(de Gruyter 2003
Curves of maximal genus in P
5Rita Ferraro
(Communicated by R. Miranda)
1 Introduction
LetCHPrbe a reduced, irreducible, non-degenerate curve, not contained in a sur- face of degree <s; when d ¼degC is large with respect to s, the arithmetic genus
paðCÞis bounded by a functionGðd;r;sÞwhich is of typed2s2þOðdÞ.
The existence of such a bound for CHP3 was announced by Halphen in 1870 and proved by Gruson and Peskine in [6] ford>s2s; for curves inPr,rd4, the bound is stated and proved in [2] (ford >r22s Qr2
i¼1ððr1Þ!sÞ1=ðr1iÞ).
The existence of curves of maximal genus, i.e., curves, the genus of which attains the bound, is known inP3ford >s2s, inP4ford>12s2and inPr,rd5, at least fordgs. [6] contains a precise description of those curves inP3 which do not lie on surfaces of degree<sand whose genus is maximal: they are arithmetically Cohen–
Macaulay, lie on a surfaceSof degreesand they are directly linked to plane curves.
[1] contains the description of curves inP4of maximal genusGðd;4;sÞ.
The complete description of curvesCHP5, not contained in a surface of degree
<s, whose genus isGðd;5;sÞhas been given by the author in her PhD dissertation [3].
The main result of this note is the classification Theorem 1.1, which holds forsd9.
Due to the long list of cases, some proofs are given only in some specific examples. We already know [2] that such curves must be arithmetically Cohen–Macaulay and they must lie on a surface S of degree s, whose general hyperplane section G is a ‘‘Castelnuovo curve’’ inP4, i.e., a curve inP4 of maximal genus. Whensd9 the surfaceSlies on a rational normal 3-foldX of degree 3 inP5, which can be singular.
Analogously to [6] and [1], we describe our curvesC of genusGðd;5;sÞin terms of the curveC0obtained by linking CwithSand a hypersurfaceF of minimal degree passing throughCand not containingS.
We state now the main theorem. In Propositions 4.1, 4.2 and 4.3 we will give a closer description of Cases (2), (3) and (4) of the theorem.
Theorem 1.1. Let CHP5 be an integral non-degenerate curve of degree d not con- tained in a surface of degree<s and let sd9.Suppose that the arithmetic genus of C is maximal among those curves not contained in a surface of degree <s, i.e., paðCÞ ¼ Gðd;5;sÞ.Assume d>2s3Q3
i¼1ð4!sÞ1=ð4iÞ.
Then C is arithmetically Cohen–Macaulay and lies on an irreducible surface S of degree s contained in a cubic rational normal3-fold XHP5.
Put d1¼smþe, 0cecs1 and s1¼3wþv, v¼0;1;2;if e<wð4vÞ, writee¼kwþd, 0cd<w;ifedwð4vÞ,writeeþ3v¼kðwþ1Þ þd, 0cd<
wþ1.
Then C is contained in a hypersurface F of degree mþ1,not passing through S.If C0is the curve linked to C by F and S,we have:
(1) when k¼3,then C0¼q;i.e.,C is a complete intersection on S;
(2) when k¼2,then C0is a plane curve;
(3) when k¼1,then C0lies on a surface of degree2;
(4) when k¼0,then C0lies on a surface of degree3.
The proof is based on the analysis of the Hilbert function of a general hyperplane sectionZ ofC. The main technical problem that one does not find in the previous cases (r¼3;4) is that for describingC0 we have to perform a linkage by a complete intersection on the scroll X, which is in general, if X is singular, a non-Gorenstein scheme. To this purpose the author has proved in [4] and [5] some general results to which we will refer in these note.
In the last section we prove the following result.
Theorem 1.2. For all s;d with sd4 and d>2s3Q3
i¼1ð4!sÞ1=ð4iÞ,there exists a smooth curve CHP5 of degree d and arithmetic genus Gðd;5;sÞwhich does not lie on a sur- face of degree<s.Moreover C is contained in an irreducible surface S of degree s.
It should be observed that in [1] the authors do not analyze the regularity of the produced extremal curves in P4 and that in [2] the produced examples of curves of maximal genusGðd;r;sÞinPrfordgsare in general singular.
With the same techniques used for the classification inP5 it is possible to classify curves inPr of maximal genusGðd;r;sÞfor everyrandsd2r1. In [5] the author has given an example of the classification procedure for curves of maximal genus Gðd;r;sÞinPrand of the construction of such smooth extremal curves.
Acknowledgments. The paper has been written while the author was supported by a Post-Doc scholarship of Universita´ di Roma Tre. The author thanks Ciro Ciliberto for his patience and help.
2 Weil divisors onX
We will see in the next section that curvesCHP5of maximal genusGðd;5;sÞwhich we want to classify lie on a cubic rational normal 3-foldXHP5about which we need to fix some notation and mention some results. A rational normal 3-foldXHP5 is the image of a projective bundlep:PðEÞ !P1overP1, via the morphism j defined by the tautological line bundleOPðEÞð1Þ, whereEis a locally free sheaf of rank 3 on P1of one of the following three kinds:
(1) E¼OP1ð1ÞlOP1ð1ÞlOP1ð1Þ. In this case XGP1P2 is smooth and it is ruled byy1disjoint planes; we putX ¼Sð1;1;1Þ.
(2) E¼OP1lOP1ð1ÞlOP1ð2Þ. Here X is a cone over a smooth cubic surface in P4 with vertex a point V and it is ruled byy1 planes intersecting atV; we put X¼Sð0;1;2Þ.
(3) E¼OP1lOP1lOP1ð3Þ. X is a cone over a twisted cubic in P3 with vertex a linel, and it is ruled byy1 planes intersecting atl; we putX ¼Sð0;0;3Þ.
Let us writePðEÞ ¼XX~. The morphism j: ~XX!X is a rational resolution of sin- gularities, called thecanonical resolutionofX. It is well known that the Picard group PicðXX~Þ of XX~ is isomorphic to Z½HH~ lZ½RR, where~ ½HH ¼ ½O~ XX~ð1Þ is the hyperplane class and½RR ¼ ½p~ OP1ð1Þis the class of the fiber of the mapp: ~XX !P1. The inter- section form onXX~ is determined by the rules:
H~
H3¼3; RR~HH~2¼1; RR~2HH~ ¼0; RR~3¼0: ð2:1Þ The cohomology of the invertible sheaf OXX~ðaHH~þbRRÞ~ associated to a divisor@ aHH~þbRR~inXX~ can be explicitly calculated using the Leray spectral sequence. In par- ticular, for ad0 and bd1, the dimensionh0ðOXX~ðaHH~ þbRRÞÞ~ does not depend on the type of the scroll and it is given by the formula ([4] 3.5):
h0ðOXX~ðaHH~þbRRÞÞ ¼~ 3 aþ2 3
þ ðbþ1Þ aþ2 2
: ð2:2Þ
Let H and R be the strict images in X of HH~ and RR~ respectively, i.e. the scheme- theoretic closures jðHH~jj1XSÞand jðRR~jj1XSÞ, whereXS denotes the smooth part ofX.
