Math. Ann. 308, 559–569 (1997)

Versality for canonical curves and completeintersections

James McKernanDepartment of Mathematics, South Hall, University of California at Santa Barbara,Santa Barbara, CA 93105, USA (e-mail: mckernan@math.ucsb.edu)

Received: 10 January 1994

Mathematics Subject Classi�cation (1991): 14

1 Introduction

Let X be a smooth subvariety of Pr , over an algebraically closed �eld k. Inan attempt to understand the geometry of X , one would like to understandthe singular hyperplane sections of X , and how the family of all hyperplanesections behave around a singular one. Classically this involved the study ofin ection points on curves. If X is arbitrary nothing much can be said, sosuppose that X is general, meaning that it belongs to an open subset of theHilbert scheme of Pr .

It is natural to expect the singularities of hyperplane sections are isolatedand versally deformed by the nearby hyperplane sections. If this property holds,we shall call X hyperplane versal. Versality is de�ned in (1.2), and roughlymeans that every possible deformation appears.Apart from the fact that versality is an interesting property in its own right,

there is often a normal form for a versal deformation, and it is a useful propertyfor enumerative questions.We are able to prove the following results:

1.1 Theorem. Let C be a general canonical curve in Pg−1; where the char-acteristic of k is either zero; or larger than g.Then C is hyperplane versal.

(A canonical curve, means a Gorenstein curve embedded by its dualisingsheaf.) As a very special case, this implies the classical result that the generalcurve has simple Weierstrass points. In [10] and [11], it is shown a generalsurface in P3 is hyperplane versal, and in [8], a slightly weaker result is provedfor a threefold in P4. Unfortunately, for most r and X (probably everythingnot covered in (1.3)), X is not hyperplane versal, due to the appearance ofnon-simple singularities (those with moduli). One can then slightly weaken the

560 J. McKernan

notion of hyperplane versal to hyperplane semi versal (cf. (2.6)). Semi versalis a new notion, which should be of interest independently of the rest of thepaper.

1.2 Theorem. Let X be a general complete intersection in Pr ; of hypersur-faces of high degree; in characteristic zero.Then X is hyperplane semi versal.

Although semi versal is a weaker condition than versality, the two notionsdo coincide for simple singularities, so that if X is hyperplane semi versal, thenany simple singularity is versally deformed. (The isolated simple hypersurfacesingularities have been classi�ed in characteristic zero, they are the A-D-Esingularities, see [4].) It is then easy to conclude:

1.3 Corollary. Let X be a general complete intersection in Pr ; of dimension n;of hypersurfaces of high degree; in characteristic zero. Suppose either n = 1;or n = 2 and r 5 7; or r 5 6.

Then X is hyperplane versal.

Note that (2) of (1.6) is an example of a general smooth curve in P3 whichis not even hyperplane semi versal.Now we give a short sketch of the proof of these results. For a canonical

curve C we give a simple geometric criteria for versality, (2.1) and (2.2). Wethen degenerate C to the union of two rational normal curves (4.1). (1.1) thenfollows by projection and induction, (4.4) and (4.7).To prove (1.2) and (1.3), we introduce a new idea. Previously the meth-

ods of di�erential geometry, such as moving frames and Fundamental formshave been used. However we give a static characterisation of versality in termsof zero dimensional schemes (2.1). We then de�ne another zero dimensionalscheme, the characteristic scheme (2.3), which essentially linearises the prob-lem. (1.2) is then a standard application of Bertini, and the existence of theHilbert scheme.To prove (1.3), we then have to estimate the dimension of the locus of

certain zero dimensional schemes, (3) and (6) of (2.5). Here we use a non-trivial result that an isolated hypersurface singularity can be recovered from itsmoduli algebra, see [7].

Some standard de�nitions

1.4 Notation. We recall some standard notation. Let X be a scheme. Wesay a subset S of X is constructible, if S is the �nite union of locallyclosed subsets. Let z be a zero dimensional scheme. The length of z, isthe dimension of the artinian ring Oz, considered as a vector space overk. If X is projective, then Hilb(X ) denotes the Hilbert scheme of X , andHilbl(X ) the subscheme corresponding to length l zero dimensional subschemesof X .

