hilb 0, 1(R), as algebraic set, is a chain of nonsingular rational curves of the for C 1 (Q 2,Q 1 1 ) C 1 1 (Q 2,Q 1 2 ) C 2 (Q 1,Q 1 1 ) C 1; it has ordinary nodes at Q 2,..., Q 1 and is sooth elsewhere. Each coponent Ci projects isoorphically to its iage in Hilb 0 (R) and to a point Q 1 i in Hilb 1 (R), and vice-versa for C 1 i. (iii) The flag Hilbert schee Hilb, 1 (R), as foral schee along Hilb 0, 1(R), has noral crossing singularities and at ost triple points. Each of its coponents is forally sooth, diensional and of the for and projects to Di, D 1 i nontrivially iff D, 1 i,i, i = 0,...,, i 1 i i, respectively (cf. Theore 2); D, 1 i,i i j + i j 1; eets D, 1 j,j the coponents of Hilb 0, 1(R) contained in D, 1 i,i (resp. D, 1 i,i 1 ) are Ci and C 1 i (resp. C 1 i 1 and Ci ). (iv) The relative flag Hilbert schee Hilb, 1 ( R/B), as foral schee along Hilb 0, 1(R), is forally sooth and ( + 1)-diensional over k. The natural ap Hilb, 1 ( R/B) Hilb 1 ( R/B) is a flat, locally coplete intersection orphis of relative diension 1. Proof. Assertion (i) is an eleentary consequence of the analysis in the proof of Theore 1 and its proof will be oitted. Assertion (ii) and the set-theoretic portion

10 10 Z. RAN of Assertion (iii) follow fro this. To coplete the proof of (iii),(iv), it reains to analyze the situation locally at each pair (I, I ) Hilb 0, 1(R). We will consider the case of (Q i, Q 1 i ), 1 < i <, as other cases are siilar or sipler. There we will focus ainly on Assertion (iv), as (iii) is essentially a special case of this (as will be indicated below). Consider then a pair (I S < I S ) flatly deforing (Q i, Q 1 i ) relative to B. Then we ay assue that for soe s S, I S and I S are generated by xy s and f, g (resp. f, g ) with i f = x +1 i + a j x j + i g = y i + c j x j + i 1 b j y j, i 1 d j y j, i 1 f = x i + a jx j + i 1 g = y i + c jx j + i 1 b jy j, i 1 d jy j. (For the non-relative case we take s=0.) As we saw above, the relations as in (7), or equivalently (8), are necessary and sufficient so that I S, I S are S-flat deforations of I, I respectively. In particular, these include (14) b i 1 c i = s = b i 1c i 1. The other relations can be used to eliinate soe of the paraeters. It reains to account for the condition that I S < I S. To this end it suffices to note that 1, x,..., x i 1, y,..., y i 1 for an S-free basis of R S /I S, then express f, g in ters of this basis and equate the coefficients to 0. This yields the coefficient relations a 0 = (a i a i 1)a 0 + sb 1, a j = a j 1 (a i a i 1)a j, j = 1,..., i 1, b j = ((a i a i 1)b j + sb j+1, j = 1,..., i 2, b i 1 = (a i a i 1)b i 1, c j = c j + c i a j, j = 0,..., i 1, d j = d j + c i b j, j = 1,..., i 1. These coefficient relations are equivalent to (14.1) f = (x + a i a i 1)f, g = g + c i f.