Let us consider onX the direct image ofOXX~ðaHH~ þbRRÞ, for every~ a;bAZ:
OXðaHþbRÞ:¼ jOXX~ðaHH~þbRRÞ:~
If the scroll X is smooth, then the sheavesOXðaHþbRÞare the invertible sheaves associated to the Cartier divisors@aHþbR, while when X is singular this is no longer true. In this case we have the following proposition which has been proved in [4] (Lemma 2.14, Cor. 3.10 and Theorem 3.17). The reader may refer to [10] for a survey ondivisorial sheaves associated to generalized divisors, in particular to Weil divisors.
Proposition 2.3. Let XHP5 be a rational normal 3-fold and let j : ~XX!X be its canonical resolution. Let ClðXÞ be the group of Weil divisors on X modulo linear equivalence.Then the following holds:
If X is smooth or X ¼Sð0;1;2Þwe haveClðXÞGZ½HlZ½R.The divisorial sheaf associated to a divisor@aHþbR on X isOXðaHþbRÞfor every a;bAZ.
If X¼Sð0;0;3Þwe have that H@3R andClðXÞGZ½R.The sheavesOXðaHþbRÞ
with3aþb¼d and b<3 are all isomorphic to the divisorial sheaf associated to a divisor@dR.
Remark 2.4. Since RijOXX~ðaHH~þbRRÞ ¼~ 0 for i>0 and for all aAZ and bd1, Formula (2.2) holds for h0ðOXðaHþbRÞÞ too. In particular we can use Formula (2.2) to compute dimjDj for an e¤ective divisor D@aHþbR in X¼Sð1;1;1Þor X ¼Sð0;1;2Þwithbd1. WhenX ¼Sð0;0;3Þby Proposition 2.3 we can write the divisorial sheaf associated to D@dR in the form OXðaHþbRÞ with 0cb<3;
therefore we can use Formula (2.2) to compute dimjDjfor every e¤ective divisor D inX¼Sð0;0;3Þ.
WhenX is smooth or X¼Sð0;1;2Þthe intersection form (2.1) onXX~ determines via the isomorphism PicðXXÞ~ GClðXÞ of Proposition 2.3 the intersection form on X. In particular we use the intersection number DD0H of two e¤ective divisor D@aHþbR and D0@a0Hþb0R with no common components to compute the degree of their scheme-theoretic intersectionDVD0.
WhenX ¼Sð0;0;3Þ the computation of degðDVD0Þis more complicated and it is explained in detail in [4]; here we want just to state the results that we need. So let us first introduce theintegral total transformof an e¤ective Weil divisorD@dR:
Definition 2.5.Let X ¼Sð0;0;3Þ. LetDHX be an e¤ective Weil divisor. Then the integral total transformD ofDinXX~ is:
D:¼DD~þ dqeE
where DD~@aHH~þbRR~ is the proper transform of Din XX,~ E@HH~3RR~ is the excep- tional divisor inXX~ anddqeis the smallest integerdq:¼b3.
Then let us introduce, for every e¤ective Weil divisor D in Sð0;0;3Þ, the rational numbere:¼ dqe q. We can compute the degree of the intersection schemeDVD0 of two e¤ective divisorsD@dRandD0@d0Rwith no common components using the following formula which has been proved in [4] Proposition 4.11:
degðDVD0Þ ¼ DD0HH~ if ½eþe0 ¼0;
DD0HH~þ3ðeþe01Þ þ1 if ½eþe0 ¼1:
ð2:6Þ
By abusing notation we will write the degree degðDVD0Þas the intersection number DD0H. Moreover we compute the intersection multiplicity mðD;D0;lÞofDand D0along the singular linelofXas follows:
mðD;D0;lÞ ¼DD0HDD~DD~0HH:~ ð2:7Þ Remark 2.8.To prepare the proof of the main theorem and of Propositions 4.1, 4.2, 4.3 we briefly describe all possible planes and surfaces of degree 2 and 3 contained in a rational normal 3-foldXHP5.
1. WhenX¼Sð1;1;1Þ. The only planes contained inX are the ones in the ruling ofX, otherwise the linear systemjOXðRÞjwould cut on a planepHX, which is not in the ruling, a pencil of lines which of course intersect each other, while the planes in jOXðRÞjare pairwise disjoint. The surfaces of degree 2 contained inX are either reducible and hence linearly equivalent to 2R, or irreducible and hence degener- ate, therefore linearly equivalent toHR. A reduced surfaceY@2Ris the disjoint union of two planes in P5. A surfaceQ@HR is a smooth quadric surface; the two systems of lines onQare cut by the linear systemjOXðRÞj(lines of typeð1;0Þon Q) and by the linear system jOXðHRÞj (lines of type ð0;1Þ on Q). The surfaces of degree 3 contained inXare either reducible in the union of three planes and hence linearly equivalent to 3R, or reducible in the union of an irreducible quadric surface and of a plane, hence linearly equivalent toHRþR@H, or finally irreducible, therefore degenerate, and so linearly equivalent to a hyperplane sectionH. A reduced surface@3Ris the disjoint union of three planes inP5. A reducible hyperplane sec- tion ofXis the union of a smooth quadric surface and of a plane meeting along a line of type ð1;0Þ. An irreducible hyperplane section ofX is a smooth rational normal surface inP4. Lastly we recall that a surface@aHþbR on X is irreducible when a¼0 andb¼1, ora>0 andbda(by [9], V, 2.18, passing to general hyperplane sections).
2. When X¼Sð0;1;2Þ, a plane contained in X is either one of the ruling of X, therefore linearly equivalent toR, or it is the plane p@H2R, i.e., the plane spanned by the vertex V of X and by the line image of the section defined by ProjOP1ð1Þ,!PðEÞ. The reducible surfaces of degree 2 contained in X are either linearly equivalent to 2R(when reduced they are the union of two planes meeting at the pointV), or linearly equivalent toHR(the union of pand of a planep@R meeting along a line passing through V), or linearly equivalent to 2ðH2RÞ (the plane p counted with multiplicity 2). The irreducible ones are linearly equivalent to HR. An irreducible surfaceQ@HRis a quadric cone with vertexV; the pencil of lines onQis cut by the linear systemjOXðRÞj(or equivalently by the linear system jOXðHRÞj). The surfaces of degree 3 contained in X are either reducible in the union of three planes and hence linearly equivalent to 3R(when reduced they are the union of three planes meeting at the pointV), or to 2RþH2R@H(when reduced each plane@Rmeets the plane p along a line passing through the pointV), or to Rþ2ðH2RÞ ¼2H3R, or to 3ðH2RÞ(in the last two cases the surface is not reduced). They may be also reducible in the union of an irreducible quadric cone and of a plane, hence linearly equivalent toHRþR@H (the cone and the plane meet along a line passing through V), or to HRþH2R¼2H3R(the cone and the planepmeet at the pointV). Finally they can be irreducible, therefore degener- ate and so linearly equivalent to a hyperplane sectionH, which is a rational normal surface in P4. As in the previous case, since a general hyperplane section of X is smooth, we have that a surface@aHþbR on X is irreducible when a¼0 and b¼1, ora>0 andbda.