Versality for canonical curves 561

We recall the de�nition of versality.

1.5 De�nition. Let X be an analytic isolated hypersurface singularity. Thetriple (X; S; �); is said to be a deformation of X; if X is a at family overS; and � is an isomorphism of the �bre X0 (over some special point 0 of S),with X . (We will often drop S and �; and refer to a deformation only by itstotal family X.)A deformation X of X is said to be versal, if every deformation (Y; T )

of X; is pulled back from X; in the obvious way. I.e. there are morphisms : T → S; � : Y→ X such that

Y�−−−−−→ Xy

yT

−−−−−→ S

is a �bre square.If X is an a�ne scheme; over an algebraically closed �eld, there is a

similar notion of a deformation of X; and of versality, except that we have towork in a more abstract category, cf. [12] or [3]. However there is a simplecriteria for versality, see [6]:Suppose X is an isolated hypersurface singularity, given by a polynomial

f in the variables x1; x2; : : : ; xn. The moduli algebra is de�ned to be

k[x1; x2; : : : ; xn]/(

f;@f@x1

;@f@x2

; : : : ;@f@xn

):

Suppose we have a deformation of X; with smooth base S; with localparameters t1; t2; : : : ; tr . Then any deformation of f over S has the formf + t1g1 + t2g2 + : : : + trgr + : : : ; where the dots indicate higher terms inthe t’s. The deformation is versal i� the images of the functions g1; g2; : : : ; gr

generate the moduli algebra.We de�ne the moduli scheme z; to be the subscheme of X; associated to

the moduli algebra.

1.6 Examples(1) Let C be a smooth cubic in P2, over a �eld of characteristic three.

Let c be the intersection of C with one of its ex lines. Then z is isomorphicto c, and any versal deformation space of c has dimension at least three. Itfollows that the neighbouring lines intersected with C never versally deformc. Similarly for nearly all smooth curves in Pr over a �eld of characteristic atmost r + 1.(2) Start with a curve of high degree d in P2. Let V be the Veronese in

P5, and consider the general projection down to P3. We obtain a smooth curve

562 J. McKernan

C, sitting inside the projection V ′ of the Veronese. Hyperplane sections of Ccorrespond to a generic three dimensional linear system of conics in P2. Sincea �nite number of these are double lines, and any double line cuts out d lengthtwo schemes, C is not hyperplane versal.Moreover, C is also a general subvariety of P3. To prove this is a little

involved, and we will only sketch the details. Let C′ be a general curve in P3,which specialises to C. Then C′ has the same degree and genus. By a resultof Harris (see Chapter 3, Exercise Batch G, of [1]), C′ must also lie on anirreducible surface S of degree at most four, as the asymptotic genus of C′

is d2=2. As C does not lie on a cubic or quadric, neither can C′ and so Sspecialises to V ′.

Let D ⊂ S be the one dimensional support of the conductor, and letD′ be the inverse image of D on T , the normalisation of S. Suppose D isempty. Then S is normal. Note that KS = 0, by adjunction. Working up to

constants, C moves in a family of dimension(d+ 22

). Hence so does C′ in

S. By standard deformation theory, h0(NC′=S) grows like d2=2 + (3=2)d. Thus−KS · C′ grows like 3d, a contradiction. Then D specialises to the singularlocus of V ′, three non-coplanar lines which meet in a point. Thus D is a curveof degree at most three and if D has degree three then D is not planar. Thus Sis covered by a net of conics (intersect S with a quadric containing D and con-sider the residual curve) and it is then a classical result that S is the projectionof the Veronese.(3) Let S be a general smooth surface in P8, of large degree. Then a �nite

number of hyperplane sections will be curves with ordinary fourfold points.Such singularities have a versal deformation space of dimension at least nine,and so S is never hyperplane versal.

