On k-gonal loci in Severi varieties on general K 3 surfaces and rational curves on hyperkähler manifolds

J. Math. Pures Appl. 101 (2014) 473–494

www.elsevier.com/locate/matpur

On k-gonal loci in Severi varieties on general K 3 surfaces and rational curves on hyperkähler manifolds

Ciro Ciliberto

, Andreas Leopold Knutsen

aDipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00173 Roma, Italy bDepartment of Mathematics, University of Bergen, Postboks 7800, N-5020 Bergen, Norway

Received 8 September 2012 Available online 4 July 2013

Abstract

In this paper we study the gonality of the normalizations of curves in the linear system|H|of a general primitively polarized complexK3 surface (S, H )of genus p. We prove two main results. First we give a necessary condition onp, g, r, d for the existence of a curve in|H|with geometric genusgwhose normalization has ag^r_d. Secondly we prove that for all numerical cases compatible with the above necessary condition, there is a family ofnodalcurves in|H|of genusgcarrying ag_k¹and of dimension equal to theexpected dimensionmin{2(k−1), g}. Relations with the Mori cone of the hyperkähler manifold Hilb^k(S)are discussed.

2013 Elsevier Masson SAS. All rights reserved.

Résumé

Dans cet article on étudié la gonalité des normalisations des courbes dans le système linéaire|H|d’une surface généraleK3 complexe principalement polarisée (S, H ) de genrep. On démontre deux resultats principaux. Premièrement on donne une condition nécessaire surp, g, r, d à l’existence d’une courbe dans|H|de genre géometriqueg dont la normalisation a ung^r_d. Deuxièmement on démontre que pour tous les cas numérique compatible avec la condition nécessaire ci-dessus, il existe une famille de courbesnodalesdans |H| de genreg qui possèdent ung_k¹ et dont la dimension est égale à ladimension attendue min{2(k−1), g}. On discute aussi des relations avec le cône de Mori de la variété hyperkählerienne Hilb^k(S).

2013 Elsevier Masson SAS. All rights reserved.

Keywords:K3 surfaces; Severi varieties; Deformation theory; Punctual Hilbert schemes

Introduction

Let(S, H )be a primitively polarized complexK3 surface of genusp�2, i.e.Ω_S²�O^S,h¹(O^S)=0 andH is a globally generated, indivisible, divisor (or line bundle) withH²=2p−2. The main objective of this paper is to study the gonality of the normalization of curves, specifically ofnodalcurves, in the linear system|H|, when(S, H ) is general in its moduli space, that is,(S, H )belongs to a Zariski open dense subset.

* Corresponding author.

E-mail addresses:[email protected](C. Ciliberto),[email protected](A.L. Knutsen).

0021-7824/$ – see front matter 2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.matpur.2013.06.010

LetV_|H|,δ(S)⊆ |H| be theSeveri varietyof curves with δ�p nodes. It is a classical result thatV_|H|,δ(S)is a nonempty, locally closed, smooth variety of dimension g=p−δ, which is the geometric genus of the curves in V|H|,δ(S). The moduli morphismV|H|,δ(S)→M^gis finite to its image (seeProposition 1.2below).

We consider V_|H^k _|,δ(S)⊆V_|H|,δ(S)the subvariety of curves whose normalizations carry ag_k¹. By Brill–Noether theory, if g�2(k−1)thenV_|H|,δ^k (S)=V|H|,δ(S)so the interesting range is g >2(k−1). A count of parameters, carried out in Section1.4, suggests that theexpected dimensionofV_|H|,δ^k (S)is 2(k−1). In any event, if nonempty, V_|H^k _|,δ(S)has dimension at least 2(k−1)(seeProposition 1.5).

Our first main result isTheorem 3.1, which yields a necessary condition for the normalization of a curveC∈ |H| of geometric genusg(with any type of singularities) on a primitively polarized K3 surface(S, H )of genusp with no reducible curves in|C|(e.g., with Pic(S)�Z[H]) to possess ag_d^r, namely that

ρ(p, αr, αd+δ)�0, whereα:=

�gr+(d−r)(r−1) 2r(d−r)

� ,

andρis the usualBrill–Noether number. In the case ofg_k¹’s, settingr=1,d=kandδ:=p−g, the above necessary condition reads

δ�α�

p−δ−(k−1)(α+1)�

, whereα:=

� p−δ 2(k−1)

�

. (1)

Theorem 3.1 is a strong improvement of[13, Thm. 1.4]and its proof is based on the vector bundle approach à la Lazarsfeld[28].

Our second main result deals with nonemptiness and dimension ofV_|H^k _|,δ(S), and with properties of its general member, and proves that the bound(1)is optimal:

Theorem 0.1.Let(S, H )be a general primitively polarizedK3surface of genusp�3and letδandk be integers satisfying0�δ�pandk�2. Setg=p−δ. Then:

(i) V_|H^k _|,δ(S)�= ∅if and only if (1)holds;

(ii) when nonempty, V_|H^k _|,δ(S) has an irreducible component of the expected dimension min{2(k−1), g} whose general element is an irreducible curve C with normalization C˜ of genus g such that dim(W_k¹(C))˜ = max{0, ρ(g,1, k)=2(k−1)−g};

(iii) in addition, wheng�2(k−1)(resp.g <2(k−1)), any(resp. the general)g_k¹onC˜ has simple ramification and all nodes ofCare non-neutral with respect to it.

As for statement (i), we note that nonemptiness whenp is even,δ� ^p₄ and k� ^p₂ +1−δhas been proved by Voisin[35, pf. of Cor. 1, p. 366]by a totally different approach.

Theorem 0.1yields that, for fixedδ >0, thegeneralcurve in (some component of) the Severi varietyV_|H|,δ(S)has the gonality of a general genusgcurve, i.e.�(g+3)/2�, but, for allksatisfying(1), there are substrata of dimension 2(k−1)of curves of lower gonalityk. This is in contrast to the caseδ=0, where the gonality is constant and equal to �(p+3)/2�. Parts (ii) and (iii) ofTheorem 0.1 also yield thatC˜ has gonality min{k,�(g+3)/2�} and enjoys properties of a general curve of such a gonality.

