JohannesHuisman Ontheenumerativegeometryofrealalgebraiccurveshavingmanyrealbranches

(de Gruyter 2003

On the enumerative geometry of real algebraic curves having many real branches

Johannes Huisman

(Communicated by C. Scheiderer)

Abstract.LetCbe a smooth real plane curve. Letcbe its degree andgits genus. We assume thatChas at leastgreal branches. Letdbe a nonzero natural integer strictly less thanc. Lete be a partition ofcdof lengthg. Letnbe the number of all real plane curves of degreedthat are tangent togreal branches ofCwith orders of tangencye1;. . .;eg. We show thatnis finite and we determinenexplicitly.

Key words.Enumerative geometry, real algebraic curve, real branch,M-curve,ðM1Þ-curve, Picard group, real space curve, real plane curve, conic, cubic, quartic, quintic.

2000 Mathematics Subject Classification. 14N10, 14P99

1 Introduction

Enumerative real algebraic geometry has known a growing attention throughout the last decade or so (see [10] for a survey). In his work on intersection theory, Fulton attracted attention to the number of real solutions of enumerative problems in alge- braic geometry [1, p 55]. As an example, he explicitly posed the question of how many of the 3264 conics tangent to five given real conics can be real. He proved that all of them can be real (unpublished). Independently, Ronga, Tognoli and Vust also proved this fact [7]. Sottile proved analogous results [9]. Fulton’s question naturally leads to the more general question of how many real curves of degreed are tangent to a certain number of real branches of a given real plane curve of degreec.

In this paper we answer the above question for any smooth real plane curve of any degreecthat has many real branches. Let us explain what we mean by a real plane curve having many real branches. By Harnack’s Inequality [2], a smooth real plane curve Cof degreechas at mostgþ1 real branches, wheregis the genus ofC, i.e., g¼¹₂ðc1Þðc2Þ. Harnack also showed that, for any natural number c, there is a smooth real plane curve of degree c havinggþ1 real branches. Such real plane curves are also known as M-curves. They are subject to intensive study ever since Hilbert included them in his 16th problem. We say that a smooth real plane curve has many real branchesif its genusgis at least 1 and if it has at leastgreal branches. To

put it otherwise, a smooth real plane curve has many real branches if it is either a nonrationalM-curve or a nonrationalðM1Þ-curve.

Before we can state one of our main results, we need to fix some terminology. Letn be a natural number. Apartitionofnis an elementeofN^l, for some natural integer l, such that

n¼e₁þ þe_l and e₁d de_ld1:

The integerlis called thelengthof the partition. Letebe a partition ofnof lengthl.

Letkbe a natural number. We say thatkis amemberofeif there is aniAf1;. . .;lg withe_i¼k. The partitioneofnisevenif all its members are even. Ifkis a member of e, then the multiplicity ofk ineis the number of indices iAf1;. . .;lg such that e_i¼k. Amultiplicityofeis a multiplicity of a member ofe.

LetCbe a smooth real algebraic curve inP²having many real branches. Letcbe its degree and letgbe its genus. Letd be a nonzero natural integer. Letebe a parti- tion ofcd of lengthg. A real algebraic curveDinP² of degreed is said tohave tan- gencyetogreal branches ofCifDis tangent togreal branches ofCwith orders of tangencye₁;. . .;e_g.

Theorem 1.1.Let C be a smooth real algebraic curve inP²having many real branches.

Let c be its degree and let g be its genus. Letebe a partition of cðc1Þof length g.

Let nbe the number of real plane curves of degree c1 having tangency eto g real branches of C.Then,nis finite.Moreover,n00if and only ifeis an even partition.In that case,

n¼

g!

m1! . . .mr!Y^g

i¼1

e_i if C is anðM1Þ-curve;and ðgþ1Þ!

m1! . . .mr!Y^g

i¼1

e_i if C is an M-curve;

8>

>>

><

>>

>>

:

where m1;. . .;mrare the multiplicities ofe.