2 Some interesting characterisations of versality

We will say that a zero dimensional scheme z in Pr is linearly normal, if itis not the projection of another scheme.Normally it is hard to check versality since we have to write down the

functions g1; g2; : : : ; gr of (1.5). However:

2.1 Lemma. Let V be a linear system on a smooth variety X; and D a divisorin V; with isolated singularities. Suppose z is the moduli scheme of D.Then the family corresponding to V versally deforms D; i� the map of

sheaves

V ⊗ OX → Oz

is surjective.In particular if V embeds X in projective space; versality is equivalent to

the condition that z is linearly normal.

Versality for canonical curves 563

Proof. If the functions g0; g1; g2; : : : ; gr are a basis of V , where g0 de�nesD; then the family corresponding to V is given locally by the vanishing ofg0 + t1g1 + t2g2 + : : :+ trgr . Now just apply the de�nitions.

For an arbitrary projective variety, (2.1) does not seem to admit a nicereformulation. However for a canonical curve it does:

2.2 Lemma. Let C be a smooth canonical curve in Pg−1. Suppose that forevery hyperplane H in Pg−1; the support of H ∩ C spans H; and the charac-teristic of k is either zero or larger than g.Then C is hyperplane versal.

Proof. Let D =∑

�ipi be the divisor in C, where the pi are distinct, cor-responding to a canonical hyperplane section. Clearly the cardinality s of thesupport of D is at least g, and so �i 5 g. But then D′ =

∑(�i − 1)pi, where

D′ is the moduli scheme, considered as a divisor on C.By assumption, the points p1; p2; : : : ; ps span H . Thus

1 = h0(C;OC

(KC −

∑ipi

))= h1

(C;OC

(∑ipi

))= h1(C;OC(KC − D′)) :

Looking at the long exact sequence, associated to the restriction exact sequence

0→ OC(KC − D′)→ OC(KC)→ OD′ → 0 ;

we see that the map H 1(C;OC(KC−D′))→ H 1(C;OC(KC)) is an isomorphismof one dimensional vector spaces. But then the map H 0(C;OC(KC))→ OD′ issurjective, and we may apply (2.1).

2.3 De�nition-Lemma. Suppose f de�nes an isolated hypersurface singularityX0; with local co-ords x1; x2; : : : ; xn. Suppose the moduli scheme z has ideal I;in the local rings of the singular points. Let J be the ideal{

g ∈ I∣∣∣∣ @g@xi ∈ I; for every i

}and let c be the corresponding zero dimensional scheme. We will call c; thecharacteristic scheme. Then

(1) z ⊂ c ⊂ X0;(2) if Y is another hypersurface containing c; then the moduli scheme of

Y contains z;(3) z only depends on X0; and(4) c only depends on X0.

564 J. McKernan

Proof. (1) and (2) follow immediately from the de�nitions. (3) is an easyapplication of the chain rule (cf. [7] or [2]). To prove (4), by (3) we onlyneed to show that c depends on z; which again follows from the chain rule.

For the rest of this section, we will assume k has characteristic zero.

2.4 Example. Suppose X0 has an Ak -singularity, so that f may be written asx21 + x22 + : : : x2r−1 + xk+1r . Then z is de�ned by the ideal (x1; x2; : : : ; xr−1) + mk

and c is de�ned by the ideal (x1; x2; : : : ; xr−1)2 +mk+1; where m is the maximalideal of the singular point.

We �x some notation for (2.5) and (2.6). Let � : X→ S be a deformationof the isolated hypersurface singularity X0. Suppose � is projective and let kand l denote the length of the moduli and characteristic scheme c respectively.Let M denote the algebraic group of all (r+1)×(r+1) matrices, and let Mq bethose matrices of rank at most r+1−q. It is well known that Mq has codimen-sion q2 in M . Note that M has a natural action on Pr . Let H = Hilbl(Pr); andlet Hq denote the sublocus of H , parametrising schemes lying in a (r−q)-plane.