3. When X¼Sð0;0;3Þthe situation is simpler since ClðXÞ ¼Z½R. A plane con- tained in X is a plane of the ruling. A surface of degree 2 contained in X is always linearly equivalent to 2Rand it is always reducible in the union of two planes meeting
along the line l (the vertex of X). A surface of degree 3 contained inX is linearly equivalent to 3Rand it is reducible in the union of three planes meeting at l if the proper transform is linearly equivalent to 3RR, or it is irreducible, and hence a singular~ rational normal surface inP4, if the proper transform is linearly equivalent toHH~. In this case, since a general hyperplane section ofX is singular, by [9], V, 2.18 we have that a surfaceSHXis irreducible if its proper transform isSS~@aHH~ þbRR, with~ a¼0 andb¼1, ora>0 andbd0.
Remark 2.9.Finally we want to describe how plane curves contained inX look like.
Since X is an intersection of quadrics inP5, a plane curve of degreed3 contained inX must lie in a plane ofX. Therefore we are interested just in lines and conics.
1. LetX ¼Sð1;1;1Þ. A linerHX which is not contained in a planep@Rof the scroll is the base locus of a pencil of quadric surfaces@HR, i.e., it is a line of type ð0;1Þ(take the pencil of hyperplane sections passing throughrand a fixed plane@R intersectingr). A conicCHX which does not lie onp@Ris contained in a quadric surfaceQ@HR(take a hyperplane section passing through C and a plane@R meetingC) Therefore it is a hyperplane section ofQ, i.e., a curve of typeð1;1ÞonQ.
2. LetX¼Sð0;1;2Þ. Every linerHX is contained in a plane of the scroll. In fact ifrpasses through V, then it is obviously contained in some planep@R. Ifrdoes not pass throughV, then the plane spanned byrandV is contained inX. A conic CHX which is not contained in a plane of the scroll and that passes through V is reducible in the union of two lines. IfCdoes not pass throughV, then the cone over C with vertexV is a quadric coneQ@HR, therefore C is a hyperplane section ofQ.
3. Let X¼Sð0;0;3Þ. A line rHX is always contained in a plane of the scroll p@R. In fact it is contained in the hyperplane section passing through r and the singular line l of X, which splits in the union of three planes@R. A conic CHX which does not lie onp@Ris a hyperplane section of a surface@2R, i.e., it is the union of two lines meeting at a point.
3 Preliminaries
We start by recalling a few results of [2]. From now on, let C be an integral, non-degenerate curve of degree d and arithmetic genus paðCÞ in P5, with d >2s3Q3
i¼1ð4!Þ1=ð4iÞ. AssumeC is not contained in a surface of degree <sand de- finem;e;w;v;k;das in the statement of Theorem 1.1.
Then the genus paðCÞis bounded by the function:
Gðd;5;sÞ ¼1þd
2ðmþw2Þ mþ1
2 ðw3Þ þvm
2 ðwþ1Þ þr;
where r¼d2 ðwdÞ if e<wð4vÞ and r¼2ew2ð3vÞ d2ðwdþ1Þ if ed wð4vÞ([2], Section 5).
IfZis a general hyperplane section ofCandhZis the Hilbert function ofZ, then the di¤erenceDhZmust be bigger than the functionDhdefined by:
DhðnÞ ¼
0 if n<0 or n>mþwþe;
3nþ1 if 0cncw;
s if w<ncm;
sþk3ðnmÞ if m<ncmþd;
sþk3ðnmÞ 1 if mþd<ncmþwþe;
8>
>>
>>
<
>>
>>
>:
wheree¼0 ife<wð4vÞande¼1 otherwise.
Proposition 3.1. If paðCÞ ¼Gðd;5;sÞ, then DhZðnÞ ¼DhðnÞfor all n and C is arith- metically Cohen–Macaulay. Moreover Z is contained in a reduced curveGof degree s and maximal genus Gðs;4Þ ¼12wðw1Þðw2Þ þwv in P4 (Castelnuovo’s curve).
Since d>s2,Gis unique and,when we move the hyperplane,all these curvesGpatch together giving a surface SHP5of degree s through C.
Proof.See [2] 0.1, 6.1, 6.2, 6.3. r
S is a ‘‘Castelnuovo surface’’ inP5, i.e., a surface whose general hyperplane sec- tion is a curve of maximal genus inP4. These surfaces are classified in [7].
Proposition 3.2.S is irreducible and when sd9lies on a cubic rational normal3-fold X inP5where it is cut by a hypersurface G of degree wþ1.As a divisor on X the surface S is linearly equivalent toðwþ1ÞH ð2vÞR(or wHþR if v¼0).
Proof. S is irreducible becauseC is irreducible and is not contained in a surface of degree<s.
If a general hyperplane section G of S is a special Castelnuovo curve in P4 of degrees, then it lies on a rational normal cubic surfaceW in P4 which is the inter- section of the quadric hypersurfaces containing G, hence also Z; since C is arith- metically Cohen–Macaulay these quadrics must lift to quadric hypersurfaces in P5 containing C, hence also S. The intersection of these quadric hypersurfaces is a rational normal cubic 3-foldX inP5 whose general hyperplane section isW.
MoreoverGlies on a hypersurface of degreewþ1 which does not containW; such a hypersurface must lift to a hypersurface G of degreewþ1 inP5, containingC, hence containingSsinced >s2, and not containingX. r Proposition 3.3.There exists a hypersurface F of degree mþ1,passing through C and not containing S.
Proof. For a general hyperplane section G of S, the Hilbert function hG is known (see e.g. [8]); in particular we have DhGðnÞ ¼DhZðnÞ when 0cncm and hence h0ðICðnÞÞ ¼h0ðISðnÞÞwhen 0cncm. Forn¼mþ1 one computesDhZðmþ1Þ<
DhGðmþ1Þand this impliesh0ðICðmþ1ÞÞ>h0ðISðmþ1ÞÞ. r We recall here the definition of geometrical linkage.
Definition 3.4.LetY1,Y2,Y be subschemes of a projective spaceP. ThenY1andY2 aregeometrically linkedbyYif
(1) Y1 and Y2 are equidimensional, have no embedded components and have no common components, and
(2) Y1UY2¼Y, scheme theoretically.
Definition 3.5.CallC0the curveresidualtoConSbyF; degC0¼se1. CallC00 the curveresidualtoConX byF andG.
We note that C0 is well defined since S is irreducible and F does not contain S, degC0¼sðmþ1Þ d ¼se1. Moreover since degC0<degCthe curveC0does not containC, which is irreducible; thereforeC andC0 are geometricallylinked by SVF. AlsoC00is well defined andC0HC00:
Ifs¼3wþ3ðv¼2Þ, then
C00¼C0:
Ifs¼3wþ2ðv¼1Þ, we can choose the plane p1@Rlinked toSonX byGsuch that it is not contained inF, then
C00¼C0þC1; whereC1Hp1is a plane curve of degreemþ1.
