Existence and the reducibility of the Hilbert scheme of linearly normal curves in \mathbb{P}^r of relatively high degrees
aa r X i v : . [ m a t h . AG ] J a n Existence and the reducibility of the Hilbertscheme of linearly normal curves in P r of relatively high degrees Changho Keem
Abstract.
Let H d,g,r be the Hilbert scheme parametrizing smooth ir-reducible and non-degenerate curves of degree d and genus g in P r . We denote by H L d,g,r the union of those components of H d,g,r whosegeneral element is linearly normal. In this article we show that H L d,g,r ( d ≥ g + r −
3) is non-empty in a wider range of triples ( d, g, r ) beyondthe Brill-Noether range. This settles the existence of the Hilbert scheme H L d,g,r of linearly normal curves in the range g + r − ≤ d ≤ g + r , r ≥ d, g, r ) with g + r − ≤ d ≤ g + r for which H L d,g,r is reducible or (irreducible). Mathematics Subject Classification (2010).
Primary 14C05, Sec-ondary 14H10.
Keywords.
Hilbert scheme, algebraic curves, linearly normal, linearseries.
1. An overview, preliminaries and basic set-up
Given non-negative integers d , g and r ≥
3, we denote by H d,g,r the Hilbertscheme of smooth curves parametrizing smooth irreducible and non-degenera-te curves of degree d and genus g in P r .In two recent works [6, 22], the authors studied the Hilbert scheme of linearlynormal curves in P r , r ≥
3. As a preliminary attempt toward a reasonablesettlement of the
Modified Assertion of Severi – which was first addressedby Severi in [28] and later modified in [22] and [9, p.489 (2), (3)]– the main
This work was started when the author was enjoying the hospitality and the stimulatingatmosphere of the Max-Planck-Insitut f¨ur Mathematik (Bonn). The author was supportedin part by National Research Foundation of South Korea (2019R1I1A1A01058457).
Changho Keemproblem treated in [6, 22] was focused on the irreducibility of the Hilbertscheme of linearly normal curves as follows.
Modified Assertion of Severi.
A nonempty H L d,g,r is irreducible for anytriple ( d, g, r ) in the Brill-Noether range ρ ( d, g, r ) = g − ( r + 1)( g − d + r ) ≥ , where H L d,g,r is the union of those components of H d,g,r whose general elementis linearly normal.We briefly make a note of the results in [6] regarding this business as follows. Theorem 1.1 (Ballico et al.). (1) H L g + r,g,r is non-empty and irreducible. (2) H L g + r − ,g,r is non-empty and irreducible for g ≥ r + 1 and is empty for g ≤ r . (3) Every non-empty H L g + r − ,g,r is irreducible. (4) Every non-empty H L g + r − ,g,r is irreducible for g ≥ r + 3 and is non-empty for g ≥ r + 3 . By the Riemann-Roch formula, there is no complete linear series of degree d and dimension r in case d ≥ g + r + 1. Therefore in this range we have H L d,g,r = ∅ and the Modified Assertion of Severi makes sense only if g − d + r ≥ g with respect to the dimension r of the ambientprojective space P r for which H L d,g,r ( d ≥ g + r −
1) is non-empty is sharp.On the other hand, even though the irreducibility of H L d,g,r holds beyond theBrill-Noether range for d = g + r −
2, the optimal range of g (with respectto r ) for which H L g + r − ,g,r is non-empty has not been explicitly described in[6]. Instead, the following scattered results have been shown; cf. [6, Remark2.4].(1) H L g + r − ,g,r = ∅ for g ≤ r + 2, r ≥ H L g + r − ,g,r = ∅ inside the Brill-Noether range ρ ( g + r − , g, r ) = g − r + 1) ≥ H L g + r − ,g,r = ∅ for some values of g ≥ r + 3 outside the Brill-Noetherrange.However for most values of g outside the Brill-Noether range, the exisitenceof a component of H L g + r − ,g,r has been left undetermined.In this paper we determine the full range of the genus g for which H L g + r − ,g,r = ∅ . We also make an attempt to get the widest possible range of g for which H L g + r − ,g,r = ∅ . As a by product, we come up with a rather extensive list oftriples ( g + r − , g, r ) for which H L g + r − ,g,r is reducible.The organization of this paper is as follows. After we briefly recall someterminologies and basic preliminaries in the remainder of this section, wexistence & reducibility of the Hilbert scheme of smooth curves 3start the next section by showing the non-emptiness of H L g + r − ,g,r for everypossible g ≥ r + 3 outside the Brill-Noether range.In the subsequent section, we demonstrate the non-emptiness of H L g + r − ,g,r beyond the Brill-Noether range. We also determine the full range of g withrespect to r for which H L g + r − ,g,r is reducible. In the final section – with anopen end – we make a discussion on a couple of related aspects of our studysuch as the case d = g + r − π ( d, r ) is the maximal possible arithmetic genus of an irreducible andnon-degenerate curve of degree d in P r . Following classical terminology, alinear series of degree d and dimension r on a smooth curve C is usuallydenoted by g rd . A base-point-fee linear series g rd ( r ≥
2) on a smooth curve C is called birationally very ample when the morphism C → P r induced by the g rd is generically one-to-one (or birational) onto its image. A base-point-freelinear series g rd on C is said to be compounded of an involution (compoundedfor short) if the morphism induced by the linear series gives rise to a non-trivial covering map C → C ′ of degree k ≥
2. Throughout we work over thefield of complex numbers.We recall several fundamental results which are rather well-known; cf. [4] or[2, § § M g be the moduli space of smooth curves of genus g .Given an isomorphism class [ C ] ∈ M g corresponding to a smooth irreduciblecurve C , there exist a neighborhood U ⊂ M g of the class [ C ] and a smoothconnected variety M which is a finite ramified covering h : M → U , as wellas varieties C , W rd and G rd proper over M with the following properties:(1) ξ : C → M is a universal curve, i.e. for every p ∈ M , ξ − ( p ) is a smoothcurve of genus g whose isomorphism class is h ( p ),(2) W rd parametrizes the pairs ( p, L ) where L is a line bundle of degree d and h ( L ) ≥ r + 1 on ξ − ( p ),(3) G rd parametrizes the couples ( p, D ), where D is possibly an incompletelinear series of degree d and dimension r on ξ − ( p ).Let e G ( e G L , respectively) be the union of components of G rd whose generalelement ( p, D ) corresponds to a very ample (very ample and complete, re-spectively) linear series D on the curve C = ξ − ( p ). By recalling that an opensubset of H d,g,r consisting of elements corresponding to smooth irreducibleand non-degenerate curves is a P GL( r + 1)-bundle over an open subset of e G ,the irreducibility of e G guarantees the irreducibility of H d,g,r . Likewise, theirreducibility of e G L ensures the irreducibility of H L d,g,r .The following facts regarding the schemes G rd and W rd are also well-known;cf. [2, Proposition 2.7, 2.8], [13, 2.a] and [4, Ch. 21, §
3, 5, 6, 11, 12].
Proposition 1.2.
