On special projections of varieties: epitome to a theorem of Beniamino Segre

(de Gruyter 2001

On special projections of varieties: epitome to a theorem of Beniamino Segre

Alberto Calabri and Ciro Ciliberto

(Communicated by A. Sommese)

Abstract. In this paper we make some comments and improvements on a theorem of Benia- mino Segre, concerning the locus of points from which an algebraic variety is not projected generically one-to-one.

1 Introduction

A well-known and useful technique in algebraic geometry is the linear projection of a given projective variety XHP^r, which we will usually assume to be irreducible, reduced and non-degenerate, i.e. not contained in any proper subspace of P^r. It is clear that the projectionpfrom a general point of the ambient space of a varietyX, which is not a hypersurface, is such that p_jX is generically one-to-one, i.e. it is bira- tional to its image. For example, if n:ˆdimX<rÿ1, by applyingrÿnÿ1 such projections, one may considerXas birationally equivalent to a hypersurface inP^n‡1. However, it may be interesting to know also what is the locusS…X†of points from whichXisprojected multiply, i.e. the locus of points from which the projection ofX is not generically one-to-one. Since Beniamino Segre already studied in [4] the prop- erties of S…X†, we will call it the Segre locusofX. More precisely, we say that the projection p_z:P^r!P^rÿ1 from a point zBX onto a hyperplane P^rÿ1 not passing throughzis aspecial projectionofXifp_zjX is not generically one-to-one. We de®ne the Segre locus of an irreducible, reduced, algebraic varietyXHP^ras:

S…X†:ˆ fzAP^rÿX :p_zis a special projection of Xg:

where Y is the Zariski closure of the subsetYinP^r. For example, if XHP^n‡1 is a hypersurface (of degree>1), thenS…X† ˆP^n‡1. In [4] Segre proved the following:

Theorem 1.Let XHP^r be an irreducible,non-degenerate,algebraic variety of dimen- sion n<rÿ1.Then the Segre locusS…X†is the union of ®nitely many linear subspaces of P^r and all of its irreducible components have dimension strictly less than n. Fur- thermore,a linear k-spacePHP^r,with0<k<n,is contained inS…X†if and only if either one of the following equivalent properties holds:

(i) X lies on an…n‡1†-dimensional cone with vertex atP;

(ii) the tangent space to X at a general point cutsPin a subspace of dimension kÿ1.

FinallyPis an irreducible component ofS…X†if and only ifPenjoys either(i)or(ii) and it is maximal under this condition.This in turn happens if and only if the maximal vertex of the cone in(i)coincides withP.

Recall that a variety XHP^r is a cone if there is a point zAX such that for every other pointxAX the line joiningxandzlies inX. In this casezis called a 0- dimensional vertex of X. The set of 0-dimensional vertices of X is a subspace of P^r called themaximal vertex ofX. Any subspace of the maximal vertex is called a vertexofX.

Theorem 1 implies the following corollary, which has somewhat unexpected appli- cations to problems in numerical algebraic geometry, as shown by Sommese, Ver- schelde and Wampler in [7]. Letcˆrÿnÿ1 and letxˆ …x1;. . .;xc†be a point of X^c. Let us denote by Cone…X;x†the cone overXwith vertex at the linear subspace ofP^rspanned byx1;. . .;xc.

Corollary 2.Let X be as in Theorem1.Then:

7

xAX^cnD

Cone…X;x† ˆXUS…X†

where D is the set of xˆ …x1;. . .;xc†AX^c such that the linear space spanned by

x1;. . .;xchas dimension strictly less than cÿ1 (if cˆ1thenDˆq).

Note thatDin the above statement is a proper subset ofX^cby the General Posi- tion Theorem (see [1], pg. 109).

Section 2 will be devoted to explain some general properties of the Segre locus, to revise Segre's proof of Theorem 1 and to prove Corollary 2.

Following Segre, we de®ne a zero- (resp. positive) dimensional component of S…X† to be a centre (resp. an axis) of X. In [5] Segre shows that for every l;n;r such that 0<l<nWrÿ2 and for everym>0, there exists an irreducible algebraic varietyXHP^rof dimensionnsuch thatXhas an axis of dimensionland moreover m centers. He also studies the possible con®gurations of the axes of a surface. By extending some of these results, in section 3 we will show the following improvement of Theorem 1:

Theorem 3.Let XHP^r be an irreducible,non-degenerate,algebraic variety of dimen- sion n<rÿ1which is not a cone.Then the Segre locusS…X†is the disjoint union of

®nitely many linear subspaces of P^r. If P1 and P2 are two distinct axes of X, then dim…P1† ‡dim…P2†Wn‡1.Moreover

(i) Ifdim…P1† ‡dim…P2†Xn,thenP1 andP2are the only axes of X,nˆrÿ2and X is the complete intersection of two cones of dimension n‡1with vertices atP1

andP₂.