The proof ofTheorem 0.1 relies on a rather delicate degeneration argument. Indeed, the general(S, H )can be specialized to the case whereS contains a smooth rational curveΓ of degreep and such thatS can be embedded into P^2p−1by the linear system|H+Γ|. This S can be in turn degenerated to a unionR of two smooth rational normal surfaces R1�P¹×P¹ and R2�F2 intersecting transversally along a smooth elliptic curve of degree 2p and such that the rational curveΓ specializes to the negative sections₂ ofR₂ andH specializes to the line bundle OR(1)⊗OR(−s2). This is proved in Section4.1. In Sections4.2and5we describe nodal curves on the limit surface Rthat fill up limit components ofV_|H|,δ(S). In Section6we describe possible limits ofV_|H|,δ^k (S); this latter analysis is one of the crucial points in the paper, and, to the best of our knowledge, is a nontrivial novelty. If nonempty, these limit varieties have the expected dimension, and this yields nonemptiness and expected dimension for the generalS containing a smooth rational curve of degreepas above, and therefore also for the generalSas in the statement of the

theorem (seeProposition 1.6). In Section7we show nonemptiness of the limit varieties and the required properties for theg_k¹in the range(1).

We note that our two-step degeneration seems to be new and has the property of independent interest that the stable model of the general hyperplane section of the limit surface is an irreducible rational nodal curve. We believe that this technique can be useful in other contexts. Specifically, forK3 surfaces this can be used to study Severi varieties of nodal curves, also in|nH|forn >1.

Besides its intrinsic interest for Brill–Noether theory and moduli problems, the subject of this paper is related to Mori theory and rational curves on the 2k-dimensional hyperkähler manifold Hilb^k(S)parametrizing 0-dimensional lengthk-subschemes of theK3 surfaceS. A curve onSwith ag_k¹on its normalization determines a rational curve on Hilb^k(S). For the importance of rational curves on hyperkähler manifolds see, e.g.,[25,26,6,22,21,20,37,36,38]and Section8. In particular, rational curves determine the nef and ample cones.

The curves onSinTheorem 0.1determine a family of rational curves in Hilb^k(S)of dimension 2(k−1), which is the expected dimension of any family of rational curves on a 2k-dimensional hyperkähler manifold. In Section2we determine their classes inN₁(Hilb^k(S))(seeLemma 2.1); in this computation the properties of theg_k¹stated in part (iii) ofTheorem 0.1play an essential role. The lowerδis, the closer the class is to the boundary of the Mori cone.

As a consequence, we obtain necessary conditions for a divisor in Hilb^k(S)to be nef or ample (seeProposition 8.3).

For infinitely manyp, k we prove that the classes of the rational curves in Hilb^k(S)we obtain from Theorem 0.1 withδminimal satisfying(1)(which we calloptimal classes) generate extremal rays of the Mori cone of Hilb^k(S) (seeCorollary 8.6and Proposition 8.9). After the appearance of the first version of this paper on the web, this has been verified also in the casesp�2(k−1), whereδ=0, in[4](seeProposition 8.7). To determine the Mori cone of Hilb^k(S)for allp, kone would have to extend our results to the nonprimitive cases|nH|,n >1. This is a difficult task, but should in principle be possible to treat with similar methods. We plan to do this in future research.

In Section8we also relate our work to some interesting conjectures of Hassett and Tschinkel on the Mori cone of Hilb^k(S)(see in particularRemark 8.10) and of Huybrechts and Sawon on Lagrangian fibrations (see in particular Corollary 8.13).

Throughout this paper we work overC. As usual, and as we did already in this Introduction, we may sometimes abuse notation and identify divisors with the corresponding line bundles, indifferently using the additive and the multiplicative notation.

1. Severi varieties,K3 surfaces andk-gonal loci

1.1. Severi varieties andk-gonal loci

Let S be a connected, projective surface with normal crossing singularities and let |H| be a base point free, complete linear system of Cartier divisors onSwhose general element is a connected curveH with at most nodes as singularities, located at the singular points ofS. We will setp=pa(H ).

For any integer 0�δ�p, we denote byV|H|,δ(S)the locally closed subscheme of|H|parametrizing the universal family of curvesC∈ |H| having only nodes as singularities, exactly δ of them (called themarked nodes) off the singular locus ofS, and such that the partial normalizationC˜ at theseδnodes is connected (i.e., the marked nodes arenot disconnecting nodes). We setg=p−δ=pa(C). If˜ Sis smooth theV_|H|,δ(S)’s are calledSeveri varietiesof δ-nodal curves in|H|onS. We use the same terminology in our more general setting.

Letg�3 be an integer. We denote byMgthemoduli space(or stack) of smooth curves of genusg, whose dimen- sion is 3g−3. We recall thatM^gis quasi-projective and admits a projective compactificationM^g, parametrizing all connected stable curves of arithmetic genusg.

One has themoduli morphism

ψS,H,δ:V_|H|,δ(S) M^g (2)

sendingC∈V|H|,δ(S)to the isomorphism class of the stable modelCof the partial normalizationC˜ ofC at theδ marked nodes. We writeψrather thanψS,H,δif no confusion arises. Ifψis generically finite to its image, we say that V_|H|,δ(S)hasmaximal number of modulig.

One can consider the stratification ofM^gin terms of gonality

M¹g,2⊂M¹g,3⊂ · · · ⊂M¹g,k⊂ · · · ⊂M^g, where

M¹g,k:=�

[Y] ∈M^g�

�Y possesses ag_k¹� ,

called thek-gonal locusinM^g, is irreducible, of dimension 2g+2k−5 wheng�2(k−1), whereasM¹g,k=M^g when g�2(k−1)(see e.g.[1]). Recall thatψ (C)∈M¹g,k if and only if the partial normalizationC˜ ofC at theδ marked nodes is stably equivalent to a curve that is the domain of an admissible cover of degreekto a stable pointed curve of genus 0 (see[18, Theorem (3.160)]). For any integerk�2, we define

V_|H^k _|,δ(S):=�

C∈V|H|,δ(S)�

�ψ (C)∈M¹g,k

�,

which has a natural scheme structure. This is called thek-gonal locusinsideV_|H|,δ(S).

1.2. K3surfaces

We will mainly consider the case in whichSis a smooth, projectiveK3 surface, endowed with a globally generated primitive, i.e. indivisible, divisorH withp=pa(H )�2. We call(S, H )aprimitive(or primitively polarized)K3 surface of genus p. We denote byK^p the moduli space(or stack) of primitiveK3 surfaces of genus p, which is smooth and irreducible of dimension 19, and the general element(S, H )is such thatH is very ample. Furthermore, Pic(S)is generated by the class ofH for all(S, H )outside a countable union of Zariski closed proper subsets (the Noether–Lefschetz divisors).