If s¼3wþ1ðv¼0Þ and S@ðwþ1ÞH2R, we can choose the divisor@2R linked toS onX byGsuch that it is the union of two distinct plane p1 and p2 not contained inF, then
C00¼C0þC1þC2;
whereC1Hp1andC2Hp2are two plane curves of degreemþ1.
Ifs¼3wþ1 andS@wHþR, we can choose the divisorq@HRlinked toS byX andGsuch that it is not contained inF, then
C00¼C0þCq; whereCqis the intersection ofqandF.
C andC00 are geometrically linked byXVFVG since they are equidimensional, have no common components (Cis irreducible andC00does not containC) and with no embedded components (XVFVGis arithmetically Cohen–Macaulay).
Definition 3.6. CallZ;Z0;Z00;W general hyperplane sectionsofC;C0;C00;X respec- tively.
The next result is Lemma 4.4 of [5]. It is the main tool in the classification procedure.
Lemma 3.7. Let C00HX and Z00HW be as in Definition 3.5 and Definition 3.6 respectively.Then for icminfw;mg:
h0ðIC00jXðiHþRÞÞdh0ðIZ00jWðiHþRÞÞ ¼ Xy
r¼mþwiþ1
DhðrÞ:
Moreover if h0ðIZ00jWðði1ÞHþRÞÞ ¼0 and h0ðIZ00jWðiHþRÞÞ ¼h>0, then h0ðIC00jXðði1ÞHþRÞÞ ¼0and h0ðIC00jXðiHþRÞÞ ¼h.
The next result is a formula which relates the arithmetic genera of the curvesC,C00 andY ¼XVFVG.
Lemma 3.8.Let X be smooth or X ¼Sð0;1;2Þ.Let C,C00and Y ¼XVFVG be as usual.Then we have the following relation:
paðC00Þ ¼ paðCÞ paðYÞ þ ðmþw1Þ degC00þdegðRjC00Þ þ1 ð3:9Þ
Proof.We apply [5] Proposition 4.6. r
4 The classification
At this point we are able to prove the main theorem. The techniques that we use to prove Theorem 1.1 are the same for the three cases: X ¼Sð1;1;1Þ, X ¼Sð0;1;2Þ and X¼Sð0;0;3Þ; therefore we do not want to give a proof for all cases. On the other side the analysis is slightly di¤erent case by case, therefore, to be impartial, we will give the proof of Theorem 1.1(2) in caseX¼Sð0;1;2Þ, of Theorem 1.1(3) in case X ¼Sð1;1;1Þ and of Theorem 1.1(4) in case X¼Sð0;0;3Þ. For a complete proof the reader may consult [3]. We will give a more precise description of such curves C0in the next propositions. The reader may go back to Remark 2.8 where we have described planes and surfaces of degree 2 or 3 contained in X, and to the previous section where we have introduced some notation.
Proof of the Theorem1.1. (1) Letk¼3. This happens if and only ife¼s1. It fol- lows degC0¼se1¼0 and we are done.
(2) Letk¼2 and letX¼Sð0;1;2Þ. We treat separately the casesv¼0;1;2.
Let v¼2, i.e., S@ðwþ1ÞH. Thene¼1 and we haveeþ1¼2ðwþ1Þ þdwith 0cd<wþ1. By Lemma 3.7 we compute
h0ðIC0jXðRÞÞ ¼1:
Hence C0 is contained in a plane p@R and has degree wþ1ddegC0¼ 3wþ2ed1.
Let v¼1, i.e., Sþp1@ðwþ1ÞH. If e¼0 (i.e., if e<3w) we have e¼ 2wþd with 0cd<w. By Lemma 3.7 we compute h0ðIC00jXðRÞÞ ¼0 and
h0ðIC00jXðHþRÞÞ ¼3. SinceðHþRÞ p1H ¼1 and degC1¼mþ1>1, all the surfaces@HþR containing C00 split in the plane p1IC1 and in surfaces@H containingC0. Therefore
h0ðIC0jXðHÞÞ ¼3;
i.e., C0 is contained in a plane p and has degree wþ1ddegC0¼3wþ1e>1.
Whenv¼1 we havee¼1 only ife¼3w. In this case degC0¼1, i.e.,C0 is a line.
Let v¼0, then e¼0 and we have e¼2wþd with 0cd<w. By Lemma 3.7 we compute h0ðIC00jXðRÞÞ ¼0 and h0ðIC00jXðHþRÞÞ ¼2. In case Sþp1þp2@ ðwþ1ÞH, since ðHþRÞ p1H¼ ðHþRÞ p2H ¼1 and degC1¼degC2 ¼ mþ1>1, we find that
h0ðIC0jXðHRÞÞ ¼2:
If d<w1, then degC0¼3we¼wd>1; since ðHRÞ ðHRÞ H ¼1, the linear system jIC0jXðHRÞj has a fixed part which is necessarily the plane p@H2R. Ifd¼w1, then C0 is a line. Therefore C0 is contained in the plane p@H2Ror it is a line.
In case Sþq@ðwþ1ÞH, sinceðHþRÞ qH ¼3 and degCq¼mþ1>3, we find that
h0ðIC0jXð2RÞÞ ¼2:
ThereforeC0is contained in a planep@R, which is the fixed part ofjIC0jXð2RÞj.
(3) Letk¼1 and letX be smooth.
Let v¼2 and e¼0; we have e¼wþdwith 0cd<w. By Lemma 3.7 we com- puteh0ðIC0jXðRÞÞ ¼0 and
h0ðIC0jXðHþRÞÞ ¼3:
Since degC0¼se1d5 and ðHþRÞ ðHþRÞ H ¼5 we deduce that the linear system jIC0jXðHþRÞj has a fixed part which has degree at most 2 as one can easily verify (if we suppose, for example, that the fixed part is L@H, then h0ðOXðHþRLÞÞ ¼h0ðOXðRÞÞ ¼2, and we have a contradiction since h0ðIC0jXðHþRLÞÞ ¼3). Therefore the fixed part can be of the following types:
(a)p@R. In this caseC0is the union of a plane curveC10 onpand of a curveC20 contained in the base locus of a net of hyperplane sections, i.e. in a planes. Ifs@R, then the fixed part ofjIC0jXðHþRÞjis@2Rand we are in the next Case (b). The other possibility is thatsdoes not belong toX. Since degC0dwþ3 andpSH ¼ wþ1 this is possible only when degC0¼wþ3, i.e.,d¼w1 andC20 is a curve of typeð1;1Þon a quadric surface@HR. In this caseC0 is contained in the surface of degree twopUs.
(b)Y@2R. Then jIC0jXðHþRÞj ¼Yþ jOXðHRÞj(in fact h0ðOXðHRÞÞ ¼ 3). SincejOXðHRÞjis free,C0is contained in the surface of degree twoY@2R.