For non-negative integers d , g and r , let ρ ( d, g, r ) := g − ( r + 1)( g − d + r ) be the Brill-Noether number. Changho Keem(1)
The dimension of any component of G rd is at least λ ( d, g, r ) := 3 g − ρ ( d, g, r ) . Moreover, if ρ ( d, g, r ) ≥ , there exists a unique component G of e G which dominates M (or M g ). (2) Suppose g > and let X be a component of G d whose general element ( p, D ) is such that D is a birationally very ample linear series on ξ − ( p ) .Then dim X = 3 g − ρ ( d, g,
2) = 3 d + g − . (3) G d is smooth and irreducible of dimension λ ( d, g, if g > , d ≥ and d ≤ g + 1 . Occasionally we will use the following upper bound of the dimension of anirreducible component of W rd ; cf. [17]. Proposition 1.3 ( [17, Proposition 2.1]) . Let d, g and r ≥ be positiveintegers such that d ≤ g + r − and let W be an irreducible component of W rd . For a general element ( p, L ) ∈ W , let b be the degree of the base locusof the line bundle L = | D | on C = ξ − ( p ) . Assume further that for a general ( p, L ) ∈ W the curve C = ξ − ( p ) is not hyperelliptic. If the moving part of L = | D | is (a) very ample and r ≥ , then dim W ≤ d + g + 1 − r − b ; (b) birationally very ample, then dim W ≤ d + g − − r − b ; (c) compounded, then dim W ≤ g − d − r . Remark 1.4. (i) In the Brill-Noether range ρ ( d, g, r ) ≥
0, the unique com-ponent G of e G (and the corresponding component H of H d,g,r as well)which dominates M or M g is called the “principal component”.(ii) In the range d ≤ g + r inside the Brill-Noether range ρ ( d, g, r ) ≥
0, theprincipal component G which has the expected dimension λ ( d, g, r ) isone of the components of e G L (cf. [13, 2.1 page 70]), and hence e G L and H L d,g,r are non-empty in this range.Inside the family of plane curves of degree d in P which is parametrizednaturally by the projective space P N with N = d ( d +3)2 , let Σ d,g ⊂ P N bethe Severi variety of plane curves of degree d and geometric genus g . Thefollowing theorem of Harris is fundamental; cf. [4, Theorem 10.7 and 10.12]or [14, Lemma 1.1, 1.3 and 2.3] . Theorem 1.5. Σ d,g is irreducible of dimension d + g − λ ( d, g,
2) +dim P GL (3) . Denoting by G ′ ⊂ G d the union of components whose general element ( p, D )is such that D is birationally very ample on C = ξ − ( p ), we remark thatan open subset of the Severi variety Σ d,g is a P GL(3)-bundle over an opensubset of G ′ . Therefore, as an immediate consequence of Theorem 1.5, theirreducibility of Σ d,g implies the irreducibility of the locus G ′ ⊂ G d and viceversa. We make a note of this observation as the following lemma.xistence & reducibility of the Hilbert scheme of smooth curves 5 Lemma 1.6.
Let G ′ ⊂ G d be the union of components whose general element ( p, D ) is birationally very ample on C = ξ − ( p ) . Then G ′ is irreducible.
2. Existence of H L g + r − ,g,r In this section we prove the following theorem which provides the full rangeof the genus g for which H L g + r − ,g,r = ∅ . Theorem 2.1. (i) H L g + r − ,g,r is non-empty, irreducible, generically reducedof the expected dimension g − ρ ( g + r − , g, r ) + dim P GL ( r + 1) and has the expected number of moduli for every g ≥ r + 3 . (ii) H L g + r − ,g,r = ∅ if g ≤ r + 2 .Proof. (i) The irreducibility of a non-empty H L g + r − ,g,r was shown already in[6, Theorem 2.3] and the non-emptiness is assured by Remark 1.4 (ii) in theBrill-Nother range ρ ( g + r − , g, r ) = g − r + 1) ≥
0. Therefore we work inthe range g ≤ r + 1 which is outside the Brill-Noether range.We briefly recall the procedure in [6, Theorem 2..3] showing the irreducibilityof e G L (and hence the irreducibility of H L g + r − ,g,r ) under the assumption thatthere exists of a component G ⊂ e G L ⊂ e G ⊂ G rg + r − . The dimension estimatedim G = λ ( g + r − , g, r ) = 4 g − r − W ⊂ W rg + r − be the component containing the image of the naturalrational map G ι W rg + r − with ι ( D ) = |D| . Denoting by W ∨ ⊂ W g − r thelocus consisting of the residual series (with respect to the canonical series onthe corresponding curve) of those elements in W , i.e. W ∨ = { ( p, ω C ⊗ L − ) : ( p, L ) ∈ W} , we let f W ∨ ⊂ W g − r be the union of those components W ∨ corresponding to W arising from each component G ⊂ e G L . By the above dimension estimate,we havedim W ∨ = dim W = dim G = 4 g − r − λ ( g − r, g,
1) = dim G g − r . It was shown eventually that f W ∨ is birationally equivalent to the irreduciblelocus G g − r (cf. Proposition 1.2(3)), hence f W ∨ is irreducible and so is e G L .In order to demonstrate the existence of a component G ⊂ e G L ⊂ e G ⊂ G rg + r − ,it is enough to show that for a general ( p, g g − r ) ∈ G g − r – which is non-empty(cf. [4, Proposition 6.8, p.811]) – the residual series D = | K C − g g − r | is veryample (and complete) g rg + r − on C = ξ − ( p ). Note that D = | K C − g g − r | isvery ample if and only ifdim | g g − r + u + v | = dim | g g − r | = 1 Changho Keemfor every u + v ∈ C . Since ρ ( g + r − , g, r ) = ρ ( g − r, g,
1) = g − r + 1) < , G g − r dominate the proper irreducible closed sublocus M g,g − r ⊂ M g gener-ically consisting of general ( g − r )-gonal curves with a unique base-point-free and complete pencil g g − r . Recall that the Clifford index e of a general( e + 2)-gonal curve of genus g ≥ e + 2 is computed only by the unique pencilcomputing the gonality as long as e = 0, i.e. there does not exist a g s s + e with2 s + e ≤ g − s ≥
2; cf. [5, Theorem] or [19, Corollary 1]. Taking e +2 = g − r ,deg | g g − r + u + v | = g − r + 2 = e + 4 and hence dim | g g − r + u + v | = 1 forany u + v ∈ C .Finally one may argue that H L g + r − ,g,r is generically reduced as follows. Notethat a component H of the Hilbert scheme H L d,g,r is generically reduced ifand only if the the corresponding component G ⊂ e G L ⊂ e G ⊂ G rd is genericallyreduced. Since we have a sequence of birational maps G = f G L W W ∨ κ G g − r , G is birationally equivalent to G g − r which is generically reduced by Propo-sition 1.2 (3); cf. [4, Proosition 6.8, p.811] or [2, Proposition 2.7]. Therefore H L g + r − ,g,r is generically reduced.We recall that a component H ⊂ H d,g,r is said to have the expected numberof moduli if the image of natural functorial map H π M g has the dimensionmin(3 g − , g − ρ ( d, g, r )) . In our current situation, the image of the irreducible H L g + r − ,g,r under thefunctorial map is M g,g − r which has dimension3 g − ρ ( g − r, g,
1) = 3 g − ρ ( g + r − , g, r ) = dim G . (ii) is the just the repetition of the statement [6, Remark 2.4 (i)] which followseasily from Castelnuovo genus bound. (cid:3) Remark 2.2. (i) For some particular g such as g = 2 r + 1 or g = 2 r whichis just below the Brill-Noether range, one may show the existence of H L g + r − ,g,r utilizing the existence of a certain reducible and smoothablecurve C ⊂ P r of degree d = g + r − g ; cf. [7,Corollary 1.3, Remark 1.5]. For g = r + 3 which is just above the emptyrange, the existence of H L g + r − ,g,r = H L r +1 ,r +3 ,r = H r +1 ,r +3 ,r is assured by the existence of extremal curves of degree d = g + r − r + 1 and the Hilbert scheme H r +1 ,r +3 ,r is irreducible by [8, Theorem1.4(i)].(ii) For r = 3 and d = g + r − g + 1, it is known that H L g +1 ,g, = H g +1 ,g, = ∅ xistence & reducibility of the Hilbert scheme of smooth curves 7and is irreducible of the expected dimension 4( g + 1) for g ≥
6, whereas H g +1 ,g, = ∅ for g ≤
5; cf. [24, Proposition 2.1] and references therein.(iii) For r = 4 and d = g + r − g + 2, it is also known that H L g +2 ,g, = H g +2 ,g, = ∅ and is irreducible for g ≥ g ≤
6; cf. [21, Corollary2.2].(iv) For r = 5. it is not clear if H L g +3 ,g, is the only component of H g +3 ,g, .As far as the author knows, the irreducibility of H d,g, even for d ≥ g +5has not been completely settled yet.