If V|H|,δ(S)�= ∅, then it is regular, i.e. it is smooth and of theexpected dimension g. Indeed, the marked, not disconnecting nodes of the curves inV|H|,δ(S)impose independent conditions to the linear system|H|(see e.g.[8]).

IfV_|H|,δ(S)�= ∅andδ^�< δ, thenV_|H|,δ(S)⊂V_|H|,δ^�.

Remark 1.1.The latter holds forV|H|,δ(S)also whenS is a connected surface with local normal crossing singular- ities, trivial dualizing bundle, h¹(S,O^S)=0 andH a globally generated, primitive divisor onS. Indeed, the usual arguments (like in[8]) apply with no change.

By a result of Mumford’s (cf. [30, Appendix]), for all δ�p the Severi varieties V_|H|,δ(S) are nonempty.

Chen extended this to Severi varietiesV_|mH|,δ(S)withm >1 (cf.[7]).

The following proposition is related to the rigidity results in[17].

Proposition 1.2.Let(S, H )∈K^p. The differential of the moduli morphismψS,H,δis everywhere injective, hence all components ofV|H|,δ(S)have maximal number of moduli.

Proof. LetCbe a curve inV|H|,δ(S)and letf : ˜C→Cbe the normalization at theδnodes. We have the following exact sequence

0 T_C˜ f^∗(TS) Nf 0,

which defines the normal sheafNf to the mapf. The differential ofψ atC˜ is the coboundary mapH⁰(C, N˜ f)→ H¹(C, T˜ _C˜). Hence it suffices to prove thath⁰(C, f˜ ^∗(TS))=0, i.e. thath⁰(C, ϕ˜ ^∗(TS)_{| ˜C})=0, whereϕ: ˜S→Sis the blow-up ofS at the nodes ofC. Denote byEi the exceptional divisors, with 1�i�δ. Consider the diagram with exact rows and columns

0 0 0

0 T_S˜(− ˜C) ϕ^∗(TS)(− ˜C) ⊕^δ_i=1O_P¹(−1) 0

0 T_S˜ ϕ^∗(TS) ⊕^δ_i=1O_P¹(1) 0

0 T_{S˜| ˜C} ϕ^∗(TS)_{| ˜C} � 0,

0 0 0

(3)

where��C^2δ is a skyscraper sheaf of rank 1 supported at the 2δintersections ofC˜ with theEi’s and the rightmost sheaves in the first two rows are supported onEi�P¹, for 1�i�δ. One hash⁰(S, ϕ˜ ^∗(TS))=h⁰(S, TS)=0 and H¹(S, ϕ˜ ^∗(TS)(− ˜C))�H¹(S, T˜ _S˜(− ˜C)). Grant for the time that

H⁰(C, T˜ _{S˜| ˜C})=0. (4)

By (4), the map H¹(S, T˜ _S˜(− ˜C))→H¹(S, T˜ _S˜) is injective. Its image A corresponds to first order deformations of S˜ that keep C˜ fixed. These deformations do not move the Ei’s, and therefore A intersects the image of H⁰(S,˜ ⊕^δ_i=1O_P¹(1))→H¹(S, T˜ _S˜) in(0). This implies that the mapH¹(S, ϕ˜ ^∗(TS)(− ˜C))→H¹(S, ϕ˜ ^∗(TS)) is in- jective andh⁰(C, ϕ˜ ^∗(TS)_{| ˜C})=0 follows.

We now prove(4). A local computation shows thatϕ_∗(T_{S˜| ˜C})=TS|C, so it suffices to prove that

h⁰(S, TS|C)=0. (5)

ConsiderSembedded inP^pby|H|. By the exact sequence

0 TS|C T_P^p_|C NS/P^p|C 0,

to prove(5)one has to prove that the mapγ :H⁰(C, T_P^p_|C)→H⁰(C, NS/P^p|C)is injective.

The cohomology of the Euler sequence

0 O^S H⁰�

S,O^S(C)�∗

⊗O^S(C) T_P^p_|S 0

yields that H⁰(S, T_P^p_|S)� H⁰(P^p, T_P^p)� C^p2+2p is the tangent space to PGL(p + 1,C). Similarly H⁰(S, T_P^p_|S(−C))�C^p+1 is the tangent space to the subgroup of PGL(p+1,C) that pointwise fixes the hyperplane in whichClies.

Consider the long exact cohomology sequence associated to the Euler sequence forC, i.e.

0 H⁰(C,OC) H⁰�

S,OS(C)�∗

⊗H⁰(C, ωC)�C^p(p+1) H⁰(C, T_P^p_|C)

H¹(C,O^C) H⁰�

S,O^S(C)�∗

⊗H¹(C, ωC).

The map in the second row is dual to the surjective map H⁰�

S,O^S(C)�

⊗H¹(C, ωC)�H⁰(C, ωC)⊕C H⁰(C, ωC), hence the last map in the first row is surjective, so thatH⁰(C, T_P^p_|C)�C^p2+p−1.

Consider the commutative diagram with exact rows and columns

0 0

0 H⁰(S, T_P^p_|S(−C))�C^{p+1 α} H⁰(S, NS/P^p(−C))

0 H⁰(S, T_P^p_|S)�C^{p2+2p β} H⁰(S, NS/P^p)

H⁰(C, T_P^p_|C)�C^{p2+p−1 γ} H⁰(C, NS/P^p|C).

0

Assume we havex∈H⁰(C, T_P^p_|C)such thatγ (x)=0. Liftxtoy∈H⁰(S, T_P^p_|S). Thenβ(y)∈H⁰(S, NS/P^p(−C)).

The above geometric interpretation tells us thaty∈H⁰(S, T_P^p_|S(−C)), proving thatx=0, which implies the injec- tivity ofγ, hence(5). ✷

Later we will need to consider the substackK^�p ofKp consisting of pairs(S, H )such thatS contains a smooth rational curveΓ satisfyingΓ·H =p.

Proposition 1.3.For anyp�2,K^�pis irreducible of codimension one inK^p, and the general element(S, H )∈K^�pis such thatH is ample. Furthermore,Pic(S)�Z[H] ⊕Z[Γ]for all(S, H )outside a countable union of Zariski closed proper subsets ofKp^�.