(c) Q@HR, i.e., jIC0jXðHþRÞj ¼Qþ jOXð2RÞj. SincejOXð2RÞjis freeC0is contained in the smooth quadric surfaceQ.
Whenv¼2, then e¼1 only ife¼2w, i.e., degC0¼wþ2. In this case we write eþ1¼wþ1þdwithd¼w. By Lemma 3.7 and we computeh0ðIC0jXðRÞÞ ¼0 and
h0ðIC0jXðHþRÞÞ ¼4:
Since degC0¼wþ2d4 and ðHþRÞ ðHþRÞ H ¼5 we deduce that the linear system jIC0jXðHþRÞjhas a fixed part which is, as one can easily verify,p@R. In this caseC0is the union of a plane curve of degreewþ1 onpand of a line.
Let v¼1. Then e¼0 and we havee¼wþdwith 0cd<w. By Lemma 3.7 we computeh0ðIC00jXðRÞÞ ¼0 andh0ðIC00jXðHþRÞÞ ¼2:With the same computation we have done previously (casek¼2 andv¼1) one can deduce that
h0ðIC0jXðHÞÞ ¼2:
Since degC0>wþ1d3 and H3 ¼3 the linear pencil jIC0jXðHÞj should have a fixed part, which can be of the following types:
(a)p@R. In this caseC0is the union of a plane curve of degreewþ1¼pSH onp and of a line, which is the base locus of a pencil of quadric surfaces@HR.
This is possible only when degC0¼wþ2, i.e.,d¼w1.
(b) Q@HR, i.e., jIC0jXðHÞj ¼Qþ jOXðRÞj. Since jOXðRÞj is free C0 is con- tained in the smooth quadric surfaceQ.
The fixed part of jIC0jXðHÞjcannot beY@2Rsince in this case we would have jIC0jXðHÞj ¼Yþ jOXðH2RÞj, whileh0ðOXðH2RÞÞ ¼0.
Let v¼0, then e¼0 and we have e¼wþd with 0cd<w. By Lemma 3.7 we compute h0ðIC00jXðRÞÞ ¼0 and h0ðIC00jXðHþRÞÞ ¼1. In case Sþp1þp2@ ðwþ1Þ, one easily deduces that
h0ðIC0jXðHRÞÞ ¼1;
i.e., C0 is contained in a smooth quadric surface Q@HR. In case Sþq@ ðwþ1ÞH, one finds that
h0ðIC0jXð2RÞÞ ¼1;
thereforeC0is contained in a reducible surface of degree twoY@2R.
(4) Letk¼0 and letX ¼Sð0;0;3Þ. Thene¼0 and we writee¼dwith 0cd<w.
Letv¼2. By Lemma 3.7 we compute
h0ðIC0jXð4RÞÞ ¼2:
Since degC0d2wþ3d7 and 4R4RH¼6 by (2.6), we deduce that the linear systemjIC0jXð4RÞjhas a fixed part.
We exclude that the fixed part isp@R. Indeed in this caseC0would be the union of a curve contained in pof degree at most pSH ¼wþ1, and of a curve con-
tained in the base locus of a pencil of hyperplane sections of degree at mostH3¼3.
But this is not possible since degC0dwþ5.
AlsoY@2Ris not possible. In fact in this caseC0would be contained inY(since the reduced singular line of X, which is the base locus of the residual system jIC0jXð4RÞj YHjOXð2RÞj, is contained in Y). This cannot happen since this would imply dimðjIC0jXð4RÞjÞ ¼dimðYþ jIC0jXð2RÞjÞ ¼2 while we know that dimðjIC0jXð4RÞjÞ ¼1.
(a) The only possibility is that the fixed part isL@3R. In this case we claim that C0is contained inL. In this case we prove thatC0is contained in a surface of degree 3 which is a hyperplane section ofX.C0is the union of a curve contained inLand possibly of the reduced singular line l ofX, which is the base locus of the residual systemjIC0jXð4RÞj Y ¼ jOXðRÞj. IfLis reducible, then lHLand we are done. If Lis irreducible, i.e.,Ldoes not containl, thenC0would containlwith multiplicity 1, but this is not possible, as the following computation shows. LetSS~@ðwþ1aÞHH~þ 3aRR~ð0cacwÞ and FF~@ðmþ1bÞHH~þ3bRR~ð0cbcmÞ be the proper trans- forms ofSandF inXX. When~ ad1 andbd1 by (2.7)C0containslwith multiplicity mðF;S;lÞ ¼3ab.
Letv¼1. By Lemma 3.7 we compute
h0ðIC00jXð4RÞÞ ¼1:
Since by (2.6) 4Rp1H ¼2 and degC1¼mþ1>2, a surface@4R which con- tains C00 splits in the union of p1@Rand a surface@3R@H which contains C0. Therefore we have:
h0ðIC0jXðHÞÞ ¼1:
Letv¼0. By Lemma 3.7 we compute
h0ðIC00jWð7RÞÞ ¼4 if e¼w1;
h0ðIC00jWð7RÞÞ ¼3 if e<w1:
(
Since by (2.6) we find 7Rp1H ¼7Rp2H¼3 and degC1¼degC2 ¼mþ1>
3, we deduce that
h0ðIC0jWð5RÞÞ ¼4 if e¼w1;
h0ðIC0jWð5RÞÞ ¼3 if e<w1:
(
We claim that the linear systemjIC0jWð5RÞjhas a fixed part. To prove the claim we need first to analyze whenC0may contain the singular linelofXas a component.
Let SS~@aHH~ þ ð3w3aþ1ÞRR~ð0<acwÞ and FF~@ðmþ1bÞHH~þ3bRR~ð0cbc mÞbe the proper transforms ofSandF inXX. When~ bd1 by (2.7)C0containslwith multiplicity mðF;S;lÞ ¼3bðwaÞ þbdb. On the other hand C0 is contained in the schemeSVDfor someDAjIC0jWð5RÞj; sinceSVDcontainslwith multiplicity mðD;S;lÞ ¼2ðwaÞ þ1 if DD~@HH~þ2RR~ or mðD;S;lÞ ¼5ðwaÞ þ2 if DD~@5RR,~ thenmðF;S;lÞshould be less or equal than these values. Therefore whenbd1, since
3bðwaÞ þb>2ðwaÞ þ1, we exclude the possibility DD~@HH~þ2RR; when~ bd2, since 3bðwaÞ þb>5ðwaÞ þ2, we exclude the possibility DD~@5RR. Hence~ C0 may contain l only if b¼1, i.e.,FF~@mHH~þ3RR, and the divisors in the linear sys-~ tem jIC0jXð5RÞj are all reducible in the union of five planes. In this case C0 con- tainslwith multiplicitymðF;S;lÞ ¼3ðwaÞ þ1, which has to be less or equal than mðF;5R;lÞ, that is 5 by (2.7). This is possible only if either a¼w1, or a¼w.
Whena¼w1 we haveSS~@ðw1ÞHH~þ4RR~andC0 containslwith multiplicity 4.
Whena¼wwe haveSS~@wHH~þRR~andC0containsl with multiplicity 1.