3. Existence and the reducibility of H L g + r − ,g,r for low g In this section we turn to the next case and extend the existence of the Hilbertscheme H L g + r − ,g,r of linearly normal curves beyond the Brill-Noether range.We also show the reducibility for almost all the genus g in the range r + 5 ≤ g ≤ r + 2whereas every non-empty H L g + r − ,g,r is known to be irreducible for g ≥ r +3;cf. Theorem 1.1 or [6, Theorem 2.5]. We first collect several known relevantresults regarding the Hilbert scheme of smooth curves in P and P . Remark 3.1. (i) Somewhat stronger results hold for curves in P . For r = 3 and d = g + r − g , by [20, Theorem 2.1] H g,g, = H L g,g, is irreducible for g ≥ g ≤
7. Even though the non-emptiness of H g,g, for g ≥ P of degree d = g for g ≥ g ≤ π ( g,
3) if g ≥
9, where π ( d, r )is the so-called second Castelnuovo genus bound for curves of degree d in P ; cf. [9, Theorem 11.5, p 467]. For g = 8, there are smooth curvesof type (3 ,
5) on a smooth quadric surface which form an irreduciblefamily of the expected dimension.(ii) For r = 4 and d = g + r − g + 1, the irreducibility of H g +1 ,g, has not been fully determined yet to the best knowledge of the author.It is only known that a non-empty H L g +1 ,g, is irreducible unless g =9 by [22, Theorem 2.1]. Regarding the existence of H L g +1 ,g, and theirreducibility of H g +1 ,g, as well – especially outside the Brill-Noetherrange ρ ( g + 1 , g,
4) = g − < H g +1 ,g, = H L g +1 ,g, = ∅ for g ≤ g = 9, H , , = H L , , = ∅ and is reducible with two compo-nents of dimensions 42 and 43. Changho Keem(c) For g = 10, H , , = H L , , = ∅ and is irreducible of the expecteddimension 46.(d) For g = 12, H , , = ∅ and is reducible with two components ofthe same expected dimension 54, whereas H L , , is irreducible; notethat in this case H L , , = H , , .(e) For the other low genus g = 11 , ,
14 outside the Brill-Noetherrange, H g +1 ,g, = H L g +1 ,g, = ∅ and is irreducible, generically reduced of the expected dimension by [23,Theorem 2.7], whose the existence is guaranteed by [25, Theorem 0.1].We now shift our attention to curves in P r , r ≥
5. By the Castelnuovo genusbound and [6, Theorem 2.5 and Remark 2.6], we know the following: ⋄ H L g + r − ,g,r = H g + r − ,g,r = ∅ for g ≤ r + 4. ⋄ Every non-empty H L g + r − ,g,r is irreducible for g ≥ r + 3.For low genus g = r + 5 which is just above the empty range g ≤ r + 4, H L g + r − ,g,r = H g + r − ,g,r = ∅ which consists of extremal curves (if r ≥
4) as one expects. On the otherhand, the irreducibility of the Hilbert scheme of extremal curves H L g + r − ,g,r = H L r +2 ,r +5 ,r = H r +2 ,r +5 ,r is known to be dependent on the dimension r of the ambient projective space P r by [8, Theorem 1.4], which we quote as the following theorem. Theorem 3.2 (Ciliberto). (i)
For r ≥ , H L r +2 ,r +5 ,r = H r +2 ,r +5 ,r = ∅ and is irreducible. (ii) For r = 5 , H L r +2 ,r +5 ,r = H r +2 ,r +5 ,r = ∅ and is reducible with twocomponents; one consisting of the images of smooth plane sextics underthe -tuple embedding on a Veronese surface and the other one consistingof trigonal curves on a rational normal scroll. In the next case g = r + 6, we are confronted with the following ratherunexpected phenomenon. Indeed what we shall see next is that there existstriples ( g + r − , g, r ) for which H g + r − ,g,r = ∅ beyond the obvious empty range g ≤ r + 4. The following is the first suchkind the author knows of. However, there are further such examples of thiskind (and even more) if the degree of curves gets lower with respect to thedimension of the ambient projective spaces; cf. Remark 4.3. Proposition 3.3. (i) H g + r − ,g,r = ∅ for g = r + 6 and r ≥ . xistence & reducibility of the Hilbert scheme of smooth curves 9(ii) For g = r + 6 and ≤ r ≤ , H L g + r − ,g,r = H g + r − ,g,r = ∅ which is irreducible of the expected dimension having the expected num-ber of moduli.Proof. (i) Let C ⊂ P r ( r ≥
10) be a smooth, non-degenerate and irreduciblecurve of genus g = r + 6 with a very ample hyperplane series D = g r r +3 . Wemay assume that D is complete, otherwise we have g = r + 6 ≤ π (2 r + 3 , r + 1) = r + 4which is an absurdity. Consider the residual series E = | K C − D| = g . Notethat g = r + 6 (cid:9) ( e − e − e ≤
7. Hence E is compounded, with the non-empty base locus ∆ = { q } and C is either(a) trigonal with E = | g + q | or(b) bielliptic with E = | φ ∗ ( g ) + q | , where C φ → E is the double covering ofan elliptic curve E .If C is trigonal, let q + t + s ∈ g be the unique trigonal divisor containingbase point q of E . We then havedim |D − t − s | = dim | K − g − q − t − s | = dim | K − g | ≥ r − . If C is bielliptic,dim |E + φ ∗ ( u ) | = dim | φ ∗ ( g + u ) + q | ≥ , for any u ∈ E , implyingdim |D − φ ∗ ( u ) | = dim | K − E − φ ∗ ( u ) | = dim | K C − φ ∗ ( g + u ) − q | ≥ r − . Both are contradictory to the assumption D being very ample.(ii) For 5 ≤ r ≤ H L r +3 ,r +6 ,r = H r +3 ,r +6 ,r is irreducible of the expecteddimension λ (2 r + 3 , r + 6 , r ) + dim P GL( r + 1)and the proof is almost parallel to the proof of [6, Theorem 2.5]. However, wepresent a shortened version of a proof for the convenience of readers. Let e G L be the union of irreducible components G of G rg + r − whose general elementcorresponds to a pair ( p, D ) such that D is very ample and complete linearseries on C := ξ − ( p ) – as was defined in the paragraph preceding Proposition1.2. Assuming the existence of a component G ⊂ e G L ⊂ e G ⊂ G rg + r − , we notethat the moving part of a general element E ∈ W ∨ ⊂ W g − r +1 = W W ∨ is defined as in the proof of Theorem 2.1) is birationally veryample by the proof of (i) and is base-point-free since g = r + 6 ≥ (cid:9) (6 − − . Using Proposition 1.3(b), one has the exact dimension estimatedim G = dim W = dim W ∨ = 4 g − r − λ ( g + r − , g, r ) . (1)Let f W ∨ be the union of the components W ∨ of W g − r +1 corresponding toeach G ⊂ e G L . We also let G ′ be the union of irreducible components of G g − r +1 whose general element is a base-point-free and birationally very ample linearseries. We recall that, by Lemma 1.6 and Proposition 1.2(2), G ′ is irreducibleand dim G ′ = 3( g − r + 1) + g − g − r − . By the equality (1),dim W ∨ = dim G = 4 g − r − G ′ . (2)Then one may argue that there is a natural generically injective dominantrational map f W ∨ κ G ′ with κ ( |E| ) = E as well as another natural rationalmap G ′ ι f W ∨ with ι ( E ) = |E| , which is an inverse to κ (wherever it isdefined). It then follows that f W ∨ is birationally equivalent to the irreduciblelocus G ′ , hence f W ∨ is irreducible and so is e G L . Since H L g + r − ,g,r is a P GL( r +1)-bundle over an open subset of e G L , H L g + r − ,g,r is irreducible of the expecteddimension.As was suggested by the proof of the irreducibility as we outlined above, weshow the existence as follows.(1) In particular for r = 9, H L r +3 ,r +6 ,r = H L , , is the family of smooth plane septics embedded in P by the 3-tupleembedding.(2) The existence of H L r +3 ,r +6 ,r for the other 5 ≤ r ≤ S π → P which is the blowing up of P at 9 − r (5 ≤ r ≤
8) generalpoints P , · · · P − r with exceptional divisors E , · · · , E − r . Denoting by( a ; b , · · · , b − r ) the linear system | π ∗ ( aL ) − P − ri =1 b i E i | – where L is aline in P – we note that (7; 2 , · · · ,