Proof. This is standard: the proof follows much the same arguments as, e.g., [10, Proposition (3.2)], using [31, Theorem 1.14.4]and[11, pp. 271–272]. ✷

We note thatK^�pis a Noether–Lefschetz divisor inK^p. 1.3. Universal Severi varieties and degenerations

For anyp�2, and 0�δ�p, one can consider a stackV^p,δ(see[14, Proposition 4.8]), called theuniversal Severi variety, which is pure and smooth of dimension 19+g, endowed with a morphismφp,δ:V^p,δ →K^p and its fibres are so described

V^{p,δ ⊃}

φ_p,δ

V_|H|,δ(S)

K^{p �} (S, H ).

Some fibers may be empty, but there is a dense open substackK^◦pofK^p over which the fibers are nonempty and the morphismφp,δ:V^p,δ→K^◦pis smooth on all components ofV^p,δ, each dominatingK^◦p.

In a similar way one can consider thek-gonal universal locusVp,δ^k ⊆V^p,δ.

We will need this in a more general setting. Suppose we have a proper flat family of surfacesf :S→D, whereD is a disk. Assume that

• Sis smooth, endowed with a line bundleH;

• f is smooth overD^∗=D− {0};

• ift∈D^∗, then the fibreSt off overt is aK3 surface;

• the fibreS₀off over 0 is a local normal crossing divisor inS;

• the line bundleHt:=H|St determines a complete linear system|Ht|of dimensionpfor allt∈Dand(St, Ht)∈ K^pfor allt∈D^∗.

SinceV^p,δ is functorially defined, we havef-relative Severi varieties φf;p,δ:V^f;p,δ →D^∗, withV^f;p,δ locally closed inP(f∗(H)), such that the fibre ofφf;p,δ overt isV|Ht|,δ(St)for allt∈D^∗. We will drop the indexδwhen δ=0.

Lemma 1.4.LetC₀∈ |H₀|be an element ofV_|H0|,δ(S₀), withδnot disconnecting nodesq₁, . . . , qδ off the singular locus ofS0. ThenC0sits in the closure ofVf;p,δ inP(f∗(H))andVf;p,δ dominatesD.

Proof. We have a commutative diagram with exact rows and column

0

0 T_[C0](V_|H0|,δ(S0))�H⁰(C0, N_C^�

0/S₀) C^p�T_[C0](|H0|)� H⁰(C0, NC₀/S₀) ^α ⊕^δ_i=1T_q¹_i

0 H⁰(C₀, N_C^�

0/S) C^p+1�T_[C0](V^f;p)� H⁰(C₀, NC₀/S) ^β ⊕^δ_i=1T_q¹_i,

H⁰(C0,O^C0)

0 whereN_C^�

0/S₀ and N_C^�

0/S are the equisingular normal sheaves at the marked nodes ofC inS₀and S, respectively.

By hypothesis,αis onto, hence so isβ. ThusH⁰(C₀, N_C^�

0/S), which is the tangent space at[C₀]of the space of equi- singular deformations at theδnodes ofC₀inS, has dimension dim(V|H0|,δ(S₀))+1, and the assertion follows. ✷ 1.4. K3surfaces andk-gonal loci

Let(S, H )∈K^p be general. By Brill–Noether theory,V_|H^k _|,δ(S)=V_|H|,δ(S)ifδ�p−2(k−1).

Proposition 1.5.Let (S, H )be in Kp. Assume g:=p−δ�2(k−1). Then for any irreducible componentV of V_|H^k _|,δ(S)one hasdim(V )�2(k−1).

Proof. Consider the morphismψin(2). LetV be an irreducible component ofV_|H|,δ^k (S)andV^� theg-dimensional, irreducible component ofV_|H|,δ(S)containing it, so that

∅ �=ψ (V )⊆ψ� V^��

∩M¹g,k.

SetW=ψ (V )andW^�=ψ (V^�), so thatW is an irreducible component ofW^�∩M¹g,kand, byProposition 1.2, one has dim(W^�)=g. Then

dim(V )�dim(W )�dim� W^��

+dim� M¹g,k

�−dim(M^g)=2(k−1). ✷

The proof of Proposition 1.5 shows that the expected dimension of an irreducible component of V_|H^k _|,δ(S) is min{2(k−1), p−δ}.

It is convenient to have arelative versionofProposition 1.5. Letf :S→Dbe as in Section1.3. One can define thef-relativek-gonal locusVf^k;p,δ⊆V^f;p,δ overD^∗.

Proposition 1.6.LetV₀be a component ofV_|H^k

0|,δ(S₀). Ifdim(V0)=2(k−1), thenV₀is contained in an irreducible componentV ofV_f^k_;p,δ dominatingD^∗, withdim(V)=dim(V0)+1.

Proof. Similar to the proof ofProposition 1.5, usingLemma 1.4(see also[12, Prop. 5.11 and its proof]). ✷ 2. Rational curves in the Hilbert scheme of points of aK3 surface

If S is a K3 surface, then Hilb^k(S) is a hyperkähler manifold, also called anirreducible symplectic manifold (see e.g. [5,25]). The cohomology group H²(Hilb^k(S),Z) is endowed with the Beauville–Bogomolov quadratic formqand one has the orthogonal decomposition

H²�

Hilb^k(S),Z�

�H²(S,Z)⊕_⊥Z[ek], (6) where�k:=2ekis the class of the divisor (still denoted by�k) parametrizing nonreduced 0-dimensional subschemes [5]. Equivalently,�kis the exceptional divisor of theHilbert–Chow morphismµk:Hilb^k(S)→Sym^k(S). The embed- ding ofH²(S,Z)intoH²(Hilb^k(S),Z)in(6)is given by sending a classF∈H²(S,Z)to the class inH²(Hilb^k(S),Z) determined by all subschemes whose support intersects a representative ofF. By abuse of notation we will still denote byFthis class inH²(Hilb^k(S),Z). The restriction of the Beauville–Bogomolov form toH²(S,Z)is the cup product onS, andq(ek)= −2(k−1). Accordingly,(6)induces an orthogonal decomposition (see[5])

Pic�

Hilb^k(S)�

�Pic(S)⊕_⊥Z[ek]. (7)

Given a primitive classα∈H₂(Hilb^k(S),Z), there exists a unique classwα∈H²(Hilb^k(S),Q)such thatα·v= q(wα, v), for allv∈H²(Hilb^k(S),Z), and one sets (cf. e.g.[20])

q(α):=q(wα). (8)

This gives aQ-valued form on homology, and we have H2�

Hilb^k(S),Z�

�H2(S,Z)⊕_⊥Z[rk], (9) whererk is the homology class orthogonal toH²(S,Z)and satisfyingek·rk= −1, see e.g.[22, §1]. As explained in [20, Ex. 4.2],rk is the class of a fiber of the Hilbert–Chow morphism, i.e., it is the class of the rational curve in�k

corresponding to the curve lying above 2x1+x2+ · · · +xk−1in Sym^k(S), for anyk−1 distinct pointsx1, . . . , xk−1

ofS. The embedding of H₂(S,Z)inH₂(Hilb^k(S),Z)is given by sending the class of a cycleY to the class of the cycle

�ξ∈Hilb^k(S)�

�Supp(ξ )= {p1, . . . , pk−1, y}, y∈Y� , wherep1, . . . , pk−1are distinct fixed points ofSoffY.