(ii) Ifdim…P1† ‡dim…P2† ˆn‡1thenS…X† ˆP1UP2. In the last section 4 we discuss some open problems.

2 Properties of the Segre locus

In this paper, XHP^r will always be an irreducible, reduced, algebraic variety of dimensionn, whereP^rˆP^r…C†is the projectiver-dimensional space over the com- plex numbers. If x is a smooth point ofX, we will denote by T_X;x the projective tangent space toXatx. IfZis any subset ofP^r, we denote, as usual, by Span…Z†the smallest linear subspace ofP^r containingZ.

If Y is a variety in P^r and P is a subspace of P^r of dimension l, we denote by Cone…Y;P†the cone overYwith vertex atP, that is the Zariski closure of the union of all theP^l‡1's joiningPwith a point inYÿPVY.

LetXbe a cone. IfPis a vertex of dimensionlofXandYis the intersection ofX with a P^m independent of P, for r>mXrÿlÿ1, then X ˆCone…Y;P†. If mˆ rÿlÿ1, thenPis the maximal vertex ofXif and only ifYis not a cone.

We can now list some basic properties of the Segre locus:

Lemma 4. Let X be an irreducible, reduced, projective variety of dimension n in P^r. Then:

(i) for every xAX,S…X†JCone…X;x†JSpan…X†;

(ii) dimS…X†Wn‡1;

(iii) if X ˆSpan…X†,i.e. if X is a linear subspace ofP^r,thenS…X† ˆq;

(iv) dim…S…X†† ˆn‡1 if and only if dim…Span…X†† ˆn‡1, i.e. if and only if S…X† ˆSpan…X†;

(v) if P is a subspace of P^r such that Span…X†VPˆq, then S…Cone…X;P†† ˆ Cone…S…X†;P†;

(vi) if P is a hyperplane of P^r such that XVP is irreducible and reduced, then S…X†VPJS…XVP†;

(vii) if zAP^rÿ …XUS…X††, set pˆpz, and suppose that for an irreducible compo- nent Z ofS…X†,p…Z†is not contained in p…X†.Then p…Z†JS…p…X††. In par- ticular,if zAP^ris a general point,thenp…S…X††JS…p…X††.

Proof. (i) and (ii): By de®nition of the Segre locus, if zAS…X†, the line Ljoining z with a general point xAX intersects X also at another point y, hence zALJ Cone…X;x†JSpan…X†. Moreover dim…Cone…X;x††Wn‡1.

(iii): For every zAP^rÿX, any line through zintersects Xin almost one point, thuspzis not a special projection ofX.

(iv): By (i) and (iii) we may assume that Span…X† ˆP^randr>n. If dim…S…X†† ˆ n‡1, thenS…X† ˆCone…X;x†for everyxAX by (i). Hence the maximal vertex of the coneS…X†coincides with Span…X†, thusS…X† ˆP^r. Ifrˆn‡1, then a general

line through any point zBX intersects Xin d points, where dˆdeg…X†>1, thus zAS…X†, hence S…X† ˆP^r. Finally, if S…X† ˆSpan…X†, then dim…Span…X†† ˆ dim…S…X†† ˆn‡1 by (ii) and (iii).

(v): We may assume that Span…X†andPspan the whole ofP^r.

LetzAS…X†and letwbe a general point on the line joiningzand a pointvAP.

Since p_zis a special projection ofX, the line Ljoiningzwith a general pointxAX intersectsXat another pointyAX. Notice thatLlies in Span…X†, henceLVPˆq.

Thus the lines joiningvwithxandyare distinct. Therefore the line joiningwwith a point of a general linevxof Cone…X;P†intersects this cone at another point lying on the linevy. This means thatwAS…Cone…X;P††.