2) is very ample on S ; cf. [16, V, 4.12,Ex.4.8] and hence contains an irreducible and non-singular member C ,which is a curve of genus g = 12 (7 − − − X b i ( b i −
1) = 15 − (9 − r ) = r + 6and degree d = (3; 1 , · · · , · (7; 2 , · · · ,
2) = 21 − − r ) = g + r − P r under the embedding induced by the anti-canonical linear system(3; 1 , · · · , H L r +3 ,r +6 ,r M g factors through G = e G L ⊂ G r r +3 and the functorial map Σ ,r +6 M g factors through G ′ ⊂ G . Since G bir ∼ = W bir ∼ = W ∨ bir ∼ = G ′ , these two functorial maps have the same image in M g . By [27, Theorem 4.2],Σ ,r +6 has the expected number of moduli and since ρ (2 r + 3 , r + 6 , r ) = ρ (7 , r + 6 , , H L r +3 ,r +6 ,r also has the expected number of moduli. (cid:3) Almost all the components of H L d,g,r which we have seen so far in this paperhave the minimal possible dimension X ( d, g, r ) := λ ( d, g, r ) + dim P GL( r + 1)and have the expected number of moduli.For the next g = r + 7, H L r +3 ,r +6 ,r is non-empty but reducible for every r ≥
5. Moreover if r ≥
15, no component of H L r +3 ,r +6 ,r has the expectednumber of moduli, unlike the previous case g = r + 6. Theorem 3.4. (i)
For r ≥ and g = r + 7 , H L g + r − ,g,r = H L r +4 ,r +7 ,r = ∅ and is reducible without any component of the expected dimension. More-over all the components have more than the expected number of moduli. (ii) For ≤ r ≤ and g = r + 7 , H L g + r − ,g,r = H L r +4 ,r +7 ,r = ∅ and is reducible with at least one component of the expected dimensionhaving the expected number of moduli, together with another componenthaving more than the expected number of moduli.Proof. (i) We deal with the case r ≥
15 first. Let D = g r r +4 ∈ G ⊂ e G L ⊂ G r r +4 be a complete and very ample linear series on a smooth curve C of genus g = r + 7. The residual series E = | K C − D| = g is compounded since g = r + 7 (cid:9)
21 = (8 − − r ≥ . Let δ be the degree of the base locus ∆ of the compounded E . We list up allthe possible cases as follows.2 Changho Keem(1) δ = 0, E = 2 g and C is 4-gonal. Recall that on a general 4-gonal curve C with a unique g , | K C − g | is very ample by [18, Corollary 3.3]. Hence G dominates the irreduciblelocus M g, and dim G = dim M g, . Since λ ( d, g, r ) = 3 g − ρ ( d, g, r ) = r + 22 (cid:8) dim M g, = 2 g + 3 = 2 r + 17the component G over the locus M g, as well as the corresponding com-ponent of H L r +4 ,r +7 ,r over G – which we denote by H , – have morethan the expected number of moduli, with the dimension strictly greaterthan the expected dimension X ( d, g, r ).(2) δ = 0, C is a double cover C π → E of a non-hyperelliptic curve E ofgenus h = 3 and E = π ∗ ( g ) = π ∗ ( | K E | ).Let X n,h be the locus in M g corresponding to curves which are n-foldcoverings of smooth curves of genus h . In the current case, there exista sequence of natural rational maps G W W ∨ X , , which are generically injective into a component of X , .Recall that for a covering C π → E and a line bundle M on the base curve E of genus h , it is well known by projection formula that H ( C, π ∗ M ) = H ( E, π ∗ π ∗ M ) = H ( E, π ∗ ( π ∗ M ⊗ O C O C )= H ( E, M ⊗ O E π ∗ O C ) , and that det π ∗ O C ∼ = O E ( − D )for some divisor D on E such that 2 D is linearly equivalent to the branchdivisor B of π consisting of those points on E over which π ramifies; cf.[16, IV, Ex. 2.6]). In particular, the vector bundle π ∗ O C on E of rankequal to deg π has degree − deg D = −
12 deg B = (deg π ) · ( h − − ( g − ≤ . If π is a double covering, the rank two vector bundle π ∗ O C splits into theline bundles O E and det π ∗ O C of degree 2( h − − ( g −
1) = 2 h − g − H ( C, π ∗ M ) = H ( E, M ) ⊕ H ( E, M ⊗ O E ( − D )) , and if deg M < deg D = g + 1 − h , then H ( E, M ⊗ O E ( − D )) = 0and hence H ( C, π ∗ M ) = H ( E, M ) . (3) Notation of this kind shall be defined precisely in a more general context in the proof ofTheorem 3.9. xistence & reducibility of the Hilbert scheme of smooth curves 13We first put M = O ( K E ). Since deg M = 2 · − ≤ g − · g = r + 7 ≥
22, we have h ( C, π ∗ O ( K E )) = h ( E, O ( K E )) = h = 3 . We now show that the complete linear series | K C − π ∗ K E | is very ample.Take any T + S ∈ C and put s = π ( S ) , t = π ( T ) , S + S ′ = π ∗ ( s ) , T + T ′ = π ∗ ( t ) . We have | π ∗ ( K E ) + S + T | = | π ∗ ( K E + s + t ) − S ′ − T ′ | . Substituting M = O ( K E + s + t ) in (3), h ( C, O ( π ∗ ( K E + s + t ))) = h ( E, O ( K E + s + t )) = h + 1 = 4 . Note that | K E + s + t | is base-point-free and hence | π ∗ ( K E + s + t ) | isalso base-point-free. We finally have h ( C, O ( π ∗ ( K E ) + S + T )) = h ( C, O ( π ∗ ( K E + s + t ) − S ′ − T ′ )) ≤ h ( C, O ( π ∗ ( K E + s + t ))) − h = h ( C, O ( π ∗ K E ))and therefore | K C − π ∗ K E | is very ample.So far we have shown that G dominates a component of X , .By [4, Theorem 8.23, p.828], X n,h has pure dimensiondim X n,h = 2 g + (2 n − − h ) − , and by noting that there is a generically one-to-one natural rationalmap between G and a component of X , , we havedim G = dim W = dim X , = 2 g − . Since r + 22 = λ ( g + r − , g, r ) (cid:8) dim G = dim X , = 2 g − r + 10 , ( ⋄ )there is a component H (2 , ⊂ H L g + r − ,g,r arising from a componentof X , , which has more than the expected number of moduli by theinequality ( ⋄ ). We observe that a general element of H (2 , is 6-gonalwhereas a general element of H , is 4-gonal anddim H , = 2 r +17+dim P GL( r +1) (cid:9) dim H (2 , = 2 r +10+dim P GL( r +1) . Therefore it follows that the locus H (2 , is not contained in the bound-ary of H , by semi-continuity. This shows that H , and H (2 , aredistinct irreducible components.Now we are only left with the following three possibilities.(3) δ = 0 and E induces a double covering onto a curve of genus h = 2.(4) δ = 2, E = | g + q + s | and C is trigonal.(5) δ = 2 and E = | φ ∗ ( g ) + q + s | for a double covering C φ → E onto anelliptic curve E .4 Changho KeemHowever for the above three cases, the residual series of the compounded E is not very ample and we omit the easy verification, which is similar to theproof of Proposition 3.3 (i).(ii) For 5 ≤ r ≤
14, there is a possibility that E = g ∈ W ∨ ⊂ W – which isthe residual series of a complete very ample D ∈ G ⊂ G rg + r − – is birationallyvery ample or very ample. In this case we may proceed in a similar way as wedid in the proof of Proposition 3.3(ii), to come up with an irreducible family H Σ ,g ⊂ H L g + r − ,g,r of minimal possible dimension X ( d, g, r ). If H Σ ,g = ∅ ,the residual series of the hyperplane series corresponding to a general elementof H Σ ,g is a birationally very ample, base-point-free net g inducing a planecurve of degree 8 with only nodal singularities; by abuse of language, onemight describe this situation as “the family H Σ ,g corresponds to theSeveri variety Σ ,g ”. We now show that the family H Σ ,g is non-empty. The issue is whether theresidual series of the linear system cut out by lines on the plane nodal curve E corresponding to a general member of the Severi vareity Σ ,r +7 is a completeand very ample g rg + r − . For r = 14 – which is a trivial case – Σ ,r +7 = P is the family of smooth octics and the residual series of the linear series E = | L | cut out by lines on a smooth plane octic is the linear series 4 E = | L | ,which certainly is very ample. Hence H Σ ,g is the family of smooth planeoctics embedded by the 4-tuple embedding into P = P of genus g anddegree d with g = r + 7 = 21 = (8 − − , d = g + r − · . For the existence of smooth curves in lower dimensional projective space P r ,5 ≤ r ≤