The decomposition(9)induces

N1�

Hilb^k(S),Z�

�Pic(S)⊕_⊥Z[rk].

IfR≡D−yrk inN1(Hilb^k(S),Z), withD∈Pic(S), then wR=D− y

2(k−1)ek, and by(8), one has

q(R)=D²− y²

2(k−1). (10)

We mentioned in the Introduction the importance of rational curves on hyperkähler manifolds. The relation with the topic of this paper is that a curve C on a K3 surface whose normalization C˜ possesses a g¹_k gives rise, in an obvious way, to an irreducible rational curveRin Hilb^k(S). Indeed, theg_k¹= |A|onC˜ induces aP¹(C,A)⊂Sym^k(C)˜ and this is mapped to an irreducible rational curveR(C,A)⊂Sym^k(S)by the composed morphism

Sym^k(C)˜ Sym^k(C) Sym^k(S).

The irreducible rational curveR=R(C,A)⊂Hilb^k(S)is the strict transform (µk)⁻¹_∗ (R(C,A))by the Hilbert–Chow morphism.

LetCbe an element ofV_|H^k _|,δ(S), and assume that its normalization carries ag_k¹such that

all the nodes ofCare non-neutral with respect to theg_k¹; (11)

theg¹_k has only simple ramification. (12)

LetRbe the corresponding rational curve in Hilb^k(S).

Lemma 2.1.Under hypotheses(11)and(12), the class¹ofRinN1(Hilb^k(S),Z)isH−(g+k−1)rk.

Proof. Write R=H−yrk, so thaty=ek·R. Since all nodes are non-neutral and theg_k¹ has simple ramification everywhere, by Riemann–Hurwitz we have

y=ek·R=1

2�k·R=1

2(2g+2k−2)=g+k−1. ✷ The particular casep=9,δ=2 andk=4 is treated in[20, Ex. 4.5].

Remark 2.2.The conclusion ofLemma 2.1holds even without hypothesis(12). Furthermore, also hypothesis(11) can be weakened: ifδ^� is the number of non-neutral nodes ofC, then the class of Ris H−(g^�+k−1)rk, where g^�:=p−δ^��g. For a detailed proof, we refer e.g. to[23, §3.3 and 5.3](see also[24, §6.2]).

3. Necessary conditions for existence of linear series on normalizations

Consider the usualBrill–Noether numberρ(g, r, d)=g−(r+1)(r+g−d).

Theorem 3.1. Let(S, H )∈K^p such that all elements in |H| are reduced and irreducible (e.g.,Pic(S)�Z[H]).

Assume that C∈ |H| is a curve whose normalization possesses ag_d^r. Letg be the geometric genus of C and set δ=p−gandα:= �gr+(d−r)(r−1)

2r(d−r) �. Then

ρ(p, αr, αd+δ)�0, i.e., δ�α�

rg−(d−r)(αr+1)�

. (13)

Proof. Letν: ˜C→Cbe the normalization ofCand letAbe a line bundle onC˜ such that|A| =g_d^r. Then, for any positive integerl, the sheafAl:=ν∗(lA)is torsion free of rank one onCwithh⁰(Al)=h⁰(lA)�lr+1 and degAl= deg(lA)+δ=ld+δ(see, e.g.,[13, Prop. 3.2]). We haveρ(A^l)=ρ(pa(C), h⁰(A^l)−1,degA^l)�ρ(p, lr, ld+δ) and we claim that

ρ(p, lr, ld+δ)=l²r(d−r)−l(gr+r−d)+δ�0. (14) To prove(14), letA^�l denote the globally generated part ofA^l, that is, the image of the evaluation mapH⁰(A^l)⊗ O^C→A^l. If(14)does not hold, thenρ(A^�l)�ρ(A^l) <0. In particular,h¹(A^�l) >0. The kernel of the evaluation map

H⁰(A^l)⊗O^S A^�l 0

is a vector bundle, whose dual bundleEl has rankh⁰(A^�l)=h⁰(Al)�lr+1 and satisfiesc1(El)=Candc2(El)= degA^�l �degA^l =ld+δ (see [15]). One has χ (E^l ⊗El^∗)=2(1−ρ(A^�l))�4 (see e.g. [28, §1]). Furthermore, dualizing the sequence definingEl, we obtain

1 There is an erroneous fraction of 1/2 in the corresponding formula fork=2 in[12, (6.7)], due to a trivial computational mistake in the line above the formula:P¹_�.e= −2 should have been−1, whereP¹_�isr_kin our notation.

0 H⁰(A^�l)^∗⊗O^S E^{l q} ext¹_O

S(A^�l,O^S) 0.

Ash⁰(ext¹_O

S(A^�l,O^S))=h¹(A^�l) >0 (by[15, Lemma 2.3]), this proves thatE^l is globally generated off a finite set.

Thus, as in the proof of [28, Lemma 1.3], the linear system|C| would contain a reducible curve, a contradiction.

This proves(14).

The polynomial inl in(14)attains its minimum forl₀=^gr+r−d_2r(d−r). The inequality(14)holds for the closest integer tol0, which isα. This proves(13). ✷

The property of|H|not containing reducible curves is a dense, open property inK^p. Therefore, the theorem proves the “only if” part inTheorem 0.1(i).

Remark 3.2.(a) The proof does not requireC˜to be smooth, only to possess ag^r_d. Thus the proof works for anypartial normalizationC˜ ofCpossessing ag_d^r, withg:=pa(C)˜ instead of the geometric genus ofC.

(b) Fixp,dandr. If(13)holds for a givenδ, then it holds for allδ^��δ. Indeed, for all positive integersland for anyδ^��δ, we haveρ(p, lr, ld+δ^�)�ρ(p, lr, ld+δ).