11 HILBERT SCHEME OF NODAL CURVES 11 By foral anipulations, these relations iply that (15) c i 1 = c i (a i a i 1). Therefore the 2 relations (14) are replaced by the single relation (16) (a i a i 1)b i 1c i = s. Consequently the relative flag Hilbert schee is sooth here, with regular paraeters a 1,..., a i 1, a i, d 1,..., d i 1, b i 1, c i and its fibre, i.e. Hilb, 1 (R), is the noral-crossing triple point (a i a i 1)b i 1c i = 0. The 3 coponents are: D, 1 i 1,i 1, defined by a i a i 1 = 0 (which iplies b i 1 = c i 1 = 0); D, 1 i,i, defined by b i 1 = 0 (which iplies b i 1 = 0); D, 1 i,i 1, defined by c i = 0, (which iplies c i 1 = 0). Finally the relation (15) exhibits Hilb, 1 ( R/B) locally as a conic in an A 2 with coordinates a i, c i over Hilb 1 ( R/B), and therefore the projection is a flat locally coplete intersection orphis. Reark 5.1. In the case of the punctual flag Hilbert schee, we have by Reark 2.1 that the only coefficients in the above calculation not autoatically 0 are b i 1, c i, b i 1, c 1 i, and the vanishing of a i, a i 1, d 1, d 1 yield the relations (17) b i 1 = c i 1 = c i b i 1 = 0. Thus Hilb 0, 1(R) has two coponents at (Q i, Q 1 i ): one where c i = 0, which projects to {Q i } Hilb (R) and to C 1 i 1 Hilb 1 (R); the other where b i 1 = 0 which projects to C i Hilb (R) and to {Q 1 i } Hilb 1 (R). Thus we recover in ters of equations the set-theoretic picture presented in Theore 5(ii). Reark 5.2. For future reference we note that in the analogous case of (relative) deforations of (Q i, Q 1 i 1 ), f, g take the for i f = x i+1 + a jx j + i 2 i b jy j, g = y i 1 + i 2 c jx j + d jy j. The relations (15) and b i 1 = (a i a i 1 )b i 1 (see just above (15)) are replaced by (18) c i = (d i 1 d i 2)c i, b i 2 = (d i 1 d i 2)b i 1. As we shall see, this reark is useful in studying flag Hilbert schees with ore than 2 constituents.

12 12 Z. RAN Reark 5.3. Note that for any (I, I ) Hilb, 1 (R), the annihilator Ann(I /I) < R is an ideal of colength 1. This yields a orphis A : Hilb, 1 (R) Hilb 1 (R) = Spec(R). For exaple, in the situation of (14.1), we have by (the analogue of) (8), yf = b i 1g = b i 1g b i 1c i f, hence J := (x + a i a i 1, y + b i 1 c i) annihilates f od I S. Hence by (14.1) again, J annihilates I S od I S. Note that by (14.1), we have b i 1 c i = d i 1 d i 1. Therefore the value of A on the S valued point (I S, I S ) (which extends the closed point (Q i, Q 1 i )) is A (I S, I S) = (x + a i a i 1, y + d i 1 d i 1). Consequently, the closed (special) fibre of A is defined in ters of f, g, f, g by the condition that f(x, 0) = xf (x, 0), g(0, y) = g (0, y). Construction 5.4. An analogue of Construction 3.2 in the flag case ay be given as follows. Let a i, d i etc be as there and let a i, d i etc. be the analogous objects for Hilb 1. Set r = a 1 a 2, s = d 1 d 2, so that F 0 = F 0(x + r), F = F 1(y + s) and we have the relation rs = t, so that (r, s) give a copy of Spec( R)/B. Then Hilb, 1 ( R/B) ay be relalized as the subschee of C B Spec( R) B Hilb 1 ( R/B) defined by u iv i = ru i v i, v iu i+1 = sv i+1 u i a iu i+1 = sd 1 iv i+1, d 1 iv i = ra iu i, i = 1,..., 1. In extending these results to the case of longer- a fortiori, full- flags, the sae ethods apply. But in the conclusions, a couple of new twists coe up, already for flags of type. = (, 1, 2). First, the punctual Hilbert schee Hilb 0.(R) will have coponents of varying diensions (in this case, 1 and 2), roughly speaking because the paraeters in Hilb 0 (R) and Hilb 0 2(R) can vary independently. Second, the relative flag Hilbert schee Hilb. ( R/B) is not the sae locally at ). The following proof contains the rel- (Q i, Q 1 i, Q 2 i evant coputation. ) as at (Q i, Q 1 i, Q 2 i 1