13, we first fix two integers e = 8 = g − r + 1 and δ = ( e − e − − g = 21 − ( r + 7) = 14 − r. (4)Let ∆ = { p , · · · , p δ } ∈ S ym δ ( P )be a general set of δ points in P . We note that δ = 14 − r ≤ min { e ( e + 3) / , ( e − e − / } and ( e, δ ) = (6 , . (5)By a result due to Arbarello and Cornalba [1, Theorem 3.2] regarding theconfiguration of nodes on plane nodal curves in Σ e,g , there exists a curve E ∈ Σ e,g having nodes at ∆ and no further singularities. With the samechoice of a general ∆, there also exists a curve E ′ ∈ Σ e − ,h of genus h anddegree e − h = ( e − e − − δ = r + 1. The author apologizes for the complicated notation involved. For later use we make thefollowing definition. Let H Σ e,g be the irreducible family in the Hilbert scheme of smoothcurves H L g + r − ,g,r consisting of curves such that the residual series of the very ample andcomplete hyperplane series g rg + r − is a net cut out by lines on the plane curve correspond-ing to a general element of the Severi varitey Σ e,g , where e = g − r + 1. xistence & reducibility of the Hilbert scheme of smooth curves 15Let S π → P be the blowing up of P at ∆. Let { E i } δi =1 be the exceptionaldivisors with linear equivalence classes { e i } δi =1 . Let l ∈ Pic( S ) be the class of π ∗ ( L ), where L ⊂ P is a line. We consider the linear system | H | = | π ∗ (( e − L ) − δ X i =1 E i | on S . Recall that by a result of d’Almeida and Hirschowitz [11, Theorem 0],the linear system | π ∗ ( tL ) − P δi =1 E i | on S is very ample if δ ≤ ( t + 3) t − . (6)Taking t = ( e − r ≥
5, whence | H | is very ample. The linear system | C | = | π ∗ ( eL ) − δ X i =1 E i | contains the non-singular model C of the nodal plane curve E and H · C = (( e − l − δ X i =1 e i ) · ( el − δ X i =1 e i ) = e ( e − − δ = g + r − . Recall that ∆ = { p , · · · , p δ } ∈ S ym δ ( P ) is general and there is a nodalcurve E ′ ∈ Σ e − ,h with nodes at ∆. Therefore we may assume further that ∆imposes independent conditions on the linear system of curves of any degree m ≥ e −
4; cf. [3, Exercise 11, p.54] or [15, (1.51, p.31)].Hence by the projection formula, we obtain h ( S, O S ( H )) = h ( P , π ∗ ( O S ( H ))) = h ( P , π ∗ ( π ∗ (( e − L ) − δ X i =1 E i ))= h ( P , O P (( e − L − ∆)) = 1 + 12 ( e − e − − δ = r + 1 . In all, we deduce that the non-singular model C ⊂ S of the nodal plane curve E of degree e = g − r + 1 = 8 in the linear system | C | = | π ∗ ( eL ) − P δi =1 E i | is embedded into P r as a curve of degree d = g + r − | H | = | π ∗ (( e − L ) − P δi =1 E i ) | . Furthermore, by our choice of general δ points ∆ in P , which imposes in-dependent conditions on curves of any degree m ≥ e −
4, the linear system g e = | π ∗ ( L ) | C | cut out on C by lines in P is complete [3, Exercise 24, p.57].6 Changho KeemHence it follows that the residual series of | π ∗ ( L ) | C | which is | ( K S + C − π ∗ ( L )) | C | = | ( − π ∗ (3 L ) + X E i + π ∗ ( eL ) − X E i − π ∗ ( L )) | C | = | ( π ∗ (( e − L ) − X E i ) | C | = | H | C | is a complete very ample g rg + r − on C . Therefore we may deduce that thefamily H Σ ,g contains a linearly normal smooth curve of degree d = g + r − g = r + 7 in P r .At this point it is not clear if the irreducible family H Σ ,g constitutes acomponent of H L g + r − ,g,r , even though we now know H Σ ,g = ∅ . In whatfollows we argue that the family H Σ ,g corresponding to the Severi varietyΣ ,g is indeed a component of H L g + r − ,g,r different from H , or H (2 , , whichwe located in the part (i) of our proof.Recall that by a result of Coppens [10, Theorem], for a general nodal planecurve Γ of degree e and genus g (i.e. a general element of the Severi varietyΣ e,g ), the normalization C of Γ is ( e − ρ ( e − , g,
1) = 2 e − g − < . He also obtained that C has only finitelymany linear series g e − (step 4 in loc. cit.). In our current case g = r + 7 and e = 8, the condition ρ ( e − , g,
1) = 2 e − g − < e = 8has only finitely may pencils of degree 6, i.e.dim W ( C ) = 0 . Since the family H Σ ,g has minimal possible dimension X ( d, g, r ), H Σ ,g isnot in the boundary of either H , or H (2 , by noting that dim W ( C ) = 2for a general C ∈ H , and dim W ( C ) = 1 for a general C ∈ H (2 , , bysemi-continuity.By a simple dimension count, the locus H , over the family consisting ofthe linear series which is residual to 2 g (and dominating M g, ) remains acomponent for every r ≥ λ ( d, g, r ) = 4 g − r − r + 22 ≤ dim M g, = 2 g + 3 = 2 r + 17 . We further note that the family of curves H (2 , sitting over a component of X , has dimensiondim X , + dim P GL( r + 1) (cid:8) X ( d, g, r )for 5 ≤ r ≤
11 sincedim X , = 2 g − r + 10 (cid:8) r + 22 = λ ( g + r − , g, r ) , and therefore the family H (2 , is in the boundary of the component H Σ ,g .Hence, for 5 ≤ r ≤
11, we have only two components H , , H Σ ,g ⊂ H L g + r − ,g,r .For 12 ≤ r ≤ H (2 , becomes a separate component different from H , xistence & reducibility of the Hilbert scheme of smooth curves 17or H Σ ,g and there are at least three distinct components H , , H (2 , and H Σ ,g . The component H Σ ,g corresponding to the Severi variety (which ex-ists only for r ≤
14) has the expected number of moduli since Σ ,g has theexpected number of moduli by [27, Theorem 4.2]. (cid:3) Remark 3.5.
As was mentioned in Remark 3.1 (i) and (ii), the irreducibilityof H L g,g, = H g,g, holds for every g ≥
8, in particular for g = r + 7 = 10. For r = 4 and g = r + 7 = 11, H L g +1 ,g, = H g +1 ,g, is also irreducible. In fact, onechecks easily that H , , = H Σ . and H , , = H Σ . . The argument which we used in the proof of Theorem 3.4 (ii) also works for r = 3 ,
4. Hence both H g,g, and H g +1 ,g, have the expected number of modulifor g = r + 7 and r = 3 ,
4. We summarize our discussion together with theresults we obtained in Theorem 3.4 in the following table.
Table 1.
Components of H L g + r − ,g,r for g = r + 7 Dimension P r compts of expected with expected (cid:8) expected (cid:9) expecteddimension r = 3 1 1 1 0 0 r = 4 1 1 1 0 0 r = 5 2 2 2 0 06 ≤ r ≤
11 2 1 1 0 1 r = 12 ≥ ≥ ≥ ≤ r ≤ ≥ ≥ r ≥ ≥ ≥ We are now ready to prove the existence theorem which asserts that thereexists a smooth, irreducible and linearly normal curve of genus g and degree d = g + r − P r . Since the existence of H L g + r − ,g,r is completely knownfor r = 3 , r ≥
5. The followingexistence theorem is valid for all g ≥ r + 5 and every r ≥ g = r + 6; recall that there is no smooth curve with the prescribed genus g = r + 6 and degree d = g + r − P r for any r ≥
10 as we proved inProposition 3.3 (i). On the other hand, we will also see that our existencetheorem would imply that H L g + r − ,g,r is reducible for every r + 7 ≤ g ≤ r + 2with any r ≥
5. This shows that the bound g ≥ r + 3 for the irreducibility of H L g + r − ,g,r in [6] is sharp for every r ≥
5. The proof of our existence theoremis based on the following lemma concerning linear series on general k -gonalcurves.8 Changho Keem Lemma 3.6. [12, Proposition 1.1]
Assume k − g − < . Let C be a general k -gonal curve of genus g , k ≥ , ≤ m , n ∈ Z such that g ≥ m + n ( k −
1) (7) and let D ∈ C m . Assume that there is no E ∈ g k with E ≤ D. Then dim | ng k + D | = n . Theorem 3.7.