Next we concentrate on the caser=1 and setd=k, where(13)reads like(1). ThenTheorem 3.1proves the part ofTheorem 0.1stating thatV_|H^k _|,δ(S)�= ∅only if(1)holds.

Remark 3.3.Setρ=ρ(p, α, kα+δ). It is convenient to write(1)in the form ρ�0, i.e., δ�(g−k+1)²−β²

4(k−1) , (15)

whereβ:=(k−1)(2α+1)−g, i.e.,−(k−1) < β�k−1.

As an aside, we obtain a bound on theBeauville–Bogomolov self-intersectionof the rational curves in Hilb^k(S) corresponding to curves inV_|H^k _|,δ(S), which confirms for them a conjecture by Hassett and Tschinkel[20, Conj. 1.2].

Corollary 3.4. Let(S, H )∈K^p such that all elements in|H| are reduced and irreducible (e.g.,Pic(S)�Z[H]).

Assume that C ∈V_|H|,δ^k (S) is a curve whose normalization possesses a g_k¹ = |A| satisfying (11) and (12).

LetR=R(C,A)andg:=p−δ. Then

q(R)=2(p−1)−(g+k−1)²

2(k−1) =2(ρ−1)− β²

2(k−1)�−k+3

2 , (16)

withρandβas inRemark3.3.

Proof. The first equality on the left follows from(10)andLemma 2.1, the middle equality is a direct computation and the inequality follows fromTheorem 3.1andRemark 3.3. ✷

Remark 3.5.ByRemarks 2.2 and 3.2(a), the corollary also holds without assumptions(11)and(12)if we substitute the number δ with the number δ^� of non-neutral nodes of C with respect to the g_k¹. In particular, the inequality q(R)�−^k+3₂ holds for any rational curveRobtained from a curve inV_|H|,δ^k (S).

4. Chains of rational curves on unions of scrolls that are limits ofK3 surfaces

We will henceforth fix an integerp�3 and setq=2p−1.

4.1. Unions of scrolls as limits of specialK3surfaces

If(S, L)∈K^q, then|L|determines a morphismφ_|L|:S→P^q, which is an embedding for general(S, L). LetHq

be the component of the Hilbert scheme of surfaces inP^q whose general point corresponds to an embedding of anS as above. One has dim(Hq)=q²+2q+19 andHq is smooth at each point corresponding to a smoothK3 surface.

The component Hq contains points that correspond to degenerations of elements of K^q, as we will now explain (see[9, §2.2]).

LetE⊂P^q be a smooth elliptic normal curve of degreeq+1 with two distinct line bundlesLi∈Pic²(E), with i=1,2. LetR1andR2be the rational normal scrolls of degreeq−1 inP^q defined byL1andL2, respectively, i.e.

Ri is the union of lines spanned by the divisors of|Li|. ThenR₁∩R₂=E, the intersection is transversal andEis anticanonical on eachRi. MoreoverR=R1∪R2corresponds to a smooth point ofHq.

We will be concerned with the following case. Let Li∈Pic²(E), with 1�i�2, be two general line bundles.

Consider the embedding ofEgiven byL^⊗p₂ =:O^E(1). ThenR₁�P¹×P¹andR₂�F2. We letsiandfidenote the classes of the nonpositive section and fiber, respectively, ofRi, for 1�i�2. ThenO^R1(1)�O^R1(s1+(p−1)f1)and O^R2(1)�O^R2(s2+pf2). The sections2does not intersectE, hence lies in the smooth locus ofR, and it is embedded inP^qas a (degenerate) rational normal curve of degreep−2. In particular,s2is a Cartier divisor onR, so that

H0:=O^R(1)⊗O^R(−s2) (17)

is also Cartier, withH₀²=2p−2. Moreovers2·H0=p.

Lemma 4.1.There is a unique irreducible, codimension one subvarietyH^�_q ofHq containing the point corresponding toR and smooth there, such that the general point of H^�_q represents a smoothK3surface S containing a smooth rational curveΓ of degreep−2degenerating tos2whenSflatly degenerates toR. The line bundleH:=O^S(1)⊗ O^S(−Γ ) is globally generated and primitive with H²=2p−2. In particular,(S, H )is a general element of Kp^�

(cf. Section1.2).

Proof. LetX →Hqbe the universal family and considers2⊂R2⊂R⊂X. Thens2stays off the singular locus of X and by standard deformation theoretic arguments (cf. e.g.[27, II, Thm. 1.14]), it moves insideX in a familyF with

dim(F)�−K_X·s2+dim(X)−3= −K_R2·s2+dim(Hq)−1=dim(Hq)−1. (18) Sinces2 does not move on R and since, forS ∈Hq outside a countable union of Noether–Lefschetz divisors, we have Pic(S)�Z[L], with L=O^S(1), then equality must hold in(18). This implies that F is smooth at the point corresponding tos2. Hence there is a unique irreducible codimension one subvarietyH^�_q inHq, containing the point corresponding toRand smooth there, over whichs2deforms.

LetS be the surface corresponding to the general point ofH^�_q, letL=O^S(1)and letΓ be the rational curve of degreep−2 that is a deformation ofs2. Since the locus of pairs of scrolls insideHphas codimension 16, the surface Sis irreducible with at most isolated double points of typeAn for somen�1, andΓ sits in the smooth locus ofS.

SupposeS is singular. Then its minimal desingularizationπ :S^�→S is a K3 surface and we setL^�=π^∗(L). By standard Hodge theory, the subvariety ofHqcorresponding to such singular surfacesSis irreducible of codimension 1, and all elements outside a countable union of Zariski closed proper subsets have one single double point of typeA₁ (anode) and Pic(S^�)�Z[L^�] ⊕Z[N], whereNis the(−2)-curve corresponding to the node. Hence such a singularS cannot contain a smooth rational curveΓ not containing the node. This proves thatSis smooth.

To finish the proof, note that forS general inH^�_q, the linear system|H| is base point free, as|H0|is, andH²= 2p−2. SupposeH =hA, with h >1. Then p−1=h²(γ −1), where A²=2γ −2. Moreover p=H ·Γ = h(A·Γ ). Hencehdivides bothp−1 andp, a contradiction. ThusH is indivisible, whence(S, H )is general inKp^�

byProposition 1.3. ✷

LetD be a disk. We fix ϕ:D→H^�_q a holomorphic map with nonzero differential, such that ϕ(0)is the point corresponding toRand, fort∈Dgeneral,ϕ(t )is a general element inH^�_q. By pulling back the universal family on Hq, we obtain a flat familyX →D, whose total space has isolated singularities alongE in the central fibre, and is otherwise smooth. Indeed, the singular locus is the zero locus of the section inH⁰(E, T_R¹)corresponding to the section inH⁰(R, NR|P^q)that is the image of the differential ofϕat 0 (cf.[9, p. 647]or[7, Sec. 3.1]). Note thatT_R¹is a line bundle of degree 16 onE(cf.[9, p. 644]).