In Section 4 we show a similar statement for the number of real curves of degree c2 that are tangent to g real branches of a given real plane curve C of degree c (cf. Theorem 4.3). By Bezout, there are no real curves of degree strictly less thanc2 that are tangent togreal branches of a given real plane curveCof degreec.

Example 1.2.LetCbe a smooth real cubic curve inP². Since the degree ofCis equal to 3, the genus ofC is equal to 1. Moreover, CðRÞ0 q. Hence,C necessarily has many real branches. Let nbe the number of real conics tangent to one real branch ofCwith order of tangency equal to 6. Then, according to Theorem 1.1,n¼6 ifC has only one real branch, andn¼12 ifChas exactly 2 real branches. This statement is the well-known fact that a real elliptic curve has either 6 or 12 real points whose order is a divisor of 6 [8].

Example 1.3. Let Cbe a smooth real quartic curve in P². Since the degree ofC is equal to 4, the genus ofCis equal to 3. Therefore, in order to apply Theorem 1.1, we assume thatChas at least 3 real branches. The partitions of 43¼12 of length 3 and in even numbers areð8;2;2Þ,ð6;4;2Þorð4;4;4Þ. Letebe one of these partitions. Let nbe the number of real cubics tangent to 3 real branches ofCwith orders of tangency e₁;e₂;e₃. When one applies Theorem 1.1 to the current situation one obtains the fol- lowing values forn. IfChas exactly 3 real branches then

n¼

96 ife¼ ð8;2;2Þ; 288 ife¼ ð6;4;2Þ;

64 ife¼ ð4;4;4Þ:

8>

<

>: IfChas exactly 4 real branches then

n¼

384 ife¼ ð8;2;2Þ;

1152 ife¼ ð6;4;2Þ;

256 ife¼ ð4;4;4Þ:

8>

<

>:

The cases where e¼ ð4;4;4Þ have already been shown in [4]. According to my knowledge, the other cases are new.

We refer to Section 4 for examples of higher degree (see Remark 4.1 and Example 4.2).

Theorem 1.1 is an application of Theorem 2.1 below that may be of independent interest. Section 2 is devoted to the proof of Theorem 2.1. In Section 3, we give appli- cations to enumerative problems for real curves in any projective space. In Section 4, we specialize to plane curves; we prove Theorem 1.1 and formulate and prove The- orem 4.3. We also give several examples.

2 Divisor classes on real algebraic curves

Let Cbe a smooth geometrically integral proper real algebraic curve. Areal branch of C is a connected component of the set CðRÞ of real points of C. By Harnack’s Inequality [2],Chas at mostgþ1 real branches, wheregis the genus ofC. We will say thatC has many real branchesifgd1 and the number of real branches ofCis at leastg.

Theorem 2.1. Let C be a smooth geometrically integral proper real algebraic curve having many real branches.Let g be the genus of C.Let B1;. . .;Bgbe mutually distinct real branches of C and put

B¼Y^g

i¼1

Bi:

Let e₁;. . .;e_g be nonzero natural integers,and let j:B!PicðCÞ be the map defined byjðPÞ ¼clðPg

i¼1eiPiÞ,wherecldenotes the divisor class.Then,j is a topological covering of its image of degreeQg

i¼1ei.

Proof. Since Bis connected, there is a connected componentX of PicðCÞsuch that jðBÞJX. SinceBandX are of the same dimension, it su‰ces to show that the map jis unramified, in order to show thatjis a topological covering map.

Let PABand let vbe a tangent vector toB atP. Suppose that the tangent map Tjofjmapsvto 0. We have to show thatvis equal to 0, in order to show thatjis unramified.