Conversely, let zAS…Cone…X;P†† ÿCone…X;P†. The line L joining z with a general point xACone…X;P†intersects Xat another point yACone…X;P†. Notice that LVPˆq otherwise L would be contained in Cone…X;P†, whereas zB Cone…X;P†. Letp be the projection ofP^r fromPto Span…X†. Then the linep…L†

containsz⁰ˆp…z†BX andp…x†, which is a general point ofX, and it also contains p…y†AX; that implies thatz⁰AS…X†.

(vi): LetzAPVS…X† ÿPVX. By the assumption thatXVPis reduced, the line joiningzwith a general pointxAXVPintersectsX, henceXVP, at another point y, thereforexAS…XVP†.

(vii): LetwAZbe a general point. We may assume thatp…w† ˆwby choosing the hyperplane whichpprojects onto. Notice thatp…w† ˆwBp…X†by our assumption.

Let xAX be a general point and setyˆp…x†. Then the line joiningwwithxinter- sectsXat another pointx⁰(not lying on the line zx), hence the line joiningwwithy intersectsp…X†also iny⁰ˆp…x⁰†0y. The ®nal assertion easily follows.

Notice that, in view of Lemma 4, we may and will assume, without loss of gener- ality, thatXis non-degenerate. We may also ignore, from now on, the trivial cases in whichXis either a projective space or a hypersurface or a cone.

For the proof of Segre's theorem 1 we need to recall the following result from [2], Proposition 4.1:

Proposition 5.Let X be an irreducible,reduced,projective variety of dimension n inP^r and letPbe a subspace ofP^rof dimension l.The tangent space TX;xto X at a general point xAX intersectsPin a subspace of dimension h if and only if the projection Y of X fromPto aP^rÿlÿ1has dimension nÿhÿ1.This in turn happens if and only if X sits on an …n‡lÿh†-dimensional cone with vertex at P, namely on the Cone…Y;P†. In particular:

(i) in case hˆl:X is a cone with vertex atPif and only if the tangent space T_X;xto X at a general point xAX containsP;

(ii) in case hˆn:X is contained inPif and only if the tangent space T_X;xto X at a general point xAX is contained inP;

(iii) in case hˆnÿ1:X is contained in aP^l‡1containingPif and only if the tangent space TX;xto X at a general point xAX intersectsP in a subspace of dimension nÿ1.

Now we are ready for the:

Proof of Theorem 1.Notice that properties (i) and (ii) in the statement of Theorem 1 are equivalent by Proposition 5.

LetZbe an irreducible component ofS…X†. Fix a general pointxAX. For a gen- eral pointzAZ, the lineL_zjoiningzwithx intersectsXat another pointy. Notice that both T_X;x and T_X;y lie in a subspace of dimension n‡1 which is tangent to Cone…X;z† along the line L_z. Let Y be an irreducible component of the Zariski closure of the locus of such y's as zvaries inZ. Then h:ˆdimYWn and T_Y;yJ T_X;y, hence T_Y;y intersects T_X;x in a subspace of dimension hÿ1, therefore Y is contained in a subspace P_x of dimension n‡1 containing T_X;x by Proposition 5, (iii). Now Pxcontains both xandyhence it contains the line Lz, thusPxcontains the general pointzAZ, i.e.ZJPx. Notice thatPxdepends onx, otherwiseXwould be contained in PxˆP^n‡1, against the assumptions. ThereforePˆ7_xAXPx is a linear subspace of dimension lWn which containsZ. For every xAX, the tangent spaceTX;xintersectsPin a subspace of dimensionlÿ1, because bothTX;xandPlie inPx. The caselˆnis ruled out by Proposition 5 and the assumptions. Hencel<n andX is contained in a cone with vertex atP by Proposition 5, (i). Finally pz is a special projection ofXfor everyzAP, thusZˆP.

At this point the statement clearly follows.

We ®nish this section with the

Proof of Corollary 2. Let Sˆ7_xAXcnDCone…X;x†. Clearly XJS. Lemma 4, (i) implies thatS…X†J 7_xAXCone…X;x†JS. It remains to prove thatSJXUS…X†.

Assume ®rst that cˆ1. If zASÿX, then the line L joining z to a general point xAX sits in Cone…X;x†. ThereforeLintersectsXat some other pointyAX and this means thatp_zjX is not birational, i.e.zAS…X†.

Assume now that c>1 and suppose that there is a point zASÿ …XUS…X††.