2.5 De�nition-Lemma. Let Iw denote the locus of subschemes of Pr isomor-phic to a zero-dimensional scheme w. Set I qw = Iw ∩ Hq:

(1) I qw is irreducible;(2) I qw has codimension at least q in Iw; and(3) I qc has codimension at least l+ k − r in Hq:

Let C be the locus of all characteristic schemes in Pr . Set

Cp = {c | z lies in a (k − 1− p)-plane}:Let i be the codimension of Ic ∩ Cp inside Cp at c.

(4) Cp is a constructible subset of H;(5) Cp has codimension at least p in C; and(6) i is at most the modality of X0 (see [2] for a de�nition of the

modality): In particular if X0 is simple; i = 0:Similarly let CS be the locus of all characteristic schemes of length l; con-tained in �bres of �; whose associated moduli scheme has length k:(7) CS is a constructible subset of Hilb

k(X):

Proof. A standard application of the scheme Isom proves that Iw is a countableunion of constructible sets, see [9]. Since any line bundle on a zero dimensionalscheme w is trivial, any embedding of w in Pr just corresponds to a linearsubspace of P(Ow); up to projective motions. Thus Iw is covered by orbitsof open subsets of M; parameterised by linear subspaces of P(Ow) where I qwcorresponds to Mq. Hence (1) and (2).We may assume that X0 is de�ned by a polynomial f of degree l + 1;

see [2]. Let PN denote the family of hypersurfaces of degree l + 1. Let Vbe a subspace of PN containing f; and let Y be the corresponding family ofhypersurfaces of degree l+1: Choose co-ords x1; x2; : : : ; xr on Pr . Suppose Y is

Versality for canonical curves 565

de�ned by the bi-homogeneous polynomial F; and consider the ideal J which

is constructed from the ideal I =(F;

@F@x1

;@F@x2

; : : : ;@F@xr

)as in (2.3). Let Z be

the subscheme de�ned by J . If we restrict to the constructible locus where Zis �nite over its image in PN ; we obtain a family of characteristic schemesCV → T say, depending only on V .

Let Y be any �bre of Y; with moduli scheme z′ and characteristic schemec′. If Y is close to X0, then the length of z′ is at most k. Here is the crucialobservation. If c′ is isomorphic to c; then z′ must be isomorphic to z; as (2) of(2.3) implies z′ contains a scheme isomorphic to z: But then Y is isomorphicto X0; by [7].

To prove (3) and (4), we may assume q = 0. Using (1.5), it is easy tochoose V so that Y is a versal family, and V has dimension k. But then if Yis close to X0; Y is never isomorphic to X0; see [2]. As CV is the zero locusof a section of a vector bundle, cf. Sect. 3, (3) follows.Now every scheme of length l is contained in a hypersurface of degree

l+ 1. Thus CPN contains every characteristic scheme of length l. By standardproperties of the Hilbert scheme, (4) holds.Since C is covered by the Ic; (5) follows from (2). (7) follows easily

from (4) and (6) follows, by choosing S to be the family which de�nes themodality.

Motivated by 3 of (1.6), and (2.5), we modify the de�nition of versal to:

2.6 De�nition. Suppose S is smooth of dimension r: We say � is semi versal;if CS is a general subvariety of C of codimension at least r; and the functionsg1; g2; : : : ; gr span a general subspace of dimension at least r−p. Hyperplanesemi versal is de�ned accordingly.

3 Complete intersections

We will prove (1.2) and (1.3) in this section, and we will assume that thecharacteristic of k is zero. Suppose we have a vector bundle E of rank l overa variety Z; which is globally generated, i.e. the natural map of sheaves

H 0(Z; E)⊗ OZ → E

is surjective. The idea is to use a form of Bertini, which says we may �nda section � of E, whose zero scheme is transverse to any constructible subsetof Z:First we construct E. Let Z = Zl be the Hilbert scheme of zero dimensional

subschemes of Pr of length l and suppose Z is the universal family inside Y =Pr × Z . We may twist the restriction exact sequence, by OY (d) = p∗OPr (d);where p is projection to the �rst factor, to obtain:

0→ IZ(d)→ OY (d)→ OZ(d)→ 0 :

Let E = q∗(OZ(d)); where q is projection to the last factor.