13 HILBERT SCHEME OF NODAL CURVES 13 Lea 6. Set. = (, 1, 2). Then (i) as algebraic set, Hilb 0.(R) is of the for C 1 C 1 1 C, 2 2,1 C C, 2 2, 3 C 1 2 C 1. Each coponent C, 2 i,i 1 projects isoorphically to Ci C 2 i 1 and to {Q 1 i } Hilb 1 (R). Hilb, 2 (R) (ii) Hilb. ( R/B) is irreducible and is sooth except at points (Q i, Q 1 i where is has a rank-4 quadratic hypersurface singularity with local equation (19) (a i a i 1)c i = (d i 1 d i 2)c i 1, Q 2 i 1 ), proof. The set-theoretic assertion (i) follows fro Theore 5. For (ii), we will analyze Hilb. ( R/B) locally at points of the for (Q i, Q 1 i, Q 2 i ) or (Q i, Q 1 i, Q 2 i 1 ) as other cases are siilar or sipler. Beginning with the forer case, consider a deforation (I S < I S < I S) of (Q i, Q 1 i, Q 2 i ) where I S, I S are as in the proof of Theore 5 and I S is analogously defined by i 2 i 1 f = x i 1 + a j x j + b j y j, i 2 g = y i + i 1 j x j + c d j y j. Then working as in the proof of Theore 5 we find the the ( + 1) paraeters a 1,..., a i 2, d 1,..., d i 1, b i 1, c i, a i 1, a i such that all the coefficients of all our polynoials f,..., g, as well as the paraeter s, are regular expressions in these and there are no relations. This shows that Hilb. ( R/B) is sooth at (Q i, Q 1 i, Q 2 i ). In the case of (Q i, Q 1 i, Q 2 i 1 ), we ay assue i 1 i 2 f = x i + a j x j + b j y j, i 1 g = y i 1 + c i 2 j x j + and analogous considerations yield ( + 2) paraeters d j y j a 1,..., a i 1, d 1,..., d i 2, b i 1, c i, c i 1, d i 1, a i such that all the coefficients of all our polynoials, as well as the paraeter s, are regular expressions in these, and satisfying the relation c i 1 = (a i a i 1)c i = (d i 1 d i 2)c i 1 which yields the equation (19). Finally, irreducibility of Hilb. ( R/B) follows fro the fact that its natural ap to Hilb, 1 ( R/B) is flat with irreducible generic fibre and Hilb, 1 ( R/B) is irreducible by Theore 5(iv).

14 14 Z. RAN Theore 7. The (full) flag Hilbert schee fhilb ( R/B) has locally coplete intersection singularities and its natural ap to B is a local coplete intersection orphis. In particular fhilb ( R/B) is reduced and is flat over B. proof. The fact that fhilb ( R/B) has locally coplete intersection singularities follows by a downwards induction as in the proof of Lea 6. A siilar arguent shows that each ap fhilb ( R/B) fhilb 1 ( R/B) is a locally coplete intersection orphis, hence so is fhilb ( R/B) B. As in Reark 5.3, we have a natural ap fa = A A 1... A 1 : fhilb (R) Hilb 1 (R) = Spec(R) which fits in the diagra fhilb (R) Hilb (R) fa cyc Spec(R) Sy (Spec(R)) Then the closed fibre of fa is called the th punctual flag-hilbert schee of R, denoted fhilb 0 (R), and the (schee-theoretic) projection of fhilb 0 (R) to Hilb (R), i.e. the closed fibre of the cycle ap cyc, is called the th punctual Hilbert schee of R, denoted Hilb 0 (R). It is clear that fhilb 0 (R) and Hilb 0 (R) endow the siilarly denoted algebraic sets discussed previously with a schee structure. Corollary 8. The full flag punctual Hilbert schee fhilb 0 (R) = Hilb 0,...,1(R) is reduced and is the transverse union of coponents of the for C, 2,..., 2j i,i 1,...,i j, i, 1 i 1, j = in([ ], i 1, i 1) 0, 2 which projects isoorphically to Ci factors;... C 2j i j, and to a point in the other C 1, 2,..., 2j 1 i,i 1,...,i j, i, 1 i 2, j = in([ 1 ], i 1, i 1 2) 0, 2 which projects isoorphically to C 1 i... C 2j 1 i j and to a point in the other factors. The punctual Hilbert schee Hilb 0 (R), with the schee structure as above is a reduced nodal curve. proof. Except for the assertion that fhilb 0 (R), hence also Hilb 0 (R) are reduced, this is a straightforward extension of Lea 6; the constraints on j coe siply fro the fact that the coponents of Hilb 0 k (R) are Ck j, j = 1,..., k 1. For the reducedness assertion, we argue by induction on. We ay work locally at a point of the for (Q i, Q 1 i,...), as other cases are siilar or sipler. Consider a deforation of the for (I S = (f, g) < I S = (f, g ) <...) with I S, I S as in the proof of Theore 5, that yields an S-valued point of fhilb 0 (R). By induction, we ay assue f (x, 0) = x i, g (0, y) = y i,