Then the projection ofXfromzto aP^rÿ1would be ann-dimensional varietyYwhich would enjoy the following property: every P^cÿ1, which cuts Y at c independent points, contains a further point ofY. This property contradicts the General Position Theorem (see [1], pg. 109).

3 On the axes of a variety

This section is devoted to the proof of Theorem 3. This will be done in a few di¨erent steps. The ®rst one is the following:

Proposition 6.Let XHP^r be an irreducible,non-degenerate, algebraic variety of di- mension n<rÿ1 which is not a cone. Suppose that X has two distinct axesP1 and P2.ThenP1VP2ˆq.

Proof.LetT :ˆTX;xbe the tangent space toXat its general pointx. We will use the following notation: for hˆ1;2, we set lhˆdim…Ph†, ThˆTVPh, thˆdim…Th†,

iˆdim…P1VP2†, jˆdim…T1VT2†, PˆSpan…P1UP2†, lˆdim…P† ˆl1‡l2ÿi, YˆSpan…T1UT2†,yˆdim…Y†. Theorem 1 implies thatth ˆlhÿ1, forhˆ1;2.

We argue by contradiction and we assumeiX0. SinceXis not a cone, Proposition 5, (i) forcesj<i, hence:

rÿ2XnXyˆt₁‡t₂ÿj

X…l₁ÿ1† ‡ …l₂ÿ1† ÿ …iÿ1† ˆl₁‡l₂ÿiÿ1ˆlÿ1

thusPis a proper subspace ofP^r. Ifyˆl, thenYˆPhencePJT, but this is not possible because Xshould be a cone by Proposition 5, (i). It follows thatyˆlÿ1, i.e.jˆiÿ1.

If l<n, thenP would enjoy property (ii) of the statement of Theorem 1, contra- dicting the fact thatP1 andP2are irreducible components ofS…X†. The caselˆn is also excluded by Proposition 5, (iii), sincelˆnWrÿ2. Therefore we may assume lXn‡1. On the other handlÿ1ˆyWn, hencelˆn‡1 andYˆT. SinceYJ P, we ®nd a contradiction by Proposition 5, (ii).

The next step is as follows:

Proposition 7.Let XHP^r be an irreducible,non-degenerate, algebraic variety of di- mension n<rÿ1 which is not a cone. Suppose that X has two distinct axesP₁ and P₂.Thendim…P₁† ‡dim…P₂†Wn‡1.Moreover ifdim…P₁† ‡dim…P₂†Xn then nˆ rÿ2and X is the complete intersection of two cones of dimension n‡1 with vertices respectively atP₁andP₂.

Proof.We keep the above notation. SinceP1andP2are disjoint, the same is true for T1 andT2. ThusnXyˆt1‡t2‡1ˆl1‡l2ÿ1.

If l1‡l2ˆn‡1, then T ˆYJP. Proposition 5, (ii) implies that PˆP^r, thus rˆlˆl1‡l2‡1ˆn‡2, namely X has codimension 2 and X is contained in Cone…X;P1†VCone…X;P2†. Actually X is equal to the complete intersection of the two cones. Indeed, for iˆ1;2, any P^li‡1 generator of Cone…X;Pi† cuts Cone…X;P_3ÿi†along a varietyYand Cone…X;P_3ÿi† ˆCone…Y;P_3ÿi†. This implies that Y is irreducible and reduced. One moment of re¯ection shows then that the complete intersection of Cone…X;P₁† and Cone…X;P₂† itself is irreducible and re- duced, i.e. it coincides withX.

If l₁‡l₂ˆn, then yˆnÿ1. Proposition 5, (iii) forces n‡1ˆl₁‡l₂‡1ˆ lXrÿ1, hence again rˆn‡2 and, as above, X is the complete intersection of Cone…X;P₁†and Cone…X;P₂†.

The ®nal step consists in proving the following result, which, together with Prop- ositions 6 and 7, implies Theorem 3:

Proposition 8.Let XHP^r be an irreducible,non-degenerate, algebraic variety of di-

mension nˆrÿ2which is not a cone.Suppose thatP1 and P2 are two distinct irre- ducible components ofS…X†.ThenS…X†VSpan…P1UP2† ˆP1UP2.

Proof.Again we keep the above notation. We ®rst treat the casel₁‡l₂Wn.