566 J. McKernan

3.1 Lemma. E is globally generated; for d= l:

Proof. We just have to check that

H 0(Pr ;Pr(d))⊗ OZ → E

is surjective. But R1q∗(IZ(d)) vanishes, since IZ is a at OZ-module, andHi(Pr ; Iz(d)) vanishes, for any scheme of length l.

Proof of 1.2. It is easy to see that there is a constant M (which dependsonly on r), such that any characteristic scheme c of length greater than M , orany non-isolated singularity, contains a scheme isomorphic to the characteristicscheme of an Ar+1-singularity, cf. (2.4).

Suppose d= M: By (3.1) and Bertini, we may �nd a section of E, whosezero locus is transverse to C1 ∩ Hilbl(Pr); for any l 5 M: Since the modulialgebra of an Ar+1-singularity has length r + 1, the corresponding hypersur-face X contains no Ar+1-singularities, by (3) and (6) of (2.5). But then X ishyperplane semi versal, by (5) of (2.5).The general case proceeds by induction, replacing Pr with X .

Proof of 1.3. In the indicated ranges, if X is hyperplane semi versal, then Xhas only simple singularities, see [2]. But then X is automatically hyperplaneversal, by (6) of (2.5).

3.2 Remark. The astute reader will realise that the de�nition of semi versal isa little ad hoc. A more satisfying de�nition would be to require that the familyof zero dimensional schemes contained in �bres of �; meet C transversally.Hopefully this modi�ed de�nition is equivalent to (2.6).

4 Rational and canonical curves

We will prove (1.1) in this section. First a smoothing result.

4.1 Lemma. Let C = C1 ∪ C2 be the union of two rational normal curves inPr ; with only nodes as singularities; joined at r + 2 points.

Then C can be smoothed.

Proof. Observe that C as an abstract curve is stable. As �Mg (the moduli spaceof stable curves of genus g) is irreducible, there is a surface S over a curve �;such that one �bre C0 is isomorphic to C; and the general �bre is smooth, ofgenus g. Now we may embed S in Pr×� via the relative dualising sheaf.Then C0 is embedded as a non-degenerate curve in Pr ; which is project-ively equivalent to C; as the dualising sheaf has degree r = r+2− 2 oneach Ci:

By (2.2), to prove (1.1), we only need to exhibit a smooth canonical curveC in Pg−1 = Pr ; such that:

(4.2) For every hyperplane H in Pg−1, the support of H ∩ C spans H .

Since (4.2) is an open condition, by (4.1), it is enough to prove:

Versality for canonical curves 567

(4.3r) For two general rational normal curves in Pr , joined in r + 2 points,(4.2) holds.

In this context, note that general means only that the points where the tworational curves meet are general.We will be interested in the following curves and their degenerations,

in Pr :

(i) a rational normal curve of degree r;(ii) the smooth projection of a rational normal curve from Pr+1;(iii) the projection of a rational normal curve from Pr+1; with one node.

4.4 Lemma. Let C be a curve of type (i), (ii), or (iii).Then there is a degeneration Ct of C to C′ ∪ L; where L is a line

and C′ is a rational normal curve of degree r − 1 in case (i) or degree rotherwise; joined in a single point; such that each Ct passes through r + 2�xed points of Pr in general position; and L contains precisely two of thesepoints.

Proof. Let � : S → P1 × A1 be the blow up of P1 × A1 at a single point,and E be the exceptional divisor. Let p and q be the composition of �and projection to the factors A1 and P1 respectively, and L the line bun-dle q∗OP1 (d) ⊗ OS(−E); where d is r in case (i), and r + 1 otherwise.Choose r+4 disjoint sections �; �1; �2; : : : ; �r+2; � of p; such that only the �rstthree intersect E. Let F be the vector bundle p∗L in case (i) and (ii), andotherwise de�ne F to be the subvector bundle of p∗L which identi�es thesections � and �:Now any vector bundle is trivial on A1; and so we have

F ' W ⊗ OA1 ;for some vector space W . Let V be the vector space W in case (i) and (iii),and otherwise de�ne V to be a general hyperplane in W: Finally embed S inPr = P(V ∗)×A1; and use the fact that any two sets of r+2 distinct points inlinear general position are conjugate under the action of the projective lineargroup.