Since B¼Q

B_i, P¼ ðP1;. . .;P_gÞ and v¼ ðv1;. . .;v_gÞ, where P_iAB_i and v_i is a tangent vector toB_i at P_i. Let T ¼SpecðR½eÞ, whereR½eis the R-algebra of dual numbers [3]. Each pairðPi;v_iÞdetermines a morphism

fi:T !C⁰¼C_SpecðRÞT;

the image of each fi is a relative Cartier divisorDi ofC⁰=T [6]. Ifxi is a local equa- tion for Pi onC, then xilieis a local equation forDionC⁰, for someliAR. We have to show thatli¼0 fori¼1;. . .;g, in order to show thatv¼0.

Recall [3] that one has a short exact sequence

0!H¹ðC;OCÞ !PicðC⁰Þ !PicðCÞ !0:

In fact, this short exact sequence is naturally split since Ccan be identified with the special fiber of C⁰=T. Let Dbe the relative Cartier divisor P

e_iD_i on C⁰=T. Con- sider the class clðDÞ of the divisor D in PicðC⁰Þ. The hypothesis that Tj maps v onto 0 implies that clðDÞis contained in the image of the natural section of the map PicðC⁰Þ !PicðCÞ. Hence, the natural projection from PicðC⁰ÞontoH¹ðC;OCÞmaps clðDÞonto 0. Now, let us compute the image of clðDÞby this natural projection.

Recall [3] thatH¹ðC;OCÞcan be identified with the cokernelRof the natural map

K! 0

QAC

K=OQ;

whereOQis the local ring ofCatQandK is the function field ofC. Since ðxilieÞ^ei ¼x_i^eieilix_i^ei1e¼x_i^ei 1eili

1 xi

e

;

the image of clðDÞinH¹ðC;OCÞis equal to the elementr¼ ðr_QÞofRdefined by

r_Q¼ eil_i 1

x_i if Q¼P_i;

0 otherwise

8<

:

Take some iAf1;. . .;gg and let us show that l_i¼0. By the Riemann–Roch Theorem, there is a nonzero di¤erential form oonC such thatohas a zero at the points Pj, j¼1;. . .;g, j0i. Since the divisor of ois of even degree on each real branch ofC,ohas at least 2 zeros on each of the real branchesBj, j¼1;. . .;g, j0i.

Since ohas exactly 2ðg1Þzeros, it follows thatodoes not vanish on Bi. In par- ticular,odoes not vanish atPi. Let tbe the trace map fromH¹ðC;WCÞinto R[3].

Sincer¼0 inR, one hastðroÞ ¼0. From the definition of the trace map, it follows that the residue of eili1

xiovanishes atPi. Therefore, li¼0. This proves thatj is unramified.

In order to finish the proof, we show the statement concerning the topological degree ofj. Choose a base pointOABand writeO¼ ðO1;. . .;O_gÞ. Let

c:B!PicðCÞ

be the map defined by lettingcðPÞbe the divisor class clðP

P_iO_iÞ. By [5, Theorem 3.1],cis a homeomorphism onto the neutral component PicðCÞ⁰of PicðCÞ. Lettbe the translation byclðP

eiOiÞon PicðCÞ. Clearly,tmaps the imageX ofjhomeo- morphically onto PicðCÞ⁰. In order to show that the degree ofjis equal toQ

e_i, we show that the self-maptjc¹ of PicðCÞ⁰has degreeQ

e_i.

Each factor B_i of B gives rise to an element b_i of the first homology group H₁ðB;ZÞ. Clearly, b₁;. . .;b_g is a basis of H₁ðB;ZÞ. Then, cðb₁Þ;. . .;cðb_gÞ is a basis of H1ðPicðCÞ⁰;ZÞ. Since the multiplication-by-ei map on PicðCÞ⁰ induces the multiplication-by-eimap onH1ðPicðCÞ⁰;ZÞ,

ðtjc¹Þðcðb_iÞÞ ¼eicðb_iÞ:

It follows thattjc¹ is of degreeQ

e_i. r

Corollary 2.2. Let C be a smooth geometrically integral proper real algebraic curve having many real branches.Let g be the genus of C.Let B₁;. . .;B_gbe mutually distinct real branches of C.Let S be a complete linear system on C of degree e.Let e_i be the degree mod 2of S on Bi, for i¼1;. . .;g. Let e1;. . .;eg be nonzero natural integers satisfying

X^g

i¼1

e_i¼e:

Letnbe the number of divisors D of the formPg

i¼1eiPi,for some PiABi,that belong to S.Then,nis finite.Moreover,n00if and only if

ei1ei ðmod 2Þ for i¼1;. . .;g:

In that case,n¼Qg i¼1e_i. Proof. Let B¼Q

B_i, as before, and let j:B!PicðCÞ be defined by jðPÞ ¼

clðP

e_iP_iÞ. LetX be thej-image ofB. Suppose there is an integeri, 1cicg, such that ei2ei ðmod 2Þ. Then, clðSÞdoes not belong toX. Hence, there is no divisorD in S of the form P

eiPi for some PiABi. In that case, n¼0. Assume, now, that eieiðmod 2Þfor alli¼1;. . .;g. Then, clðSÞbelongs toX. According to Theorem 2.1, the number of PABsuch thatjðPÞ ¼clðSÞis equal toQ

ei. It follows that the number of divisorsDinSof the formPg

i¼1eiPi, for somePiABi, is equal toQg i¼1ei. Therefore,n¼Qg

i¼1ei. r

3 Enumerative problems for real space curves

Let C be a smooth geometrically integral real algebraic curve in Pⁿ, where nd2.

Let d be a nonzero natural number. We say that the linear system of all real hyper- surfaces of degree d in Pⁿ cuts outa complete linear system on C if the restriction map

H⁰ðPⁿ;OðdÞÞ !H⁰ðC;OðdÞÞ is an isomorphism.

LetBbe a real branch ofC. Then,Bis a compact connected smooth real analytic curve in the real projective space PⁿðRÞ. Since the fundamental group of PⁿðRÞis isomorphic to Z=2Z, two cases can occur: B is contractible inPⁿðRÞ or B is not.

In the latter case, we say thatBis apseudo-lineofC. In the former case,Bis anoval ofC.

Corollary 3.1.Let nd2 be an integer.Let C be a smooth geometrically integral real algebraic curve in Pⁿ.Let c be its degree and let g be its genus. Suppose that C has many real branches and let B1;. . .;Bgbe mutually distinct real branches of C.Let d be a nonzero natural integer such that the linear system of all real hypersurfaces of degree d inPⁿ cuts out a complete linear system on C.Letebe a partition of cd of length g.

Letnbe the number of real hypersurfaces D inPⁿof degree d such that D is tangent to the real branches B1;. . .;Bg of C with order of tangencye₁;. . .;e_g,respectively.Then, nis finite.Moreover,n00if and only if one of the following conditions is satisfied:

1. d is even andeis an even partition,or

2. d is odd and,for all i¼1;. . .;g,B_i is an oval of C if and only ife_iis even.

Furthermore,ifn00thenn¼Qg i¼1e_i.

Proof. Let S be the linear system onC cut out by all real hypersurfaces of Pⁿ of degreed. By hypothesis,S is complete. Moreover,nis equal to the number of divi- sorsDinSof the formP

e_iP_i, for someP_iAB_i. We determine the latter number.

The degree of S is equal to e¼cd. Let e_i be the degree mod 2 of S on B_i. If d is even thene_i10ðmod 2Þfori¼1;. . .;g. Ifdis odd thene_i10ðmod 2Þif and only if B_i is an oval of C. By Corollary 2.2, the number of divisors Din S of the form Pe_iP_i, for some P_iAB_i, is finite and is nonzero if and only if condition 1 or 2 is satisfied. Moreover, in that case, this number is equal toQ

e_i. r

Corollary 3.1 is a generalization of [4, Theorem 3.1], where we have counted only hypersurfaces tangent togreal branches with one and the same order of tangency to each of these branches.