Now we are able to prove thatjIC0jXð5RÞj has a fixed part. Let us suppose first thatC0containsl. In this case if the linear systemjIC0jXð5RÞjhas no fixed part, then C0 is supported on the line l. Our previous computation implies that degC0c4, while we know that degC0d7. Therefore jIC0jXð5RÞjhas a fixed part as claimed.
Let us suppose that C0 does not contain l. If jIC0jXð5RÞj has no fixed part, then the generic element Din jIC0jXð5RÞj is irreducible and has proper transform DD~@ H~
Hþ2RR, therefore for~ D;D0in the linear system we havemðD;D0;lÞ ¼1. In this case, since by (2.6) 5R5RH ¼8, the base locus of a pencil injIC0jXð5RÞjnot supported onlhas degree 7. Since degC0d7 andh0ðIC0jXð5RÞÞ ¼3 we have a contradiction.
ThereforejIC0jXð5RÞjhas a fixed part.
We claim first that this fixed part cannot bep@R. In this caseC0 would be the union of a curve C10Hp and of a curve C20 contained in the base locus of a linear subsystem jajHjIC0jXð4RÞj of projective dimension 3 if e¼w1;2 if e<w1.
We want to prove thatjaj has a fixed part. Let us suppose first thatC0 containsl, with multiplicity 4 (if a¼w1) or 1 (if a¼w) by our previous computation. If jajhas no fixed part then C20 is supported onl, and since the component ofC10 dis- joint from the line l has degree equal to pp~SS~HH~ ¼a, we should have degC0¼ 4þw1¼wþ3 (ifa¼w1) or degC0¼1þw(ifa¼w), but this is not possible since degC0¼se1¼3we>2wdwþ3. Therefore we have a contradiction andjajhas a fixed part. With similar arguments it is easy to prove thatjajhas a fixed part if we suppose thatC0does not containl. The fixed part ofjIC0jXð5RÞjcan be of the following types:
(a) Y@2R. In this caseC0¼C10UC20, whereC10HY andC20 is contained in the base locus of a linear systemjbjHjOXðHÞjof projective dimension 3 ife¼w1;2 ife<w1. Ife¼w1, then C20 is a linerHp@R. ThereforeC0 is contained in the cubic surfaceYUp@3R. If e<w1, then C20Hs is a plane curve contained in a plane s. If C0 contains l, the divisors in jbj are reducible in the union of 3 planes, therefore s@Ris a fixed part forjIC0jXð5RÞj and we are in the next Case (b). If C0 does not contain l, then degC10c2w; therefore if ecw3 we have degC20d3 which implies s@R. If e¼w2 and degC10¼2w (i.e.,SS~@wHH~ þRR),~ then degC20¼2 and s may be a plane not contained inX. C0 is contained in the cubic surfaceYUs.
(b) L@3R. Here we must put e<w1, since for e¼w1 we have h0ðIC0jXð5RÞÞ ¼4, whileh0ðOXð5RLÞÞ ¼h0ðOXð2RÞÞ ¼3. In this caseC0is con- tained in a cubic surfaceL@3R, a hyperplane section ofX. r In the next propositions we give a closer description of Case (2), (3) and (4) of
Theorem 1.1. We omit the proofs which are quite long and repetitive, the reader may look at [1] Proposition 8 and Proposition 9 to have an idea of the general arguments used. WhenX ¼Sð0;1;2Þwe use the genus Formula (3.9) to understand whether the curveC0does pass through the vertexV or does not; when we don’t specify, it means thatC0may indi¤erently pass or not throughV. In caseX ¼Sð0;0;3Þ, when it is not specified otherwiseC0does not containlas a component. IfC0containsl, the explicit computation for the multiplicity of the singular linelas a component ofC0is similar to the one appearing in the proof of Theorem 1.1 Part (4). For some notation used in the statements of the following propositions the reader may go back to Remark 2.9.
Proposition 4.1.Let k¼2.Then C0is a plane curve of degree se1 and we have the following possibilities:
1.When v¼2 or v¼0 and S@wHþR,then C0 has degree1cdegC0cwþ1 (resp. 1cdegC0cw) and it is contained in a plane p@R. If X ¼Sð0;1;2Þ and v¼2,C0does not pass through the vertex V of X.
2.When v¼1:
(a) If X ¼Sð1;1;1Þ,then C0 has degree1cdegC0cwþ1 and it is contained in a plane p@R. If degC0¼1 there is the further possibility that C0 lies on a plane s which does not belong to the scroll,i.e.,it is a line of typeð0;1Þ.
(b) If X ¼Sð0;1;2Þ and 1cdegC0cw, i.e., 2w<ec3w, then C0 lies either in p@R or in p@H2R.IfdegC0¼wþ1,i.e.,e¼2w,then C0Hp@R and passes through the vertex V of X.
(c)If X ¼Sð0;0;3Þand1cdegC0cw, then C0is contained in a plane p@R.If e¼2wðdegC0¼wþ1Þthere are no curves of maximal genus on Sð0;0;3Þ.
(d)IfdegC0¼2 there is the further possibility that C0lies on a planeswhich does not belong to the scroll. In this case C0 is a conic hyperplane section of a quadric surface@HR.
3.When v¼0and S@ðwþ1ÞH2R:
(a) If X ¼Sð1;1;1Þ and e¼3w1, then C0 is a line of type ð0;1Þ, while if e03w1there are no curves of maximal genus on Sð1;1;1Þ.
(b)If X ¼Sð0;1;2Þand1cdegC0cw1 (i.e., 2w<ec3w1),then C0is con- tained in p@H2R.WhendegC0¼1,i.e.,ife¼3w1,there is the further possi- bility that C0is a line contained in a planep@R and passing through V.Ife¼2w(i.e., degC0¼w)there are no curves of maximal genus on Sð0;1;2Þ.
Proposition 4.2.If k¼1,then C0 is a curve of degree se1contained in a surface of degree two and we have the following possibilities:
1.If v¼2or v¼0and S@wHþR, the surface may be reducible in the union of two planesp1@R andp2@R.In this case C0¼C10UC20,where C10is a curve of degree 2wþ1eif v¼2 (resp. of degree2weif v¼0)onp1and C20 is a curve of degree wþ1 (resp. w)onp2.If X ¼Sð0;1;2Þand S@wHþR,then C0 passes through the vertex V.If X ¼Sð0;0;3Þ,C10 intersects C20in2wþ1e(resp.2we)points on l.
2.If X¼Sð0;0;3Þ,v¼1ande>w(i.e., degC0<2wþ1),the surface is reducible in the union of two planesp1@R andp2@R.In this case C0¼C10UC20,where C10is a curve of degree2wþ1eonp1 and C20 is a curve of degree w onp2.C10 intersects
C20 in2wþ1epoints on l.If v¼1ande¼w there are no curves of maximal genus Gð5;d;sÞin X ¼Sð0;0;3Þ.