Suppose we have one curve of type (i) and another of type (i), (ii) or(iii) in Pr ; joined in r + 2 points. We may degenerate such a con�guration ina family Ct; to C0 = C1 ∪ C2 ∪ L2; where the �rst curve C1 and the pointsof attachment are �xed and the second degenerates to C2 ∪ L2; as in (4.4).Note that C1 and C2 meet in r points, and L2 meets C1 in two points. Wewill call these degenerations type (a), (b) or (c), respectively. Type (d) issimilar, but we assume that the �rst curve is the union L1 of a line and arational normal curve C1 of degree r − 1; joined at a single point, and thesecond curve is of type (i), which we again degenerate to a curve C2 ∪ L2 asin (4.4). Note that C1 and C2 meet in r points, and all other curves meet in apoint.

568 J. McKernan

4.5 Lemma. Suppose we have a degeneration Ct of type (a), (b), (c) or (d),together with a family of hyperplanes Ht . Let �t be the span of the supportof Ht ∩ Ct; and � be the at limit of �t; as t goes to zero.

Then � contains �0.

Proof. This is clear, unless H0 contains a component of C0.But, as � contains any isolated point of H0 ∩C2; H0 ∩C1 or H0 ∩ L1; and

any k 5 r + 1 points of a rational normal curve span a (k − 1)-plane, it iseasy to check the remaining cases.

4.6 Lemma. Let C1 ∪ C2 be the union of two rational normal curves in Pr ;joined in r+2 points. If r = 4; then we can choose C2 so that for any pointp ∈ C2\C1; p does not lie on a tangent of C1.

Proof. Let T be the surface swept out by the tangent lines of C1. It is aclassical result that through any r+3 points of Pr ; there is at most one rationalnormal curve C2 of degree r.

Thus there is an (r − 1)-dimensional family of rational normal curvesC2 meeting C1 in r + 2 �xed points and only a two dimensional familymeet T .

4.7 Lemma. (4.3r) holds for r = 2:

Proof. We will prove (4.3r) by induction. The case r = 2 is trivial. We assumer = 3; and (4.3r−1) holds.

Let C = C1 ∪ C2 be the union of two general rational normal curves inPr ; joined in r + 2 points, H a hyperplane, and � the span of the support ofC ∩ H . The idea is either to �nd a good projection or to degenerate and thenproject.Suppose there is a point p of H ∩ C1; such that p does not lie on a line

contained in H; which is also tangent to one of C1 or C2. Let primes denoteimages under projection from p. By assumption the image of p on C1 is notcontained in H ′. Thus �′ contains H ′ ∩C′. Now C′

1 is a rational normal curveof degree r−1, and C′

2 has at worse a node. If C′2 has degree r; then degenerate

in a family of type (b) or (c). By (4.5) and induction (throwing away the line),we are done.Using (4.6), if r = 4; and otherwise doing the case r = 3 by hand, we

may therefore assume that H is tangent to at least one of C1 and C2 at everypoint of C ∩H: Now degenerate C1 ∪C2; �rst in a family of type (a) and thenof type (d). Suppressing primes, we obtain a curve C = C1 ∪C2 ∪ L1 ∪ L2; thecentral �bre of a type (d) degeneration and a hyperplane H; where it is easy topick the degeneration so that H contains L1 and L2. Now project from L1∩L2;and apply induction and (4.5) twice.

Acknowledgement. I would like to thank Clint McCrory, Theodore Shifrin and RobertVarley, for suggesting that (1.1) should be true, for their encouragement and their hos-pitality during a trip to Georgia. I would also like to thank Joseph Harris, Sean Keel andespecially Lung-Ying Fong for some helpful conversations and suggestions.

Versality for canonical curves 569

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