As immediate consequences of Corollary 3.1, we mention the following two state- ments.

Corollary 3.2.Let nd2 be an integer.Let C be a smooth geometrically integral real algebraic curve in Pⁿ having many real branches.Let c be its degree and let g be its genus.Let d be a nonzero even natural integer such that the linear system of all real hypersurfaces of degree d in Pⁿ cuts out a complete linear system on C. Let e be a partition of cd of length g.Letnbe the number of real hypersurfaces inPⁿof degree d having tangencyeto g real branches of C.Then,nis finite.Moreover,n00if and only ifeis an even partition.In that case,

n¼

g!

m1! . . .mr!Y^g

i¼1

e_i if C is anðM1Þ-curve;and ðgþ1Þ!

m1! . . .mr!Y^g

i¼1

e_i if C is an M-curve;

8>

>>

><

>>

>>

:

where m1;. . .;mrare the multiplicities ofe.

Corollary 3.3.Let nd2 be an integer.Let C be a smooth geometrically integral real algebraic curve in Pⁿ having many real branches.Let c be its degree and let g be its genus.Letdbe the number of pseudo-lines of C.Let d be an odd natural integer such that the linear system of all real hypersurfaces of degree d inPⁿ cuts out a complete linear system on C.Letebe a partition of cd of length g.Letnbe the number of real hypersurfaces inPⁿ of degree d having tangencyeto g real branches of C.Then,nis finite.Moreover,n00if and only if the number of odd members ofeis equal tod.In that case,

n¼

d! ðgdÞ!

m1! . . .mr! Y^g

i¼1

e_i if C is anðM1Þ-curve;and d! ðgþ1dÞ!

m1! . . .mr! Y^g

i¼1

e_i if C is an M-curve:

8>

>>

><

>>

>>

:

where,as before,m₁;. . .;m_rare the multiplicities ofe.

4 Enumerative problems for real plane curves

Proof of Theorem 1.1. Letd ¼c1. Let us show that the linear system of all real curves of degreed inP²cuts out a complete linear system onC. The restriction map

H⁰ðP²;OðdÞÞ !H⁰ðC;OðdÞÞ

is injective sinced <candC is irreducible. The dimension ofH⁰ðP²;OðdÞÞis equal to¹₂ðdþ2Þðdþ1Þ. We need to show thatH⁰ðC;OðdÞÞis of the same dimension. The degree ofOðdÞonCis equal tocd¼2gþ2ðc1Þ. In particular, its degree is strictly greater than 2g2. Hence,OðdÞis nonspecial onC. By the Riemann–Roch Theo- rem,

dimH⁰ðC;OðdÞÞ ¼cdgþ1¼¹₂ðdþ2Þðdþ1Þ ¼dimH⁰ðP²;OðdÞÞ:

It follows that the linear system of all real curves of degreed in P² cuts out a com- plete linear system onC.

Now, there are two cases to consider: the cased is even and the cased is odd. Ifd is even then the statement of Theorem 1.1 follows from Corollary 3.2. Ifdis odd then cis even and the numberdof pseudo-lines ofCis equal to 0. Therefore, ifd is odd, the statement of Theorem 1.1 follows from Corollary 3.3. r Remark 4.1.LetC,candgbe as in Theorem 1.1. Observe that there are many par- titionseofcðc1Þto which Theorem 1.1 applies, i.e. partitionseofcðc1Þin even numbers and of lengthg. Indeed, there are as many as the number of partitions of the integerc1, ifcd4. Let us show this fact.

Let dbe any partition ofc1. Letkbe its length. Sinceg¼¹₂ðc1Þðc2Þ, the numberksatisfies

kcc1¼ 2g c2 cg:

DefineeAN^g bye_i¼2d_iþ2 ifick, ande_i¼2 ifi>k. Then, e₁þ þe_g ¼2ðc1Þ þ2kþ2ðgkÞ ¼cðc1Þ:

It follows thateis a partition ofcðc1Þin even numbers and of lengthg.