3.If X ¼Sð0;1;2Þ,SSwHþR ande0w;wþ1the surface may be reducible in the union of a planep@R and of the plane p@H2R meeting along a line r.In this case C0¼C10UC20,where C10is a curve of degree wþ1onpand C20is a curve of degree 2we1þv on p.C10and C20 meet each other in2we1þv points over r.
4. If X ¼Sð1;1;1Þ or X ¼Sð0;1;2Þ, SSwHþR and e¼w;wþ1 the surface may be an irreducible quadric surface Q@HR. In this case if X ¼Sð1;1;1Þ and e¼w,then C0 is a curve of typeðw1þv;wþ1Þon Q, (resp. of type ð2wþvÞif X ¼Sð0;1;2Þ); if e¼wþ1 then C0 is of type ðw1þv;wÞ on Q (resp. of type ð2w1þvÞ).
5. (a)When v¼2ande¼2w1the surface may be the union of a planep@R and a planes not contained in the scroll X.In this case C0¼C10UC20,where C0Hphas degree wþ1 and C20Hsis a hyperplane section of a quadric surface@HR(@2R, if X ¼Sð0;0;3Þ).
(b)When X ¼Sð1;1;1Þ,v¼2 ande¼w or v¼1ande¼2w1,as in the previ- ous case C0¼C10UC20 where C20 is a line of typeð0;1Þ.
6. If X ¼Sð1;1;1Þ, v¼1 and e0w;wþ1;2w1 or S@ðwþ1ÞH2R and e0w;wþ1there are no curves of maximal genus Gð5;d;sÞon X.
Proposition 4.3.If k¼0,then C0 is a curve of degree se1contained in a surface of degree3.We have the following possibilities:
1.The surface is a hyperplane section L of X.More precisely:
(a)ife¼0,then C0is linked to a line by the intersection SVL;
(b)ife¼1,then C0is linked to a conic by the intersection SVL;
(c)ife>1we have these cases:
i. If X¼Sð1;1;1Þ, then L splits in the union of a smooth quadric surface Q@HR with a planep@R.If SSwHþR,C0is the union of a curve C10HQ of typeðwþ1;w1þvÞand of a plane curve C20Hpof degree we>0that intersect each other in wepoints over the line r¼QVp.If v¼0and S@wHþR,C0is the union of a curve C10HQ of type ðw;wþ1Þ and of a plane curve C20Hp of degree we1 that intersect each other in we1 points over the line r (in this case,if e¼w1,C0is contained in the smooth quadric Q).
ii. If X ¼Sð0;1;2Þ, then L may split in the union of an irreducible quadric cone Q@HR with a planep@R,and we have a similar description of C0as in the pre- vious case. Otherwise L may split in the union of three planes p1@R, p2@R and p@H2R. If SSwHþR and ecw2þv, C0 is the union of a plane curve C10Hp1of degree wþ1,of a plane curve C20Hp2of degree wþ1and of a plane curve C30Hp of degree we ð2vÞ(if v¼1ande¼w1 or v¼0ande¼w2then C0is contained in the union of two planes).C10 intersects C30 in we ð2vÞpoints along r1¼p1Vp.C20intersects C30in we ð2vÞpoints along r2¼p2Vp.If v¼0 and S@wHþR, C0 is the union of a plane curve C10Hp1 of degree w, of a plane curve C20Hp2of degree w and of a plane curve C30Hp of degree we.C10 intersects C30in wepoints along r1 and C20 intersects C30in wepoints along r2.
iii.If X ¼Sð0;0;3Þ,then L splits in the union of three planespi@Rði¼1;2;3Þ.In
this case, when v¼2the proper transform of S isSS~@ðwþ1aÞHH~þ3aRR with~ 0c acwe.C0is the union of the line l counted with multiplicity3a,of a curve C10Hp1 of degree wþ1a,of a curve C20Hp2 of the same degree,and of a curve C30Hp3 of degree wea.C10 and C20 intersect each other in wþ1a points on l.C10 and C20 both intersect C30 in wea points on l.
When v¼1 the proper transform of S is SS~@ðwaÞHH~þ ð3aþ2ÞRR with~ 0cacwe1.C0 is the union of the line l counted with multiplicity 3aþ2,of a curve C10Hp1of degree wa,of a curve C20Hp2 of the same degree,and of a curve C30Hp3 of degree wae1.C10 and C20 intersect each other in a points on l.C10 and C20both intersect C30 in wae1points on l.
When v¼0the proper transform of S is SS~@wHH~þRR.~ If e¼w1,then C0 is the union of a curve C10Hp1 of degree w,of a curve C20Hp2 of the same degree,and of a curve C30Hp3 of degree we.C10 and C20 intersect each other in w points on l.C10 and C20both intersect C30 in wepoints on l.Ife<w1,then C0may contain l with multiplicity 1.In this case C0 is the union of l,of C10,of C20, and of a curve C30Hp3 of degree we1.C10 and C20 intersect each other in w points on l.C10 and C20 both intersect C30 in we1points on l.
2.If X ¼Sð1;1;1Þor X¼Sð0;1;2Þ,v¼0and S@wHþR the cubic surface may be non-degenerate and@3R.In this case C0is the union of a plane curve C10Hp1@R of degree we, of a plane curve C20Hp2@R of degree w and of a plane curve C30Hp3@R of degree w.If X ¼Sð1;1;1Þande¼w1there is the further possibil- ity that C10is a line which does not lie on a plane@R,i.e.,it is of typeð0;1Þ.
3.If X ¼Sð0;1;2Þ,e>0,v¼0and S@ðwþ1ÞH2R the cubic surface may be non-degenerate and@2H3R.In this case C0is the union of a plane curve C10Hp of degree weand of a curve C20HQ of degree2w on a quadric cone Q.
4.If X¼Sð1;1;1Þor X¼Sð0;1;2Þ,v¼0,S@ðwþ1ÞH2R ande¼w2,the surface may be reducible in the union of a quadric surface@HR(if X¼Sð0;1;2Þit may be the union of a planep@R and the plane p)and of a planesnot contained in the scroll.C0is the union of a curve C10 of typeðwþ1;w1Þin Q and of a conic ins.
If X ¼Sð1;1;1Þande¼w1,C0is the union of C10 and of a line of typeð0;1Þ.
5.If v¼0,S@wHþR ande¼w2the surface may split in the union of a plane p1@R,of a planep2@R and of a planeswhich is not contained in the scroll.C0is the union of a curve C10Hp1of degree w,of a curve C20Hp2of the same degree,and of a conic ins.If X¼Sð1;1;1Þande¼w1,C0is the union of C10,C20and of a line of typeð0;1Þ.
5 The existence
Lastly we need an e¤ective construction for curves of degreed, genusGðd;5;sÞinP5, not lying on a surface of degree<s. It should be noted that in casek¼v¼1 it is not possible to construct curves of maximal genus on asmoothrational normal 3-fold (see Proposition 4.2 Case 6)). In this case the construction is possible only on a rational normal 3-fold whose vertex is a point. Before the construction we state the following result (see [11] Lemma 1 p. 133), that we will use. Since [11] is an unpublished work we give a quick proof.