Let ‰x₀;. . .;x_rŠ be projective coordinates inP^r. We ®x the hyperplane at in®nity

P_y to be the one with equationx₀ˆ0 and we consider…x₁;. . .;x_r†as a½ne coor- dinates onA^rˆP^rÿP_y. We denote byP_ithe point at in®nity of thex_i-axis, i.e. the point ‰0;. . .;1;. . .;0Š where 1 is at the …i‡1†-th place. We may assume P₁ to be generated by the points P₁;. . .;P_l1‡1 andP₂ by the points P_rÿl2;. . .;P_r, thus P₁U P₂HP_y. Hence local a½ne equations ofXin a suitable open subsetUofA^rmay be written in the form:

x1ˆf…x2;. . .;xrÿl2ÿ1†; xrˆg…xl1‡2;. . .;xrÿ1† wherefandgare analytic functions of their variables.

Suppose that the assertion is false. Then we may assume thatS…X†VPycontains the pointCˆ ‰0;1;. . .;1;0;. . .;0;1;. . .;1Š, where the 0's appear at places 1 andl1‡ 2;. . .;rÿl2ÿ1. Let

Pˆ …f…u₂;. . .;u_rÿl2ÿ1†;u₂;. . .;u_rÿ1;g…u_l1‡2;. . .;u_rÿ1††

be a general point ofX. Then there is a point

P⁰ˆ …f…u₂⁰;. . .;u_rÿl⁰ _2ÿ1†;u₂⁰;. . .;u_rÿ1⁰ ;g…u_l⁰_1‡2;. . .;u_rÿ1⁰ ††

onXsuch that the line joiningPandP⁰hasCas its point at in®nity, i.e.:

f…u2;. . .;urÿl2ÿ1† ÿf…u₂⁰;. . .;u_rÿl⁰ _2ÿ1† ˆu2ÿu₂⁰ ˆ ˆul1‡1ÿu_l⁰_1‡1

ˆu_rÿl2ÿu_rÿl⁰ ₂ ˆ ˆu_rÿ1ÿu_rÿ1⁰ ˆg…u_l1‡2;. . .;u_rÿ1† ÿg…u_l⁰_1‡2;. . .;u_rÿ1⁰ † …†

andu_iˆu_i⁰foriˆl₁‡2;. . .;rÿl₂ÿ1. Notice thatP⁰depends onP.

By shrinking the open subset UofA^r in which we are working, we may assume that u₂⁰;. . .;u_l⁰_1‡1;u_rÿl⁰ ₂;. . .;u_rÿ1⁰ are analytic functions of u₂;. . .;u_rÿ1. We claim that u₂⁰;. . .;u_l⁰_1‡1 do not depend on urÿl2;. . .;urÿ1. We argue by contradiction and we assume this is not the case. By (*) we have:

f…u2;. . .;urÿl2ÿ1† ÿuiˆf…u₂⁰;. . .;u_rÿl⁰ _2ÿ1† ÿu_i⁰ for everyiˆ2;. . .;l1‡1. Hence we deduce that:

q

qu_j…f…u₂⁰;. . .;u_rÿl⁰ _2ÿ1† ÿu_i⁰† ˆ0 for everyjˆrÿl2;. . .;rÿ1. This reads:

X

l1‡1 hˆ2

qf qx_h

qu_h⁰ qu_j ÿqu_i⁰

qu_jˆ0

which holds for everyiˆ2;. . .;l₁‡1 and everyjˆrÿl₂;. . .;rÿ1. By linear alge- bra, this yields:

X^l1‡1

hˆ2

qf qxh ˆ1:

Notice that the tangent spaceTtoXatPhas a½ne equations:

X

rÿl2ÿ1 hˆ2

qf

qxh…xÿuh† ˆx1ÿu1; X^rÿ1

kˆl1‡2

qg

qxk…xÿuk† ˆxrÿur

and the projective hyperplane de®ned by the former equation contains P2 andC.

This implies that the tangent space toXatPintersects the spanMofP2 andCin a subspace of dimension l2. ThusM is contained in an axis and we get a contradic- tion, becauseMstrictly containsP2. This proves our claim thatu₂⁰;. . .;u_l⁰_1‡1 do not depend on urÿl2;. . .;urÿ1. Similarly one shows thatu_rÿl⁰ ₂;. . .;u_rÿ1⁰ do not depend on u2;. . .;ul1‡1.