Conversely, any partition ofcðc1Þin even numbers and of lengthgarises in this way. Therefore, the number of partitionseofcðc1Þin even numbers and of length gis equal to the number of partitions of the integerc1, ifcd4.

Example 4.2.LetCbe a smooth real quintic curve inP²having many real branches.

Here, c¼5 and g¼6 and C has at least 6 real branches. There are 5 partitions of c1¼4:

ð1;1;1;1Þ; ð2;1;1Þ; ð2;2Þ; ð3;1Þ; ð4Þ:

The corresponding partitions ofcðc1Þ ¼20 in even numbers and of length 6 are ð4;4;4;4;2;2Þ; ð6;4;4;2;2;2Þ; ð6;6;2;2;2;2Þ;

ð8;4;2;2;2;2Þ; ð10;2;2;2;2;2Þ:

Then, for example, Theorem 1.1 states that there are exactly 7680 real quartics tan- gent to 6 real branches ofCwith orders of tangency 4;4;4;4;2;2, ifCis anðM1Þ- curve. There are 53760 of such real quartics ofCis anM-curve.

Theorem 4.3.Let C be a smooth real algebraic curve inP²having many real branches.

Let c be its degree and let g be its genus. Letebe a partition of cðc2Þof length g.

Let nbe the number of real plane curves of degree c2 having tangency eto g real branches of C. Then, nis finite. Moreover, n00 if and only if one of the following conditions is satisfied:

1. c is even andeis an even partition,

2. c is odd and exactly one of the members ofeis odd.

Furthermore,in Case1,

n¼

g!

m₁! . . .m_r!Y^g

i¼1

e_i if C is anðM1Þ-curve;and ðgþ1Þ!

m1! . . .mr!Y^g

i¼1

e_i if C is an M-curve;

8>

>>

><

>>

>>

:

where,as before,m₁;. . .;m_rare the multiplicities ofe.In Case2,

n¼

ðg1Þ! m₁! . . .m_r!Y^g

i¼1

e_i if C is anðM1Þ-curve;and g!

m1! . . .mr!Y^g

i¼1

e_i if C is an M-curve:

8>

>>

><

>>

>>

:

Proof.Letd ¼c2. We again need to show that the restriction map H⁰ðP²;OðdÞÞ !H⁰ðC;OðdÞÞ

is an isomorphism. For the same reasons as above, the map is injective. The degree of OðdÞonCis equal tocd¼2gþc2. In particular, its degree is strictly greater than 2g2. Hence,OðdÞis nonspecial onC. By the Riemann–Roch Theorem,

dimH⁰ðC;OðdÞÞ ¼cdgþ1¼¹₂ðdþ2Þðdþ1Þ ¼dimH⁰ðP²;OðdÞÞ: It follows that the linear system of all real curves of degreed in P² cuts out a com- plete linear system onC.

There are again two cases to consider: the casedis even and the cased is odd. Ifd is even then the statement of Theorem 1.1 follows from Corollary 3.2. Ifdis odd then cis odd as well and the numberdof pseudo-lines ofCis equal to 1. Therefore, ifd is odd, the statement of Theorem 1.1 follows from Corollary 3.3. r

Example 4.4.LetCbe a smooth real cubic curve inP². Letnbe the number of real lines tangent to one real branch ofCwith order of tangency equal to 3. Then, accord- ing to Theorem 4.3, n¼3. This statement is the well-known fact that a real cubic curve has exactly 3 real inflection points [8].

Example 4.5.LetCbe a smooth plane real quartic curve inP². Letnbe the number of real conics tangent to 3 real branches ofCwith orders of tangency 4;2;2. IfChas exactly 3 real branches thenn¼48 by Theorem 4.3. IfChas exactly 4 real branches thenn¼192 by Theorem 4.3.