Given r ≥ , for any g ≥ r + 5 there exists a smooth, non-degenerate and linearly normal curve of genus g and degree d = g + r − in P r unless g = r + 6 and r ≥ .Proof. Since the existence (non-existence, respectively) was shown for g = r + 6 and 5 ≤ r ≤ r ≥
10, respectively) we may assume g = r + 6. Weonly need to exhibit a very ample and complete linear series D = g rg + r − ona certain smooth curve of genus g . Since g = r + 6, we have either(1) g − r + 1 = 2 k for some k ≥ g − r + 1 = 2 k + 1 for some k ≥ g − r + 1 = 2 k for some k ≥
3: Note that ρ ( k, g,
1) = 2 k − g − − r − (cid:8) k -gonal curve C . Taking n = 2 , m = 2 inLemma 3.6, the numerical condition (7) is satisfied. Furthermore, thereis no E ∈ g k with E ≤ D since k ≥
3. Hence we have dim | g k + D | = 2for any D ∈ C by Lemma 3.6 and therefore | K C − g k | = g rg + r − is very ample.(2) Suppose g − r + 1 = 2 k + 1 for some k ≥
4: Since ρ ( k, g,
1) = 2 k − g − − r − (cid:8) , we again consider a general k -gonal curve C . For a general q ∈ C andfor any r + s ∈ C , we take D = q + r + s , m = 3 and n = 2 in Lemma3.6. Again the numerical condition (7) holds and there is no E ∈ g k with E ≤ D just because k ≥
4. Hence dim | g k + D | = 2 which impliesthat | K C − g k − q | = g r g − − k − = g rg + r − is very ample. (cid:3) Remark 3.8. (i) If k = 3 and g − r + 1 = 2 k in the proof (1) of Theorem3.7, i.e. g = r + 5, we remark that curves detected by our proof aretrigonal (and extremal) curves lying on a rational normal scroll S ⊂ P r .Such trigonal curves are in the linear system | H − ( r − L | where H is a hyperplane section and L is one of the rulings of S whereas thetrigonal pencil is cut out by the rulings of the scroll S . We also notethat this family of trigonal curves is indeed dense in the correspondingcomponent of the Hilbert scheme; cf. [8, Theorem 1,4].xistence & reducibility of the Hilbert scheme of smooth curves 19(ii) For g ≤ r + 2 and g = r + 6, the family of very ample linear series F ⊂ e G L ⊂ G rg + r − over M g,k which was constructed in Theorem 3.7has dimension equal todim M g,k = 2 g + 2 k − g − − r if g − r + 1 = 2 k ordim M g,k + 1 = 2 g + 2 k − g − − r if g − r + 1 = 2 k + 1 . In both cases, we have λ ( g + r − , g, r ) = 4 g − r − ≤ dim F = 3 g − − r if g ≤ r + 2.(iii) We also remark thatdim F = dim F ∨ = 3 g − − r is same as the maximal possible dimension of an irreducible componentof W g − r +1 whose general element is compounded; cf. Proposition 1.3(c).(iv) Hence this family F is not contained in any other irreducible family F ′ ⊂ e G ⊂ G rg + r − of strictly bigger dimension such that the residualseries of a general element of F ′ is compounded. We will see in the nexttheorem that the family F as well as the others of similar kind give riseto several components of H L g + r − ,g,r .Motivated by Theorem 3.7, we show the reducibility of H L g + r − ,g,r for everygenus g in the range r +7 ≤ g ≤ r +2. This generalizes the reducible examplein [6, Example 2.7] and shows that the bound g ≥ r + 3 for the irreducibilityof H L g + r − ,g,r is sharp for every r ≥
5; cf. Theorem 1.1 (4).
Theorem 3.9.
Given an integer r ≥ and for every g in the range r + 7 ≤ g ≤ r + 2 , H L g + r − ,g,r is reducible having component(s) dominating the loci M g,k for every ≤ k ≤ [ g − r +12 ] .Proof. We retain all the notations used in the proof of Theorem 2.1. Since g ≥ r + 7, we have e := g − r + 1 ≥
8. We first fix an integer k ≥ e = g − r + 1 ≥ k and set δ := g − r + 1 − k ≥
0. On a general k -gonalcurve C with a unique g k , | K C − g k | = g g − k +12 g − − k is very ample by [18, Corollary 3.3]. Furthermore, we claim that for a generalchoice of ∆ ∈ C δ = C g − r +1 − k , the complete linear series | K C − g k − ∆ | = g g − k +1 − δ g − − k − δ = g rg + r − is very ample as long as k ≥
4. We take∆ = r + · · · + r δ ∈ C δ such that no r i is a ramification point and no pair { r i , r j } lies on the samefiber of the k -sheeted covering over P . Upon making a choice of such ∆, foran arbitrary choice s + t ∈ C , we take D = ∆ + s + t , n = 2 and m = δ + 20 Changho Keemin Lemma 3.6. By the assumption g ≤ r + 2, we see that the numericalassumption (7) in Lemma 3.6 ( g ≥ m + n ( k − D, E ) ≤ E ∈ g k and hence there is no E ∈ g k with E ≤ D as long as k ≥
4. Therefore it follows thatdim | g k + ∆ + s + t | = 2 , for any s + t ∈ C implying | K C − g k − ∆ | is indeed very ample. Let F k,δ := (2 G k + W δ ) ∨ ⊂ G rg + r − be the locus consisting of residual series of nets E ∈ W g − r +1 of the form E = | g k + ∆ | , ∆ ∈ C δ on a general k -gonal curve C ∈ M g,k . The following inequality holds by theassumption g ≤ r + 2; λ ( g + r − , g, r ) = 4 g − r − ≤ dim F k,δ = dim(2 G k + W δ )= dim G k × M g,k W δ = dim G k + δ = 3 g − ρ ( k, g,
1) + δ = 3 g − k − g −
2) + ( g − r + 1 − k )= 3 g − − r. (8)It is a priori possible that these loci F k,δ (corresponding to each k ≥ δ such that 2 k + δ = g − r + 1) might be in the boundary of some othercomponent of bigger dimension. However this would be impossible for whichwe argue as follows. Let G ⊂ G rg + r − be a component properly containingthe locus F k,δ . If the base-point-free part of the residual series E of a general D ∈ G is birationally very ample, by an elementary dimension count usingProposition 1.3 (b) λ ( g + r − , g, r ) ≤ dim W ∨ = dim G ≤ g − r + 1) + g − − b implying b = 0 and dim G = λ ( g + r − , g, r ) where b is the degree of thebase locus of E . Hence we havedim F k,δ = dim(2 G k + W δ ) ∨ = 3 g − − r (cid:8) dim G = 4 g − r − , which is not compatible with above inequality (8) or the genus assumption g ≤ r + 2.We get the same numerical contradiction if the base-point-free part of theresidual series E is very ample by using Proposition 1.3 (a). The only remain-ing possibility is that the residual series E of a general D ∈ G is compoundedand induces an n -sheeted covering C π → E onto an irrational curve E ofgenus h ≥
1. However this is not possible by Remark 3.8 (iv). Therefore wemay deduce that the irreducible locus F k,δ = (2 G k + W δ ) ∨ is indeed densexistence & reducibility of the Hilbert scheme of smooth curves 21in a component a component G ⊂ e G L , which gives rise to a component of H L g + r − ,g,r of dimension,dim M g,k + δ + dim P GL( r + 1) = 3 g − − r + dim P GL( r + 1)as a P GL( r + 1)-bundle over F k,δ .In all, we deduce that there is a component H k,δ ⊂ H L g + r − sitting over thecomponent F k,δ ⊂ e G L dominating M g,k of equal dimension 3 g − − r foreach k ’s in the range 4 ≤ k ≤ [ g − r +12 ]; H k,δ F k,δ M g,k ⊂ M g . (i) For g = r + 7, H L g + r − ,g,r is reducible by Theorem 3.4.(ii) For r + 9 ≤ g ≤ r + 2, there are at least l ( r, g ) := [ g − r +12 ] − ≥ H k,δ ’s for each k in the range 8 ≤ k ≤ g − r + 1.(iii) We assume g = r + 8 and r ≥
6. Since we are working in the range r + 5 ≤ g ≤ r + 2, we have r ≥ g = r + 8 ≤ r + 2.(a) If the residual series E = g g − r +1 = g ∈ W ∨ of a general element D ∈ G ⊂ e G L is compounded and induces a 4-sheeted covering onto P , we have already seen that there is a component H , ⊂ H L g + r − ,g,r sitting over the irreducible family F , = (2 G + W ) ∨ dominating M g, .(b) We add another specific irreducible family F (2 , ⊂ G ⊂ e G L by claimingthat for a double cover C π → E onto a non-hyperelliptic curve E genus h = 3, the linear series | K C − π ∗ ( K E ) − q | is very ample for a general q ∈ C if g = r + 8 ≥