Suppose thatl1ˆ0. Sincel2Wnÿ1 one hasrÿl2>l1‡2ˆ2, and recall that in this caseu_iˆu_i⁰foriˆ2;. . .;rÿl₂ÿ1. Hencef…u₂;. . .;u_rÿl2ÿ1† ˆf…u₂⁰;. . .;u_rÿl⁰ _2ÿ1† and therefore all the di¨erences in (*) are 0, i.e.PˆP⁰, a contradiction. Similarly if l2ˆ0. Thus we may assumel1Xl2>0. In this case, as a consequence of the above analysis, we have that the di¨erences in (*) are equal to a constantc, so the transla- tion bycin the direction of the point at in®nityC®xesX. Since Xis algebraic, this yields that Xis a cone with vertex atC, a contradiction. This ends the proof in the casel1‡l2Wn.

Suppose now thatl1‡l2 ˆn‡1, thus in particularl1;l2>1. We may assume that P₂is generated byP_rÿl2;. . .;P_r, soP₂HP_y. We may assume also that the subspace P_yVP₁ is generated byP₁;. . .;P_l1. Notice thatl₁‡1ˆrÿl₂. Then we argue by contradiction supposing that the point Cˆ ‰0;1;. . .;1Š sits in S…X†. By the same computations as before (by replacing onlyl₁withl₁ÿ1 in all the above formulae), it follows thatXis a cone, a contradiction.

Remark 9. (i) We stress that the existence of varieties of dimensionnˆrÿ2, with irreducible componentsP1 andP2 ofS…X†of dimension respectively l1 andl2, are possible for all values ofl1 andl2such thatl1‡l2Wn‡1. It is su½cient to take for Xa complete intersection of cones of dimensionn‡1 with maximal vertices at two skew subspacesP1,P2 of the prescribed dimensions.

(ii) We want to point out the following interesting phenomenon. LetXHP^rbe an irreducible, non-degenerate, algebraic variety of dimension n<rÿ1 which is not a cone and letP1, P2 be two distinct, and therefore disjoint, axes of X. Consider the

projection pfrom a point ofPˆSpan…P1UP2†not onXUP1UP2, and setY ˆ p…X†, which is a variety Ybirational to X. Using Lemma 4, (vii), one sees that, in general,p…P†JS…Y†.

4 Some open problems

In the present section we want to propose some open problems. First of all, the con-

®guration and the number of the irreducible components of the Segre locus of a variety XHP^rof dimension nWrÿ2 is still pretty much a mystery. In the case nˆrÿ2 some not exhaustive information is provided by Proposition 8. Is it possible to extend this result to the casen<rÿ2?

In general any information more detailed than the one we have given here would be welcome. In particular, Segre's theorem from [5], mentioned in the introduction, about the existence of varieties with as many centers as one wants, should be com- plemented with answers to questions like:

(i) are there bounds on the number of centers, or axes, depending on any invariant of the variety, like the (co)dimension, the degree, etc.?

(ii) is the con®guration and the number of components of the Segre locus in¯uenced by the smoothness of the variety?

A generalization of the Segre locus, that we call theGrassmann±Segre locus, can be de®ned as follows. Let XHP^r be an irreducible, projective variety of dimensionn and letmbe a non-negative integer such thatmWrÿnÿ2. IfPis a general linear subspace ofP^rof dimensionm, then the projectionp:ˆp_PofP^rtoP^rÿmÿ1 fromP restricts toXto a birational morphism ofXonto its image. IfPis still such thatp_jX is a morphism, i.e. PVXˆq, butp_jX is no longer birational to its image, then we say thatpis aspecial projectionofX. We de®ne them-thGrassmann±Segre locusofX as:

Sm…X†:ˆ fPAG…m;r†:pPis a special projection ofXg:

Of course S0…X† ˆS…X†. It would be nice to have extensions of Theorems 1 and 3 to theseGrassmann±Segre loci. For instance, we have an argument, which we do not reproduce here, based on the theory of foci of planes in P⁴ (see [3]), to the e¨ect thatS₁…X†is a ®nite set ifXis a curve. Is it always the case thatS_m…X†is a

®nite set ifXis a curve?

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Received 13 November, 2000

A. Calabri, Dipartimento di Matematica, UniversitaÁ di Roma ``Tor Vergata'', via della Ricerca Scienti®ca, 00133 Roma

E-mail: [email protected]

C. Ciliberto, Dipartimento di Matematica, UniversitaÁ di Roma ``Tor Vergata'', via della Ricerca Scienti®ca, 00133 Roma

E-mail: [email protected]