Remark 4.6. Let C, candg be as in Theorem 4.3. Ifcis even and cd4, then the number of partitionseofcðc2Þsatisfying condition 1 of Theorem 4.3 is equal to the number of partitions of¹₂ðc2Þ. This can be shown in exactly the same manner as in Remark 4.1.

Ifcis odd andcd5, then there is a finite-to-one correspondence between the set of partitionseofcðc2Þsatisfying condition 2 of Theorem 4.3 and the set of all parti- tionsd of¹₂ðc1Þ. However, the correspondence is not bijective. Indeed, letd be a partition of ¹₂ðc1Þ. Letkbe its length. Let rbe a natural integer satisfying either 1crck and drþ1<dr, or r¼g. Define eAN^g bye_i¼2diþ2 if ick andi0r, e_i¼2diþ1 ifickandi¼r,e_i¼2 ifi>kandi0r, ande_i¼1 ifi>kandi¼r.

Then,eis a partition ofcðc2Þof length gand satisfying condition 2 of Theorem 4.3. Moreover, each partition of cðc2Þof length g and satisfying condition 2 of Theorem 4.3 arises in this way.

Example 4.7.LetCbe a smooth real quintic curve inP²having many real branches.

Then,c¼5 andg¼6 andChas at least 6 real branches. There are two partitions of

1

2ðc1Þ ¼2:ð1;1Þandð2Þ. The corresponding partitions ofcðc2Þ ¼15 of length 6 and satisfying condition 2 of Theorem 4.3 are

ð4;4;2;2;2;1Þ; ð4;3;2;2;2;2Þ ð6;2;2;2;2;1Þ; ð5;2;2;2;2;2Þ:

Then, for example, Theorem 4.3 states that there are exactly 960 real cubics tangent to 6 real branches of C with orders of tangency 4;3;2;2;2;2, if C is an ðM1Þ- curve. There are 6720 of such real cubics ifCis anM-curve.

References

[1] W. Fulton, Introduction to intersection theory in algebraic geometry. Reg. Conf. Ser.

Math. 54, Amer. Math. Soc. 1996. MR 85j:14008 Zbl 0885.14002

[2] A. Harnack, U¨ ber die Vieltheiligkeit der ebenen algebraischen Curven. Math. Ann. 10 (1876), 189–198.

[3] R. Hartshorne,Algebraic geometry. Springer 1977. MR 57 #3116 Zbl 0367.14001 [4] J. Huisman, On the number of real hypersurfaces hypertangent to a given real space

curve.Illinois J. Math.(to appear).

[5] J. Huisman, On the neutral component of the Jacobian of a real algebraic curve having many components.Indag. Math. (N.S.)12(2001), 73–81. MR 1 908 140 Zbl 01663809 [6] N. M. Katz, B. Mazur,Arithmetic moduli of elliptic curves. Princeton Univ. Press 1985.

MR 86i:11024 Zbl 0576.14026

[7] F. Ronga, A. Tognoli, T. Vust, The number of conics tangent to five given conics: the real case.Rev. Mat. Univ. Complut. Madrid10(1997), 391–421. MR 99d:14059 Zbl 0921.14036

[8] J. H. Silverman,The arithmetic of elliptic curves. Springer 1986. MR 87g:11070 Zbl 0585.14026

[9] F. Sottile, Enumerative geometry for the real Grassmannian of lines in projective space.

Duke Math. J.87(1997), 59–85. MR 99a:14079 Zbl 0986.14033 [10] F. Sottile, Enumerative real algebraic geometry. Preprint 2001.

http://www.math.umass.edu/~sottile/

Received 18 October, 2001; revised 15 May, 2002

J. Huisman, Institut de Recherche Mathe´matique de Rennes, Universite´ de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

Email: [email protected]