OnX we have the pullbackLof the hyperplane bundle onP^q. The restriction toX of the total space of the flat familyFof deformations ofs2is a surfaceGcontained in the smooth locus ofX. Hence it determines a line bundle

G onX. We set H=L⊗G^∗. The restrictions ofLandHto the general fibre S ofX →Dgive the two globally generated line bundles L=O^S(1)and H specializing toO^R(1)and H0, respectively, on the central fiber R. The restriction of G to the general fibre S is the smooth rational curve Γ, specializing to s2 on the central fibre, and L�O^S(H+Γ ).

One can perform a small resolution of the singularities ofX, obtaining a new familyf :S→D, which has all properties indicated in Section1.3. The central fibreS₀however is no longerR, but a modification of it. Precisely one can work things out in such a way thatS0=R1∪ ˜R2, whereR˜2is a sequence of blow ups ofR2at the singular locus ofX, andR₁andR˜₂meet transversally alongE⊂R₁and its strict transform (still denoted byE) onR˜₂.

Since the curves onRwe will be concerned with lie off the singular points ofX, we can and will work onX orS with no distinction.

4.2. Special chains of rational curves

We now introduce the building blocks of limits onRof nodal hyperplane sections on the general surface inH^�_qin the degeneration described in Section4.1.

Letmbe a positive integer satisfyingm�p. Achain of length2m−1 is a sum of 2m−1 distinct lines f_2,1+f_1,1+f_2,2+f_1,2+ · · · +f_2,m−1+f_1,m−1+f_2,m, fi,j∈ |fi|,

wheref_1,j intersects onlyf_2,j andf_2,j+1, forj=1, . . . , m−1. The chain intersectsEin 2mpoints, consisting of mdivisors of|L2|, and 2m−2 of them lie on the intersections between two lines, whereas the remaining two are on f2,1andf2,mand will be denoted bya1andb1, respectively. This pair of points will be calledthe distinguished pair of points of the chain.

We can also define chains with the roles of R1 and R2 interchanged, but we will not need this, except for the inductive argument in the proof ofLemma 4.2right below.

We will denote byCmthe family of chains of length 2m−1. Note thatCmis a locally closed subvariety of a Hilbert scheme of curves onR.

Lemma 4.2.The map sending a chain of length2m−1to its pair of distinguished points onEis a birational, injective morphism betweenC^mand|mL₂−(m−1)L1|.

Proof. We describe the inverse map. Its existence is obvious ifm=1. We proceed by induction onm. Leta+b∈

|mL₂−(m−1)L1|. Then|L₂−a| = {a^�}and|L₂−b| = {b^�}. Thereforea^�+b^�∼(m−1)L1−(m−2)L2and we are done by induction, exchanging the roles ofL1andL2. ✷

We note that ifm�p, then any chain of length 2m−1 is contained in a hyperplane. This fact will be used in the next section.

Assume we have a chain of length 2m−1 contained in a hyperplane sectionhofR. LetΓ₁andΓ₂be the sections of the rulings onR1 andR2 contained inh. Then the distinguished points area1:=Γ1∩f2,1andb1:=Γ1∩f2,m. We will calla1+b1the2-cycle onΓ1associated to the chain. We note that

a₁+b₁∈f_∗��

�mL₂−(m−1)L1

�

�

�, (19)

wheref :E→Γ₁�P¹is the morphism determined by the linear series|L₁|. The chain intersectsΓ₁and Γ₂ in a total of 2m−1 points, distributed asm−1 onΓ1andmonΓ2. They will be calledthe nodes ofhassociated to the chain. The nodes onΓ1will be called themarked nodesofh.

Fig. 1shows a chain of length 9 contained in a hyperplane section. All intersection points between the chain and Eare marked with a box, the twodistinguished pointsare marked with filled boxes, theassociated nodesare marked with circles, the ones onΓ₁are themarked nodes.

InFig. 2we describe the stable model of the partial normalization at the associated nodes of a hyperplane section containing a chain: in the stable model all rulings are contracted, so that the two distinguished points lying onΓ₁are identified, creating a node.

Fig. 1.

Fig. 2.

5. Limits of nodal curves

Letα1, . . . , αp be nonnegative integers such that p=

p

�

j=1

j αj. (20)

Definition 5.1.We defineV^�(α₁, . . . , αp)to be the locally closed subset of|O^R(1)|consisting of nodal curvesCnot passing through the singular locus ofX and containing exactlyαjchains of length 2j−1, forj=1, . . . , p. Condition (20)implies that every element ofV^�(α₁, . . . , αp)containsp lines in|f₂|, thus it containss₂. Hence the elements of V^�(α1, . . . , αp)are in one-to-one correspondence with a locally closed subset of|H0|(cf.(17)), which we denote by V (α₁, . . . , αp).

Given a curveCinV (α1, . . . , αp), we denote byΓ1 the section of the ruling onR1 contained inC(i.e.,Cis the union ofΓ₁and of chains). The curveCcomes equipped with the subscheme of its

δ=δ(α1, . . . , αp):=

p

�

j=1

(j−1)αj (21)

marked nodeslying in the smooth locus ofR. Setg=p−δ=�p j=1αj. Proposition 5.2.Under the condition(20), we have:

(i) V (α1, . . . , αp)is a smooth component ofV|H₀|,δ(R)of theexpected dimensiong;

(ii) V (α1, . . . , αp)is a component of the flat limit ofV|H|,δ(S)with(S, H )∈K^�pgeneral.

Proof. ByLemma 4.2we have a morphism ν:V (α1, . . . , αp)

p

�

j=1

Sym^αj�

�j L2−(j−1)L1

��.

The target is irreducible of dimensiong, by(20)and(21). Take any pointηtherein. ByLemma 4.2each coordinate of ηuniquely defines a chain. The reduced intersection between the union of these chains andEis an effective divisor Dηlying in|(�

j αj)L₂| = |pL₂| = |O^E(1)|, by(20). Henceν is dominant and any fiber ofν is a point. Therefore V (α1, . . . , αp)is irreducible of dimensiong.