21. We let E = | π ∗ ( K E ) + q | = g and assume the existence of D ∈ C such thatdim |E + D | = dim | π ∗ ( K E ) + q + D | ≥ dim E + 1 = 3 . Recall that | K C − π ∗ ( K E ) | is very ample as we have seen in the courseof the proof of Theorem 3.4 and hence | π ∗ ( K E ) + D | = g . Therefore | π ∗ ( K E ) + q + D | = g is base-point-free and birationally very ample (but not very ample) in-ducing a birational morphism onto a space curve ˜ C ⊂ P . Note that g = r + 8 ≤ p a ( ˜ C ) ≤ π (11 ,
3) = 20contrary to the assumption g = r + 8 ≥
21. Therefore it follows that | K C − π ∗ ( K E ) − q | is very ample finishing the proof of the claim. By theusual dimension count, the family F (2 , ⊂ G ⊂ e G L consisting of veryample linear series of the form | K C − π ∗ ( K E ) − q | over a component of X , ⊂ M g has dimensiondim F (2 , = dim X , + 1 = 2 g − ≥ g − r − λ ( d, g, r );2 Changho Keemwhereas the inequality holds by the assumption g = r + 8 ≥
21. Sincedim F , = 3 g − − r > dim F (2 , = 2 g − F (2 , is not in the boundary of F , by (lower) semi-continuity of gonality. Hence for g = r + 8 ≥
21 there are at leasttwo components of e G L ⊂ G rg + r − ; F , dominating M g, , another one F (2 , dominating a component of X , ⊂ M g .(c) For 6 ≤ r ≤
12 (in fact for 6 ≤ r ≤
20) one may come up witha component H Σ ,g ⊂ H L g + r − ,g,r corresponding to the Severi varietyΣ ,g . Indeed we may copy the proof Theorem 3.4 (ii) with only trivialmodifications as follows. In the current situation, we have e = deg E = deg | K C − D| = 9hence the equality (4) in the proof (ii) of Theorem 3.4 becomes δ = ( e − e − − g = 28 − ( r + 8) = 20 − r (9)Then for a general ∆ = { p , · · · , p δ } ∈ S ym δ ( P ) , the inequality δ = 20 − r ≤ min { e ( e + 3) / , ( e − e − / } and ( e, δ ) = (6 , e = 9 and6 ≤ r ≤
20. Then one invokes a result by Arbarello and Cornalba [1,Theorem 3.2] to see eventually that the non-singular model C ⊂ S –where S is the blowing up of P at ∆ – of the nodal plane curve E ofdegree e = g − r + 1 = 9 in the linear system | C | = | π ∗ ( eL ) − δ X i =1 E i | is embedded into P r as a linearly normal curve of degree d = g + r − | H | = | π ∗ (( e − L ) − δ X i =1 E i ) | . To sum up, for g = r +8 and r ≥ H L g + r − ,g,r : cf. Remark 3.10. H , and H Σ ,r +8 if 6 ≤ r ≤ H , , H (2 , and H Σ ,r +8 if 13 ≤ r ≤ H , and H (2 , if r ≥ (cid:3) xistence & reducibility of the Hilbert scheme of smooth curves 23 Table 2.
Components of H L g + r − ,g,r for g = r + 8 Dimension P r compts of expected with expected (cid:8) expected (cid:9) expecteddimension r = 6 2 2 1 1 0 r = 7 2 1 2 0 08 ≤ r ≤
12 2 1 1 0 1 r = 13 ≥ ≥ ≥ r = 14 ≥ ≥ ≤ r ≤ ≥ ≥ r ≥ ≥ ≥ Remark 3.10. (i) When we were dealing with the case g = r + 8 in theproof (iii) of Theorem 3.9, we ignored the possibility that E = g mayinduce a triple covering C π → E onto an elliptic curve E and E = π ∗ ( g ). Indeed, at least for g = r + 8 ≥
16, one verifies easily that theresidual series of E = π ∗ ( g ) is very ample by using Castelnuovo-Severiinequality, etc. However for 6 ≤ r ≤
10, the family of linear series of theform {E = π ∗ ( g ) | C π → E, C ∈ X , ) } ⊂ W has dimensiondim X , + dim W ( E ) = dim X , + dim J ( E ) = dim X , + 1– where E is an elliptic curve – which is strictly less than λ ( g + r − , g, r ) = 4 g − r −
6. We also remark that at least one extra componentarising from triple coverings of elliptic curves when r ≥
11 has notbeen properly reflected in the above table, which is rather based on theargument presented in the proof for showing the reducibility.(ii) If g = 2 r + 2, r is odd and r ≥ k = r +32 , δ = g − r + 1 − k = 0 we havedim π ( H k, ) = dim M g,k = 2 g + 2 k − λ ( g + r − , g, r ) , where H k, π M g is the functorial map. For every other integer k with4 ≤ k (cid:8) r +32 , the component H k,δ has less than the expected number ofmoduli even though all the components H k,δ ’s have the same expecteddimension.(iii) If g = 2 r + 2, r is even and r ≥
8, every component H k,δ has less thanthe expected number of moduli.
4. Digressions and Miscellanies
Having sorted out some basic properties and the existence of the Hilbertscheme of linearly normal curves of degree d ≥ g + r −
3, it would be natural4 Changho Keemto move on to the next case d = g + r − d ≥ g + r −
3, theproof of the irreducibility of H L d,g,r in certain ranges relied on the followings: • The irreducibility of the family G g − r (the irreducibility of the Severivariety Σ g − r +1 ,g , respectively) in the case d = g + r − d = g + r −
3, respectively). • Dimension counts such as - in the case d = g + r − G g − r = dim G = dim W ∨ = λ ( d, g, r )for every G ⊂ e G L ⊂ G rd . • Similarly - in the case d = g + r − G ⊂ e G L ⊂ G rd such thatthe family of residual series W ∨ whose general elements are birationallyvery ample, we also has the right dimension countdim Σ d,g − P GL(3) = 3 d + g − G = dim W ∨ = λ ( d, g, r ) . However for the case d = g + r −
4, the family of the residual series of thosecomplete very ample linear series under our consideration is the locus W ∨ ⊂ W g − r +2 consisting of webs on moving curves and our overall knowledge on these lociis relatively poor compared with W g − r +1 or W g − r . We take a look at thecase r = 3 and make a note of some scattered results and relevant questionsregarding the Hilbert scheme H L g − ,g, . Remark 4.1. (i) H L g − ,g, = ∅ for g ≤ H L g − ,g, = ∅ for g ≥
9. For g = 9, the (extremal) curves on quadricsof type (4 ,
4) forms the only irreducible component of degree d = 8and genus g = 9 of the expected dimension 4 · g = 10 thereexist curves of type (3 ,
6) on smooth quadrics which forms an irreduciblefamily of dimensiondim P H ( P × P , O (3 , P H ( P , O (2)) = 36 . Moreover, complete intersections of two irreducible cubics form anotherfamily of the same expected dimensiondim G (1 , P ( H ( P , O (3))) = dim G (1 ,
19) = 36 . Hence H , , = H L , , is reducible. This example is known as the firstexample such that H d,g, is reducible which goes back to Weyr andHalphen; cf. [28, p.401], Note that the inequality g ≤ ( d − d − g − g − d, g ) = ( g − , g ) with g ≥
11, which guaranteesthe existence of a smooth curve of degree g − g in P ; cf.[9, p.489 (2), (3)].xistence & reducibility of the Hilbert scheme of smooth curves 25(ii) Using elementary linkage theory one may show the irreducibility of H g − ,g, = H L g − ,g, for low g ’s outside the Brill-Noether range g ≤ g = 11 and give an outline a proofof the irreducibility as follows. First one shows that a smooth curve ofdegree d = 10 and genus g = 11 corresponding to a general element ina component of H , , lies on quartics, which is directly linked to asmooth curve of degree e = 6 and genus h = 3 via complete intersectionsof two quartics. One then considers the locusΣ ⊂ G (1 , P ( H ( P , O (4)))) = G (1 , C ofdegree d = 10 and genus g = 11 and a sextic D of genus h = 3 whichare directly linked, together with the two obvious maps G (1 , ⊃ Σ π C I ⊂ H , , π D H , , where I is the image of Σ under π C .On the other hand, since D ∈ H , , is directly linked to a twisted cubicvia complete intersection of two cubics, one deduces that h ( P , I C (4)) = 0 , h ( P , I D (4)) = 0and hence π D is generically surjective with fibers open subsets of G (1 , D ∈ H , , we have dim P ( H ( P , I D (4))) = 12.Then by the irreducibility the Hilbert scheme H , , - which is ratherwell-known - one may see that Σ is irreducible anddim Σ = dim G (1 ,