Lemma 5.1.Let X be a smooth3-fold.LetSbe a linear system of surfaces of X and let gbe a curve contained in the base locus ofS.Suppose that the generic surface of Sis smooth at the generic point ofgand that it has at least a singular point which is variable ing.Then all the surfaces ofSare tangent alongg.
Proof.Letp: ~XX!X be the blowup ofX alonggand letEbe the exceptional divi- sor. LetSS~be the linear system inXX~ of the proper transformsfSS~¼pSEgSAS. Let S0be the linear system inEgiven byS0¼ fSS~VEgSS~A~SS. LetSASbe a surface and let x1;. . .;xnbe the points ofgwhereSis singular with multiplicitiesri;. . .;rn. LetS0¼ S~
SVE¼DþPn
1riEi be the corresponding divisor in E, whereEi is the fiber of E overxiandDis the curve that is unisecantEand that represents the tangent plane to S in the generic point ofg. The lemma is proved if we show that Dis fixed when SS~ moves inSS.~
LetS00 the movable part ofS0. Every curve in S00contains the fiber ofE over the singular point of Sthat moves in g. By Bertini’s theorem the generic curve in S00is irreducible or S00 is composed with a pencil. In the first case we conclude thatDis part of the base locus of S0 and we are done. We show now that the second case is impossible. In fact two curves ofS00intersect each other because they both contain a unisecant and a fiber, moreover by hypothesis the fiber moves, therefore the inter- section points move and describe a curve. This is not possible since two generic divi- sors of a system composed with a pencil intersect each other only in a finite number
of fixed points. r
Proof of Theorem1.2. 1. Letk¼3. In this case takeSto be the complete intersection of a smooth rational normal 3-fold XHP5 and a general hypersurfaceGof degree wþ1 inP5. The complete intersection ofSwith a general hypersurfaceF of degree mþ1 gives the required curveC.
2. Letk¼2 andv¼2. In this case we haveeþ1¼2ðwþ1Þ þdwith 0cdcw.
Let p be a plane contained in a smooth rational normal 3-fold X in P5 and let DHpbe a smooth plane curve of degreewþ1>degD¼e2w1d0 (possibly D¼q). Let us consider the linear system jIDjXðwþ1Þj, which is not empty since it contains the linear subsystem Lþ jOXðwÞj, whereL is a hyperplane section of X containing the planep. This shows also that jIDjXðwþ1Þjis not composed with a pencil because in this case every element in the system would be a sum of algebrai- cally equivalent divisors, but, for example, the elements inLþ jOXðwÞjare obviously not of this type. Since degD<wþ1 the linear system jIDjpðwþ1Þj is not empty and its base locus is the curve D; this shows that the base locus of jIDjXðwþ1Þjis exactly the curveD, becausejOXðwÞjis base points free. Therefore by Bertini’s theo- rem the general divisor injIDjXðwþ1Þjis an irreducible surfaceSof degrees, which is smooth outsideD. We claim thatSis in fact smooth at every point pofD. To see this, by Lemma 5.1, it is enough to prove that for every point pAD, there exists a surface in jIDjXðwþ1Þj which is smooth at p, and that for a generic pointqAD, there exist two surfaces injIDjXðwþ1Þjwith distinct tangent planes atq. Indeed, for every pADwe can always find a surfaceT in the linear systemjOXðwÞjwhich does not pass through p, therefore the surfaceLþTis smooth at pwith tangent planep.
Moreover a generic surface in the linear systemjIDjXðwþ1Þjwhich cutsDonphas at ptangent planeTp0p.
LetC0Hpbe the linked curve toDby the intersection ofSandp. Let us consider the linear system jIC0jSðmþ1Þj. Since degC0¼se1<mþ1, with the same arguments used above it is easy to see thatjIC0jSðmþ1Þjis not empty, is not com- posed with a pencil and has base locus equal to the curveC0. Therefore by Bertini’s theorem the generic curve Cin the movable part of this linear system is irreducible and smooth. MoreoverClies on a smooth surfaceSof degrees¼3wþ3 and it has the required numerical characters, as one may easily verify using the genus Formula (3.9), paðC0Þ ¼12ð3wþ2e1Þð3wþ2e2Þ (computed by Clebsch’s formula) and degðRVC0Þ ¼0.
3. Letk¼2 andv¼1. In this case we havee¼2wþdwith 0cdcw. LetXand pHXbe as in the previous case and let p10pbe an other plane contained inX. Let DHpbe a smooth plane curve of degree 0cdegD¼e2w<wþ1 onp. In this case by Bertini’s theorem we can find an irreducible surface S@ðwþ1ÞHR of degrees¼3wþ2 in the movable part of the linear systemjIDUp1jXðwþ1Þj, smooth outside D. With the same argument used in the previous case we prove that SUp1 is smooth at every point ofD. Namely, for every pAD, a generic surface in the lin- ear system LþL1þ jOXðw1Þj, whereL¼pþQand L1¼p1þQ1 are reducible hyperplane sections containing respectively p and p1 and such that pBQUQ1, is smooth at pwith tangent planep, while a surface in the linear systemL1þ jIDjXðwÞj which cutsDonphas tangent planeTp0p. SinceDVp1¼qthis implies thatSis smooth.
LetC0be the linked curve toDby the intersectionpVS. By Bertini’s theorem the linked curveCtoC0by the intersection ofSwith a general elementFmþ1in the linear system jIC0jSðmþ1Þj is smooth of degree d. Moreover it lies on a smooth surface S of degrees¼3wþ2 and it has the required genus, as one can compute by using Formula (3.9), where the curveC00is the union ofC0and of the curveC1Hp1cut on
p1 byFmþ1, and degðRVC00Þ ¼0.
4. Letk¼2 andv¼0. In this case we havee¼2wþdwith 0cd<w. LetXand pHX as before and letqbe a smooth quadric surface contained inX, intersectingp along a liner. LetDHpbe a smooth plane curve of degree 0cdegD¼e2w<w on p(when degD¼1 we suppose thatDdoes not coincide with the line r). In this case by Bertini’s theorem we can find an irreducible surface S@wHþRof degree s¼3wþ1 in the movable part of the linear systemjIDUqjXðwþ1Þj, smooth outside D. As in the previous cases we claim thatSis smooth. Namely, for every pAD, we can find in the movable part of our linear system jIDUqðwþ1Þja surface which is smooth at p with tangent plane equal top (take a surface of the formLþp0þT, withLas usual,p0@Ra plane ofXdisjoint frompandT a surface injOXðw1ÞHj that does not pass through p), and a surface with tangent plane Tp0p (take a sur- face of the formp0þV, whereVis a divisor inIDjXðwÞwhich cutDonp).
LetC0be the linked curve toDby the intersectionpVS; we have degC0¼3we.
The linked curveCtoC0by the intersection ofSwith a general elementFmþ1in the linear system jIC0jSðmþ1Þj is smooth of degree d and lies on the smooth surface S of degree s¼3wþ1. Its genus is maximal as one can compute by using For-