Since theδnodes are not disconnecting (cf. Section4.2), the tangent space toV_|H0|,δ(R)at a point ofV (α₁, . . . , αp) has dimensiong. Hence (i) follows. Assertion (ii) follows byLemma 1.4. ✷

The next result follows from the description of the stable model of the partial normalization at the associated nodes of a hyperplane section containing a chain in Section4.2and the fact thatCis obtained by removing the component s₂from a hyperplane section ofR.

Lemma 5.3.LetCbe a curve inV (α₁, . . . , αp)andCthe stable model of its partial normalization at itsδmarked nodes. ThenCis the image ofΓ₁�P¹by the morphism identifying each distinguished pair of points of each chain contained inC. ThusChas arithmetic genusg.

The caseδ=0 corresponds to(α₁, . . . , αp)=(p,0, . . . ,0), and thenChaspnodes. Then, in the degeneration of a general (S, H )∈Kp^� to(R, H0), the general element in|H|degenerates to an irreducible p-nodal rational nodal curve.

6. Limits ofk-gonal nodal curves

Letk�2 be an integer. We keep the notation introduced in Section5.

6.1. k-gonal nodal curves in the central fibre

Definition 6.1.We defineV^k(α1, . . . , αp)to be the closed subset ofV (α1, . . . , αp)consisting of curvesCsuch that all 2-cycles (in number ofg) onΓ1associated to a chain belong to divisors of the sameg_k¹onΓ1.

The following is a consequence ofLemma 5.3:

Lemma 6.2.LetCbe a curve inV (α1, . . . , αp). The stable model of the partial normalization ofCat itsδmarked nodes lies inM¹g,kif and only ifClies inV^k(α₁, . . . , αp). In particularV^k(α₁, . . . , αp)fills up one or more compo- nents ofV_|H^k

0|,δ(R).

Ifg�2(k−1), thenV^k(α1, . . . , αp)=V (α1, . . . , αp)byLemma 5.3. As we will see inProposition 6.5below, this also follows by a parameter count concerningg_k¹’s onP¹. By the argument as in the proof ofProposition 1.5(but also by the same parameter count as above), the expected dimension ofV^k(α₁, . . . , αp)is 2(k−1)wheng >2(k−1).

Our objective is to prove that, under suitable conditions,V^k(α1, . . . , αp)is nonempty of the expected dimension. To do so, we need some intermediate results.

6.2. Some technical results

Recall the morphismf :E→P¹determined by|L1|. We have the induced mapf⁽²⁾:Sym²(E)→Sym²(P¹).

As customary, we identify Sym²(P¹)withP²: fix an irreducible conic��P¹, and identify a divisorx+yof� with:

• the pole of the line�x, y�with respect to�, ifx�=y;

• the pointx∈�, ifx=y.

In this way�is identified with thediagonalof Sym²(P¹)and thecoordinate curve{x+y, y∈P¹}with the tangent line�xto�atx.

We denote bycj, for 1�j�n, the image viaf⁽²⁾ of the smooth rational curves in Sym²(E) defined by the pencils|j L₂−(j−1)L1|. These are distinct conics, each intersecting�in four distinct points corresponding to the ramification points of the pencils.

ConsiderQ=P¹×P¹ with the two projectionsπi:Q→P¹,i=1,2. Fix a positive integerk and look at the line bundle OQ(k, k):=p^∗₁(O_P¹(k))⊗p₂^∗(O_P¹(k)), whose space of sections is H⁰(P¹,O_P¹(k))^⊗2. The two sub- spaces Sym²(H⁰(P¹,O_P¹(k)))and∧²H⁰(P¹,O_P¹(k))are invariant (resp. anti-invariant) under the natural involution that exchanges the coordinates. Hence they are pull-backs of sections of line bundles,O⁺k andO⁻k respectively, on Sym²(P¹).

Let us focus onO⁻k. One has H⁰�

Sym²� P¹�

,O⁻k

�� ∧²H⁰�

P¹,O_P¹(k)�

�H⁰�

P²,O_P²(k−1)� .

Therefore the linear system|Ok⁻|identifies with|O_P²(k−1))|and also withP(∧²H⁰(P¹,O_P¹(k))). Under the former isomorphism, a pointgof the grassmannianG(1, k)⊂P(∧²H⁰(P¹,O_P¹(k))), which can be identified with a linear seriesg¹_k onP¹, corresponds to the degreek−1 curve inP²

C_g=�

W∈Sym²� P¹� �

�g(−W )�0� .

The family of curves{C_g}_g∈G(1,k)is irreducible of dimension 2(k−1)=dim(G(1, k)).

Lemma 6.3.For general choices ofL1, we have:

(i) no curvecj is contained in a curveC_g, forg=g_k¹onP¹;

(ii) for a generalg, the curveC_gintersects eachcj transversally in2(k−1)distinct points;

(iii) in addition, none of these2(k−1)points is fixed varyingg.

Proof. By movingL1, each coniccj moves inP²in a 1-dimensional family containing the diagonal�(obtained for L1=L2). But�is not contained in anyC_g, proving (i). Similarly, it suffices to prove (ii) for the intersection with� of aC_g, withggeneral, which is obvious, since the intersection points correspond to the ramification points ofg. Part (iii) easily follows. ✷

Lemma 4.2can be rephrased as the following lemma, whose proof is left to the reader.

Lemma 6.4.A chain of length2m−1determines and is determined by a point on the coniccm. 6.3. Nonemptiness and dimension of limitk-gonal nodal curves systems

We can now prove the desired result about nonemptiness and dimension ofV^k(α₁, . . . , αp).

Proposition 6.5.If (20)and

αj�2(k−1) for allj (22)

hold, the variety V^k(α1, . . . , αp) is nonempty of the expected dimension min{2(k−1), g}and is a component of V_|H^k

0|,δ(R). Furthermore, for the general curve inV^k(α₁, . . . , αp), the family ofg_k¹’s onΓ₁satisfying the condition in Definition 6.1has dimensionmax{0, ρ(g,1, k)}.

Proof. Consider the set of curves{C_g}_g∈G(1,k)of dimension 2(k−1)above, which is in one-to-one correspondence with the set ofg¹_k’s onΓ1�P¹. A generalC_gintersects each coniccjin 2(k−1)distinct points by (ii) ofLemma 6.3.