12) + dim H . . = 22 + 4 · . On the other hand, since every smooth curve C ∈ I lies on exactly4 independent quartics by h ( P , I C (4)) = 0, π C is generically surjec-tive with fibers open in G (1 , I = H . . isirreducible of dimensiondim Σ − dim G (1 ,
4) = 46 − · . (iii) With a little extra work, together with the method similar to the aboveone may show the irreducibility of H g − ,g, for g = 12 and g = 13.However as g gets larger, the curves in question often needs to lie onsurfaces of fairly high degrees therefore are not necessarily linked tocurves which are easier to describe or we know much of.(iv) There are some reducible Hilbert schemes H L d,g, with d ≤ g −
2, e.g. H L g − ,g, with g = 12 or H L g − ,g, with g = 14. However for the case d = g −
1, it is not so clear if H L g − ,g, is irreducible for every g = 10.6 Changho KeemFor r = 4, especially outside the Brill-Noether range ρ ( g, g,
4) = g − < H L g,g, is relatively easy to determine as follows. Remark 4.2. (i) Note that H L g,g, = ∅ for g ≤
10 by Castelnuovo genusbound.(ii) Outside the Brill-Noether range 11 ≤ g ≤
20, we list up one of thepossible candidates of an (smooth) element in H L g,g, in the followingtable. We stress that the irreducibility of H L g,g, is not known yet inmany of the cases. Furthermore, the description of curves listed belowdoes not necessarily represent the property of a general element in acomponent of H L g,g, .(iii) We use the following standard convention. S n π → P is the rational sur-face P blown up at n -points in general position. ( a ; b , · · · , b n ) denotesthe linear system | π ∗ ( aL ) − P ni =1 b i E i | on S n – where E i ’s are excep-tional divisors and L is a line in P . Q is the smooth quadric surfacein P and | ( a, b ) | denotes the linear system on Q of type ( a, b ). A DelPezzo surface in P r (3 ≤ r ≤
8) is the surface S − r embedded into P r by the linear system (3; 1 − r ) := | π ∗ (3 L ) − P − ri =1 E i | . A Bordiga sur-face in P is the surface S embedded into P by the linear system(4; 1 ) := | π ∗ (4 L ) − P i =1 E i | ; cf. [25, Theorem 1.0.2] especially for( d, g ) = (16 , , (19 ,
19) in the list below.
Table 3.
Smooth curves in H L g,g, for 11 ≤ g ≤ ( d, g ) Description Remark(11 ,
11) A curve in | H + 2 L | on a rational normal scroll in P Trigonal(12 ,
12) A smooth member in (8; 3 , ) := (8; 3 , , , ,
2) Pentagonalon a Del Pezzo, and is birational to a curve in | (5 , | on Q ⊂ P with 4 nodes(13 ,
13) A smooth member in (8; 3 , , , ,
1) on a Del Pezzo, Pentagonalbirational to a curve in | (5 , | on Q ⊂ P with 3 nodes(14 ,
14) A curve with a very ample | K C − g | Tetragonal(15 ,
15) A curve with a very ample | K C − g − q | Tetragonal(16 ,
16) On a Bordiga surface [25, Theorem 1.0.2] (17 ,
17) A curve with a very ample | K − g | = g Pentagonal(18 ,
18) A curve with a very ample | K − g − q | = g Pentagonal(19 ,
19) On a Bordiga surface [25, Theorem 1.0.2]
For r ≥
5, we have a similar picture in general as follows.
Remark 4.3. (i) For r ≥ H L g + r − ,g,r = ∅ for g ≤ r +6 by the Caselnuovogenus bound. to the author xistence & reducibility of the Hilbert scheme of smooth curves 27(ii) For g = r + 7, H L g + r − ,g,r = ∅ for every r ≥ H L g + r − ,g,r = ∅ is irreducible except for the case r = 5; cf. [8].(iii) However if g = r + 8 or g = r + 9, we have H g + r − ,g,r = H L g + r − ,g,r = ∅ for sufficiently large r , which is similar to what we have seen in Theorem3.3.(iv) The non-existence of H L g + r − ,g,r in case g = r + 8 and g = r + 9 with r >> E = g g − r +2 = g e of D ∈ G ⊂ G rg + r − has degree e = 10( e = 11, respectively) for g = r + 8 (for g = r + 9, respectively). Hence if g = r + 8 (cid:9) π (10 ,
3) = 16 (if g = r + 9 (cid:9) π (11 ,
3) = 20, respectively), E is compounded inducing a double covering onto a curve of small genus(or trigonal with non-empty base locus). However the residual series D of E is not very ample in these cases, leading to a contradiction.(v) The existence of H L g + r − ,g,r = H L r +4 ,r +8 ,r with 5 ≤ r ≤ g = r + 8can be seen by analyzing curves birationally embedded into P by E = g , which happen to lie on a quadric with some nodal singularities.Smooth curves in the very ample linear systems( a ) (8; 3 , , ,
2) ( b ) (8; 3 , ,
2) o r ( c ) (8; 3 , P r (5 ≤ r ≤
7) are curves of degree d = 2 r + 4and genus g = r + 8 whose singular models induced by E sit in P ascurves of type (5 ,
5) on a quadric with δ = 1 , , r = 8, acurve of degree e = 10 and genus g = 16 in P is the (smooth) extremalcurve of type (5 ,
5) on a quadric which is embedded into P by the 2-tuple embedding as a curve of degree d = 2 r + 4 = 20. In this case H L r +4 ,r +8 ,r = H L , , is irreducible of dimension more than expectedsincedim G = dim W ∨ = dim P H ( P × P , O (5 , P H ( P , O (2)) − dim P GL (4) > λ ( d, g, r ) . It is worthwhile to note that for ( d, g, r ) = (18 , , , P . The hyperplane section of a smooth curve of type(4 ,
6) on a smooth quadric in P has a very ample residual series whichgives rise to an additional component, which has the same dimension asthe former. We denote by Σ | ( a,b ) | ,δ the Severi variety of curves on P × P whose general element corresponds irreducible curves in the linear sys-tem | ( a, b ) | with δ nodes and no other singularities. Incidentally the twodistinct irreducible Severi varieties Σ | (5 , | , and Σ | (4 , | , correspond tothe two distinct irreducible components of H L d,g,r = H L , , .(v) Even though the irreducibility needs to be worked out further in manyof the cases, the existence of H L g + r − ,g,r = H L r +5 ,r +9 ,r for 5 ≤ r ≤
118 Changho Keemcan be seen in a similar way. Needless to say, family of curves listed inthe following table may not form a dense open set in a component.
Table 4. H L g + r − ,g,r = H L r +5 ,r +9 ,r = ∅ for 5 ≤ r ≤ ( d, g, r ) Description Remark(15 , ,
5) A member in (8; 3 , , ,
1) on a Del Pezzo in P corresponds to Σ | (5 , | , (17 , ,
6) A member in (8; 3 , ,
1) on a Del Pezzo in P corresponds to Σ | (5 , | , (19 , ,
7) A member in (9; 4 ,
4) on a Del Pezzo in P Pentagonal(21 , ,
8) A member in (9; 4 , , ,
2) on S → P corresponds to Σ | (5 , | , embedded into P by very ample | (4; 2 , , , | (23 , ,
9) A curve in | (4 , | on a smooth quadric Q ⊂ P corresponds to Σ | (4 , | , embedded into P by series | (1 , | (25 , ,
10) A member in (9; 4 ,
3) on S → P corresponds to Σ | (5 , | , embedded into P by very ample | (4; 2 , | (27 , ,
11) A member in | (5 , | on a smooth quadric Irreduciblein P embedded into P by the series | (2 , | corresponds to Σ | (5 , | , We close up with the following statement which is a summary of our discus-sion regarding the existence of the Hilbert scheme of linearly normal curves H L g + r − ,g,r in general. The proof of the existence part for g ≥ r + 10 is almostparallel with the proof of Theorem 3.7. Proposition 4.4.
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