15 HILBERT SCHEME OF NODAL CURVES 15 which iplies that f = x i + b i 1y i 1, g = y i + c i 1x i 1. Then by Reark 5.3 we get a siilar conclusion for f, g. Thus, in ters of the coordinates a 1,..., a i, d 1,..., d i 1, b i 1, c i as in the proof of Theore 2, the subschee Hilb 0 (R) Hilb (R) is siply defined by the vanishing of a 1,..., d i 1, hence is reduced as claied, and likewise for fhilb 0 (R). Reark 8.1. To picure the configuration in Corollary 7, it is ausing to display the coponents of Hilb 0 k (R), k = 2,..., as segents arranged alternately with epty spaces in an isosceles triangle, with each segent overlying an epty space: Then the coponents of fhilb 0 (R) are the colunwise products of these coponents. Reark 9. in [R] we construct, based on geoetric considerations, a space W (X/B) together with a orphis J : W (X/B) fhilb (X/B), which we will prove is an isoorphis. This proof requires that we know a priori that fhilb (X/B) is reduced. Reark 10. It sees likely that the above results go through without the assuption that the base field k is algebraically closed, provided the fibre nodes of X/B are of split type, i.e. each node p, and each of the 2 tangent directions at p, are defined over k. Beyond this however, it sees soe further analysis is needed. For instance, if p is a node defined over k of nonsplit type, i.e. with equation forally equivalent to x 2 + y 2, char(k) 2, the punctual Hilbert schee Hilb 0 at p apparently has just 1 or 0 coponents defined over k depending on whether is even or odd; the other coponents occur in conjugate pairs.

16 16 Z. RAN Correction to proof of Th 1(i). Recall that an eleent z I is inial if z I; a inial basis of I is a collection of eleents z 1,..., z n I that aps to a basis of I/I; by Nakayaa, such a collection always generates I. Now suppose first that I has a inial eleent of the for z = x α. Then z alone cannot generate I so there ust exist an eleent z = ax α + by β independent of z od I, b 0. If α < α, a 0 then z is a ultiple of z, which is a contradiction. Therefore α α hence x α and by β = z ax α α z are independent od I. Moreover for any other z = a x α + b y β I, a siilar arguent shows α α, β β, therefore z (x α, y β ), so x α, y β generate I. Then a siple diension count shows α + β = + 1. Now suppose I has no inial eleent of the for x α or y β, and consider a inial eleent z = x α + by β where α is sallest aong all eleents (or equivalently, aong all inial eleents) of I. Suppose I has another inial eleent z = x α + b y β independent of z od I. Then α α. If β < β then z (b/b )y β β z = x α is inial, contradiction. If β = β then (z z )/(b b ) = y β is inial, contradiction. If β > β, α = α then x α = z (b/b )y β β z is inial, contradiction. Finally, if α < α, β < β then z is a ultiple of z, contradiction. Thus z generates I and again a siple diension count shows z = x i + ay i.

17 HILBERT SCHEME OF NODAL CURVES 17 References [A] B. Angéniol, Failles de Cycles Algébriques- Schéa de Chow, Springer., Lecture Notes in Math. no [DS] S. Donaldson, I. Sith, Lefschetz pencils and the canonical class for syplectic 4-anifolds, arxiv:ath.sg/ [R] Z. Ran, Geoetry on nodal curves (arxiv.org/ath.ag/ ; to appear in Copositio ath. 2005).. [R2], Geoetry on nodal curves II: cycle ap and intersection calculus (arxiv.org/ath.ag/ ). [Se] E. Sernesi, Topics on failies of projective varieties, Queens Univ., 1986, (Queens papers in pure and applied Math. vol. 73). [S] I. Sith, Serre-Taubes duality for pseudo-holoorphic curves, arxiv:ath.sg/ University of California, Riverside E-ail address: