Such families give rise to, because of the uniqueg12’s on their normalizations, families of the same dimensions of irreducible rational curves in the Hilbert scheme Hilb2(S)

Indag. Mathem., N.S., 19 (2), 217–238 June, 2008

Remarks on families of singular curves with hyperelliptic normalizations^✩

by Andreas Leopold Knutsen

Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008 Bergen, Norway

Communicated by Prof. M.S. Keane

ABSTRACT

We give restrictions on the existence of families of curves on smooth projective surfacesSof nonnegative Kodaira dimension all having constant geometric genuspg2and hyperelliptic normalizations. In particular, we prove a Reider-like result that relies on deformation theory and bending-and-breaking of rational curves in Sym²(S). We also give examples of families of such curves.

1. INTRODUCTION

The object of study of this paper is families of irreducible curves with hyperelliptic normalizations (of genus2) on a smooth surfaceS. Such families give rise to, because of the uniqueg¹₂’s on their normalizations, families of the same dimensions of irreducible rational curves in the Hilbert scheme Hilb²(S). Because of the importance of rational curves and the subvarieties they cover due to Mori theory, it is natural to try to check the existence of, or bound the dimensions of, families of such curves, or alternatively, their counterparts onS.

LetSbe a smooth surface andV ⊂HilbSbe a reduced and irreducible scheme parametrizing a flat family of curves on S all having constant geometric genus p_g2and hyperelliptic normalizations.

MSC:Primary 14H10; Secondary 14E30, 14H51, 14J10

✩ Research supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme.

E-mail: [email protected] (A. Knutsen).

It is easy to see (cf. Lemma 2.1) that if_|KS|is birational, thendimV =0. This shows that the problem of bounding the dimension of a family of curves with hyperelliptic normalizations is solved for a large class of surfaces. At the same time, it is relatively easy to find obvious examples of surfaces with large families of curves with hyperelliptic normalizations: In fact, if S is any smooth surface admitting a rational2:1 mapf:S→R onto a rational surface, then we can just pull back families of rational curves on R. There are several examples of such double covers, even forp_g(S) >0, see for instance the works of Horikawa [19–22]

for surfaces of general type, Saint-Donat [35] forK3surfaces, and [6,37,38] for surfaces with smooth hyperelliptic hyperplane sections.

We note that smooth hyperelliptic curves on surfaces have been extensively studied by means of adjunction theory (see [6,37,38] to mention a few). Of course Reider’s famous result [34] can be used to prove that if C ⊂S is a smooth hyperelliptic curve andC²9, then there is a pencil|E| such that eitherE²=0 andE.C=2, orE²=1,C≡3Eand|E|has one base pointx lying onC(making the obvious modifications in the proof of [34, Corollary 2]). Unfortunately no such results seem to be available, at least as far as we know, in the case of singular curves.

In this paper we prove some results bounding dimensions of families of irre- ducible curves with hyperelliptic normalizations on smooth surfacesSin Section 2.

In particular, we show that the dimension is bounded by one if S is fibered over a smooth nonhyperelliptic curve of genus 3 (Lemma 2.3) and by two if S has maximal albanese dimension (Proposition 2.5). We also give several examples of families of irreducible curves with hyperelliptic normalizations. Then, in Section 3, we prove a Reider-like result, cf. Theorem 3.3, stating that any family of dimension 3 (resp. 5) of curves with hyperelliptic normalizations on a smooth surface S with p_g(S) >0 (resp. kod(S)0 and p_g(S)=0) forces the existence of some special divisors enjoying some particular intersection properties.

Moreover, these divisors “cut out” theg₂¹’s on the normalizations of the curves in the family.

We hope the results will find more applications and also hope that the reader will find the method of proof of interest: in fact, unlike the results on smooth curves, which use adjunction theory and/or vector bundle methods, our method uses deformations of curves and bending and breaking of rational curves in Sym²(S), a method we also used in [15]. Thus, the special divisor occurring in Theorem 3.3 is obtained as a component of a degenerated member at the border of the family. In this sense our method is perhaps more geometric and intuitive than Reider’s method.

In Section 2 we state the general setting, show that the dimension of families of curves with hyperelliptic normalizations can be bounded in various cases and give some examples of such families.

Then, in Section 3 we prove the Reider-like result, Theorem 3.3, passing by Proposition 3.1 and Lemma 3.2. Finally, we make some remarks in Section 4, including writing out the results in the case of smooth curves in Section 4.1 recovering Reider’s result, and in the case of only one singular point of multiplicity one in Section 4.2.

2. DIMENSION BOUNDS AND EXAMPLES

Consider the following assumptions:

Sis a smooth projective surface with kod(S)0andV ⊂HilbS a reduced and irreducible scheme parametrizing a flat family of irreducible curves onSof constant arithmetic and geometric genera (1)

p_aandp_g2, respectively, and hyperelliptic normalizations.

We denote byCthe algebraic equivalence class of the curves.

Note thatV as in (1) is, by default, nonempty.

We have the following elementary result, already mentioned in the introduction:

Lemma 2.1. Under the assumptions(1), if|K_S|is birational, thendimV =0. Proof. We may assume thatdimV =1. Then after compactifying and resolving the singularities of the universal family overV, we obtain a smooth surfaceT, fibered over a smooth curve, with general fiberF a smooth hyperelliptic curve of genus2, and a surjective morphismf:T →S. By adjunction|K_T|is not birational on the general fiber F. Since KT =f^∗KS +R, where R is the (effective) ramification divisor off, and f is generically1:1 on the fibers, we see that_|KS|cannot be birational. ₂

Note that, as mentioned in the Introduction, any irreducible curve C on a surfaceS with hyperelliptic normalization (of geometric genus2) gives rise to a unique irreducible rational curve RC⊂Hilb²(S). Precisely, this can be seen in the following way: Letν:C→C be the normalization. Then the uniqueg¹₂ onC induces a mapP¹→Sym²⁽C)and thisP¹is mapped to an irreducible rational curve _Cin Sym²(S)by the natural composed morphism

Sym²(C)^ν˜

→(2)Sym²(C) →Sym²(S).

The irreducible rational curveR_C⊂Hilb²(S)is the strict transform by the Hilbert–

Chow morphism μ:Hilb²(S)→Sym²(S) of this curve. Note that the Hilbert–

Chow morphism resolves Sing(Sym²(S))S and gives an obvious one-to-one correspondence between irreducible curves in Hilb²(S) not contained in the ex- ceptional locus (which is aP¹-bundle overS) and irreducible curves in Sym²(S) not contained in Sing(Sym²(S)).

The correspondence in the opposite direction, that is, from irreducible rational curves in Hilb²(S)to curves inSis more delicate and we refer to [15, Section 2] for details. Suffice it to say that irreducible rational curves in Hilb²(S)not contained in the exceptional locus give rise to curves onSwith rational, elliptic or hyperelliptic normalizations, by taking the (one-dimensional component of the) union of the supports of the points of the curve in Hilb²(S)when we consider these as length-two schemes onS.

To extend the correspondence into families, we proceed as follows (cf. also [15]).

Given (1), let ϕ:C→V be the universal family. Normalizing _C we obtain, possibly restricting to an open dense subscheme ofV, a flat familyϕ_˜:C→V of smooth hyperelliptic curves of genusp_g2, cf. [40, Theorem 1.3.2]. Letω_C/V be the relative dualizing sheaf. The morphismγ:C→P(ϕ˜_∗(ω_C/V)) over V is finite and of relative degree two onto its image (cf. also [29, Theorem 5.5(iv)]), which we denote byPV. We now have a universal familyψ:PV →V of rational curves and since the points in these curves correspond to couples of points ofS, possibly coinciding, we have a natural morphism V:PV →Sym²(S). We define

R_V :=im V (the Zariski closure). (2)

We have

(3)

where π is the natural morphism. Note that V maps no curve in the family to Sing(Sym²(S))Sby construction. Also note thatdimR_V 3, as Sym²(S)is not uniruled, since kod(S)0(see e.g. [18, Proposition 2.1]).

GivenV as in (1), we will callμ⁻_∗¹R_V ⊂Hilb²(S), the strict transform ofR_V by the Hilbert–Chow morphism, thelocus covered by the associated rational curves in Hilb²(S).

We will make use of the following consequence of Mumford’s well-known theorem on 0-cycles on surfaces [32, Corollary, p. 203], as generalized in [15, Corollary 3.2]:

Proposition 2.2. Assume that S is a smooth surface with p_g(S) >0 and R ⊂ Sym²(S)is a subvariety that is covered by a family of rational curves of dimension 3.

ThenRis a surface with rational desingularization.

Proof. This is [15, Proposition 3.6]. 2

We will now give some results bounding the dimension ofV as in (1) in various situations.

Lemma 2.3. Assume(1)and thatf:S→Bis a fibration over a nonhyperelliptic smooth curveBof genus3.

ThendimV 1 with equality holding if and only if the general fiber off is a (smooth) hyperelliptic curve andV parametrizes a subset of the fibers.

Proof. Let {Cv}v∈V be the family on S given by V and {v =Cv}v∈V be the associated family of rational curves in Sym²(S), given byψand V as in (3).

Considerf⁽²⁾:Sym²(S)→Sym²(B). SinceBis neither elliptic nor hyperelliptic, Sym²(B)does not contain rational curves. Therefore,f⁽²⁾must contract every_v to a point, sayb_v+b_v∈Sym²(B), withb_v, b_v∈B. LettingF_v:=f⁻¹b_v andF_v:=

f⁻¹b_v denote the two fibers off, we get that_v is contained in the surfaceF_v+ F_v⊂Sym²(S). Hence Supp(C_v)=F_v∪F_v⊂S. Since eachC_v is irreducible, the result follows. ₂

The following example shows that the condition that B is neither elliptic nor hyperelliptic is in fact necessary:

Example 2.4. Start with a Hirzebruch surface Fe, with e0, with PicFe Z[] ⊕Z[F], where ²= −e, F²=0 and .F =1. Choose integers α and β such that

α2 and β{2, αe+1} (4)

and take a general pencil in|α+βF|. Note that the conditions (4) guarantee that the general element in the general such pencil is in fact irreducible. Now take the blow upπ:Fe→Fealong the(α+βF )²=α(2β−eα)base points of the pencil and denote the exceptional curves byE_i. Then

K_F

e∼ −2π^∗−(e+2)π^∗F+E_i. Set

D:=π^∗(α+βF )−E_i.

ThenD²=0 and|D|is a pencil defining a fibrationg:Fe→P1. For any integer l2, choose a general _l ∈ |O_P1(2l)| consisting of distinct points and let R_l ∈

|2lD| be the corresponding (smooth) divisor. Then_l and R_l define two double covers ν and μ, respectively, that are compatible, in the sense that we have a commutative diagram:

(5)

whereT is a smooth surface,B is a smooth curve andf is induced byg in the natural way.

By Riemann–Hurwitz, the genus ofBisg(B)=l−11andBis either elliptic or hyperelliptic. As

K_T ∼μ^∗(K_F

e+lD) =μ^∗

(l−1)D+π^∗

(α−2)+(β−e−2)F ,

the conditions (4) imply kod(S)0, in fact evenpg(S) >0.

The surface_Fe, being rational, contains families of irreducible (smooth) rational curves of arbitrarily high dimensions. Pulling them back onT yields families of (smooth) hyperelliptic curves of arbitrarily high dimensions. ₂

Note that the surfaces T in the example have irregularity q(T )=h¹(OT)= g(B)=l−11. Thus the surfaces have arbitrarily high irregularity. Nevertheless, using the albanese map (cf. e.g. [3, pp. 80–88]), we can prove the following bounds.

Proposition 2.5. Assume(1)withq(S):=h¹(OS)2and letα:S→AlbSbe the albanese map.

Ifimαis a curveB, thendimV 1unlessBis hyperelliptic.

Ifimαis a surface (i.e.,Sis of maximal albanese dimension), thendimV 2. Proof. If imαis a curveB, thenB is necessarily smooth of genusq(S)2(cf.

e.g. [3, Proposition V.15]). Then we apply Lemma 2.3.

If imαis a surface T, we must havepg(S) >0 by [3, Lemme V.18]. Assume now, to get a contradiction, thatdimV 3. ThenR_V, defined in (2), is a surface with rational desingularization, by Proposition 2.2.

As above, let{C_v}v∈V be the family onS given byV and{_v=_Cv}v∈V be the associated family of rational curves in Sym²(S), given byψand _V as in (3).

Considerα⁽²⁾:Sym²(S)→Sym²(T )⊂Sym²(AlbS). Asαdoes not contractC_v, for generalv∈V, the surfaceR_Vis mapped byα⁽²⁾to a surfaceR_V⊂Sym²(T ). Let :Sym²(AlbS)→AlbS be the summation morphism. As AlbS, being abelian, cannot contain rational curves, cf. e.g. [7, Proposition 4.9.5], each rational curve ⊂R_V ⊂Sym²(T )must be contracted to a point by_|R

V, say()=p∈AlbS. Now all fibers⁻¹p, forp∈AlbS, are isomorphic to the Kummer variety of AlbS, cf. e.g. [7, Section 4.8] for the definition. As rational curves on Kummer varieties cannot move, by [33, Theorem 1], we must have that any family of rational curves onR_V has dimensiondim(R_V)2. But this contradicts the fact thatR_V has rational desingularization.

ThereforedimV 2, as desired. 2

The following result shows that equalitydimV =2is in fact attained on abelian surfaces, which have maximal albanese dimension.

Lemma–Example 2.6. Assume(1)withSabelian and thatV is not contained in a scheme of larger dimension satisfying(1).

Then dimV =2 and the locus covered by the associated rational curves in Hilb²(S)is a threefold birational to a_P¹-bundle overS.

Furthermore, such families exist if and only ifSis simple (i.e., not the product of two elliptic curves).

In particular, a bielliptic surface does not contain families as in(1).

Proof. Assume thatSis abelian. Consider the natural composed morphism α:Hilb²(S)−→^μ Sym²(S)−→ S,

where μ is the Hilbert–Chow morphism and is the summation morphism. As above, let_{v=_Cv}v∈V be the associated family of rational curves in Sym²(S) given byV. As S cannot contain rational curves (see e.g. [7, Proposition 4.9.5]), must contract each_vto a pointp_v. Therefore, the strict transformR_v=R_Cv:=

μ⁻_∗¹(_v)⊂Hilb²(S)is contained in the surfaceα⁻¹(p_v)⊂Hilb²(S). Now all such fibers ofαover points inSare isomorphic to the (desingularized) Kummer surface ofS(cf. [5, Section 7], [4] or [24, 2.3] and e.g. [7, Section 10.2] and [2, V.16] for the definition). Since a Kummer surface isK3, rational curves do not move inside it (this also follows from Lemma 4.4). Therefore, the family is given by{R_v}p∈S, proving the first assertion.

IfSis simple, then it contains irreducible curves of geometric genus two, see e.g.

[28, Corollary 2.2].

Assume that S=E₁×E₂, with each E_i a smooth elliptic curve. Then each Sym²(E_i)is an elliptic ruled surface. Any rational curve in Sym²(S)not lying in Sing(Sym²(S)) is mapped by the projections Sym²(S)→Sym²(Ei), i=1,2, to a rational curve in either Sym²(E1)or Sym²(E2), which has to be a fiber of the ruling. Therefore, the rational curve in Sym²(S)corresponds to ag₂¹on one of the elliptic fibers ofS, proving that there is no irreducible curve onSwith hyperelliptic normalization of geometric genus2.

IfS is bielliptic, there is a finite morphism f:T →S whereT is a product of two elliptic curves, cf. e.g. [3, Definition VI.19] or [2, p. 199], whenceT is abelian.

Clearly, f is unramified, asK_T ∼OT, whence so is f⁽²⁾:Sym²(T )→Sym²(S). Therefore, the family of rational curves ψ:PV →V as in (3) is pulled back, viaf⁽²⁾, to two copies of the family in Sym²(T ). By what we proved above, the corresponding families of curves onT consist of elliptic curves, whence the same holds onS. ₂

We conclude this section by giving some examples of families as in (1) of high dimensions.

Example 2.7. LetW (n)⊂ |O_P2(n)|denote theSeveri varietyof nodal, irreducible rational curves in_|OP2(n)|. Then W (n)is irreducible of dimension 3n−1, by a well-known result of Severi and Harris, cf. [11, Theorem 1.1] and [17]. For any integerb3, take a general smoothB∈ |O_P2(2b)|, so that, for anyn, the general curve inW (n)intersectsBtransversally.

Letf:S→P² be the double cover defined byB, so thatS is a smooth surface andf is branched alongB. SettingH:=f^∗O_P²(1), we haveK_S∼(b−3)H. Let V (n)⊂ |nH|be the subscheme parametrizing the inverse images of the curves in W (n)that intersectB transversally and letp_a(n)=p_a(nH )andp_g(n)denote the arithmetic and geometric genera of the curves in V (n). Then V (n) satisfies the conditions in (1) and

dimV (n)=3n−1,

p_a(n)=n(n+b−3)+1 and p_g(n)=bn−1,

the second equality by adjunction and the third by Riemann–Hurwitz. Note that the elements ofV (n)haveδ(n):=pa(n)−pg(n)=n(n−3)+2nodes (two over each of the nodes of the corresponding curves inW (n)).

Also note that curves inW (n)that are tangent toB yield families in_|nH|with lower geometric genera and lower dimensions.

Of course Hilb²(S) contains a copy of the P², which is precisely the locus in Hilb²(S)covered by the rational curves associated to the curves inV (n).

Example 2.8. Let S be an Enriques surface. Then S contains several elliptic pencils and we can always pick (at least) two such,|2E1|and|2E2|, withE₁.E₂=1, cf. [12, Theorems 3 and 3.2]. (By adjunctionE²₁=E₂²=0and it is well known that 2Eiand2E_i, whereE_idenotes the unique element of_|Ei+KS|, are the two multiple fibers of the elliptic pencils.) ConsiderH:=3E1+E2; thenH²=6 and the base scheme of|H|consists of two distinct pointsxandy, where

x=E₁∩E₂ and y=E₁∩E₂

(see [13, Proposition 3.1.6 and Theorem 4.4.1]). Letf:S→Sbe the blow up along xandyandE_xandE_ythe two exceptional divisors. SetL:=f^∗H−E_x−E_y. Then L²=4and|L|is base point free and, by [13, Theorem 4.5.2], defines a morphism of degree twoϕ_L:S→Qonto a smooth quadricQ⊂P³, which can be seen as the embedding of_F0P¹×P¹by the complete linear system_|1+2|, where1and2

are the two rulings. In particular, by construction,L∼ϕ_L^∗(OQ(1))∼ϕ^∗_L(1+2). Furthermore, by [13, Remark 4.5.1 and Theorem 4.5.2] the pencil |f^∗(2E1)| on S is mapped by ϕ_L to |₁|, so that ϕ^∗_L1∼f^∗(2E1). We therefore have ϕ^∗_L2∼ L−ϕ_L^∗₁∼f^∗(E₁+E₂)−E_x−E_y.

It follows that, for anyn1, the general smooth rational curve in_|1+n2| P²ⁿ⁺¹ yields by pullback byϕ_L a smooth hyperelliptic curve in_|f^∗((n+2)E1+ nE₂)−n(E_x+E_y)|onSof genus3n+1by adjunction (or by Riemann–Hurwitz and the description of the ramification in [13, Theorem 4.5.2]).

Pushing down toSwe thus obtain subschemes, for eachn∈N, V (n)⊂ |(n+2)E1+nE2|, such that dimV (n)=2n+1,

parametrizing irreducible curves with hyperelliptic normalizations of geometric generap_g(n)and arithmetic generap_a(n), where

p_g(n)=3n+1 and p_a(n)=1

2

(n+2)E1+nE₂2

+1=n(n+2)+1.

Note that for eachn2 all the curves in the family have precisely two singular points, located atx andy, both of multiplicityn.

Of course Hilb²(S) contains a rational surface birational to_P¹_×P¹, which is precisely the locus in Hilb²(S) covered by the rational curves associated to the curves inV (n).

One can repeat the construction with E₁ and E₂ interchanged, or with other elliptic pencils on the surface. Moreover, choosing smooth rational curves that are tangent to the branch divisor ofϕ_L, we can obtain families with lower geometric genera, that is, with more singularities.

Moreover, one can also repeat the process forH:=2E1+E2, which defines a rational2:1map ontoP², following the lines of the previous example. Note that the smooth curves in_|2E1+E2|form a two-dimensional family ofsmoothhyperelliptic curves onS, by [13, Corollary 4.5.1].

Example 2.9. LetS be a K3 surface. Then Hilb²(S) is a hyperkähler fourfold, also called an irreducible symplectic fourfold, and rational curves and uniruled subvarieties are central in the study of the (birational) geometry of Hilb²(S).

For example, a result of Huybrechts and Boucksom [8,25] implies that, if the Mori cone of Hilb²(S)is closed, then the boundaries are generated by classes of rational curves. Precise numerical and geometric properties of the rational curves that are extremal in the Mori cone have been conjectured by Hassett and Tschinkel [18].

Uniruled subvarieties of Hilb²(S)are important in several aspects: The presence of a P²⊂Hilb²(S) gives rise to a birational map (the so-called Mukai flop, cf.

[31]) to another hyperkähler fourfold and all birational maps between hyperkähler fourfolds factor through a sequence of Mukai flops (see [9,23,41,42]). Moreover, uniruled threefolds in Hilb²(S) are central in the study of the birational Kähler cone of Hilb²(S)[25].

In the particular case ofK3surfaces, the study of families of irreducible curves with hyperelliptic normalizations and the loci the corresponding rational curves cover in Hilb²(S) is therefore of particular importance. In [15] we study such families.

Let nowH be a globally generated line bundle onS and denote by|H|^hyperthe subscheme of_|H| parametrizing curves with hyperelliptic normalizations. Then, any component of|H|^hyper has dimension2 with equality holding ifH has no decompositions into moving classes, e.g. if PicSZ[H], by [15, Lemma 5.1].

As for concrete examples of such families on ageneral (in the moduli space) primitively polarizedK3surface(S, H ), here are the ones that are known to us:

(i) |H| contains a two-dimensional family of irreducible curves of geometric genus p_g=2, whose general element is nodal, by the nonemptiness of Severi varieties on K3 surfaces as a direct consequence of Mumford’s theorem on the existence of nodal rational curves on K3 surfaces (cf. [30, pp. 351–352] or [2, pp. 365–367]) and standard results on Severi varieties (cf. [39, Lemma 2.4 and Theorem 2.6]; see also e.g. [11,14]). In the particular casepa(H )=3, i.e. whenS is a smooth quartic inP³, the locus in Hilb²(S)covered by the associated rational curves is a_P¹-bundle overS, by [15, Example 7.6].

(ii)_|H|contains a two-dimensional family of irreducible nodal curves of geomet- ric genusp_g=3with hyperelliptic normalizations, by [15, Theorem 5.2]. The locus in Hilb²(S)covered by the associated rational curves is birational to aP¹-bundle, by [15, Corollary 5.3 and Proposition 3.6(ii)].

(iii) [15, Proposition 7.7] IfH²=2(d²−1)for some integerd1, then Hilb²(S) contains a uniruled 3-fold that is birational to a _P¹-bundle. The fibers give rise to a two-dimensional family of curves in_|H|with hyperelliptic normalizations of arithmetic genusp_a=p_a(H )=d²and geometric genusp_g=2d−1.

(iv) [15, Proposition 7.2] IfH²=2(m²−9m+19)for some integerm6, then Hilb²(S)contains a_P²and the Severi varieties of rational curves in_|OP2(n)|, for any n1, give rise to(3n−1)-dimensional subschemes V (n)⊂ |nH| parametrizing irreducible curves with hyperelliptic normalizations of arithmetic genusp_a(n)= p_a(nH )=n²(m²−9m+19)and geometric genusp_g(n)=2n−9.

We have now seen several examples of families as in (1) of dimension 2 on surfaces with pg(S) >0 (in Examples 2.7 and 2.9(iv) with n =1, Examples 2.9(i)–(iii) and the abelian surfaces in Lemma–Example 2.6) and on Enriques surfaces (the case mentioned in the last lines of Example 2.8).

At the same time we have seen infinite series of examples of families as in (1) of arbitrarily high dimensions3(Examples 2.7 and 2.9(iv) withn2, and Examples 2.4 and 2.8).

In the next section we will see the difference between those “small” and “big”

families.

3. A REIDER-LIKE RESULT

Consider the additional assumptions dimV

3, ifp_g(S) >0orSis Enriques, 5, otherwise.

(6)

The following result is an improvement of [15, Proposition 4.2]. In fact, the idea of the proof is essentially the same.

Proposition 3.1. Assume (1) and (6). Then there is a decomposition into two effective, algebraically moving classes

[C] = [D1] + [D2]

such that, for generalξ, η∈RV (cf.(2)), each with support at two distinct points ofS, there are effective divisorsD₁ ∼algD1andD₂ ∼algD2such thatξ⊂D₁and η⊂D₂and[D₁ +D₂] ∈V, whereV is the closure ofV in the component of the Hilbert scheme ofScontainingV.

Proof. We must have dimR_V =2 or3 by (6). If p_g(S) >0, then dimV 3 by (6), whenceRV is a surface by Proposition 2.2. IfS is Enriques, then there is an

unramified double coverf:T →Ssuch thatT is a smoothK3surface, cf. e.g. [2, VIII, Lemma 15.1(ii)]. Therefore, alsof⁽²⁾:Sym²(T )→Sym²(S) is unramified.

Hence, the family of rational curves in Sym²(S)given by V andψ as in (3) is pulled back to two copies in Sym²(T ). SincedimV 3 by (6), we conclude by Proposition 2.2 that these families only cover a surface in Sym²(T ). HenceR_V ⊂ Sym²(S)is a surface as well.

Therefore, in any case, the assumptions (6) guarantee that, for generalξ, η∈R_V, the locus of points inV parametrizing curves inPV passing throughξ andηinR_V is at least one-dimensional. For generalξ, η∈RV, letB=Bξ,η⊂V be a smooth curve (not necessarily complete) parametrizing such curves and

(7)

the corresponding restriction of (3) overB. LetBbe any smooth compactification ofB. By Mori’s bend-and-break technique, as in [27, Lemma 1.9] or [26, Corollary II.5.5] (see also [15, Lemma 2.10] for the precise statement we need), there is an extension of the right hand part of (7)

such that, for someb₀∈B\B we have( _B)_∗(ψ⁻_B¹b₀)⊇_ξ +_η, where _ξ and _ηare irreducible rational curves onR_V (possibly coinciding) such thatξ∈_ξ and η∈_η. Let now

I := {(x, ξ )∈S×RV |x∈Supp(ξ )} ⊂S×RV

be the incidence variety with projection morphismsp:I→Sandq:I→R_V. Then dimI =dimR_V =2 or3 andq is finite of degree two. Consider the commutative diagram

(8)

where the square is Cartesian. DefineπB:=p◦ _B. Note that forb∈B we have π_B(q⁻¹ψ⁻_B¹b)=π_B(ϕ˜⁻_B¹b). In particular,π_B is dominant and generically one-to-

one on the fibers overB. Thereforepmust also be dominant.

We have

B

∗

q⁻¹ψ⁻_B¹b₀

⊇q⁻¹_ξ+q⁻¹_η.

Denoting byb∈B a general point and recalling thatπBis generically one-to-one on the fibers overB, we have

C∼alg(π_B)_∗

q⁻¹ψ⁻_B¹b

∼alg(π_B)_∗

q⁻¹ψ⁻_B¹b0

⊇ p_∗ q⁻¹_ξ

+p_∗ q⁻¹_η

⊇D_ξ+D_η,

whereD_ξ:=p(q⁻¹_ξ)andD_η:=p(q⁻¹_η). Now the curves contracted bypare precisely the curves of type{x, x+D}, for a pointx∈Sand a curveD⊂S. Since S is not covered by rational curves, and ξ and η are general, their support onS does not intersect any of the finitely many rational curvesγonSwithγ .CC². If q⁻¹_ξ contained a component of the form{x, x+D}, then, by definition ofq, we would have

q⁻¹ξ= {x, x+y}y∈D∪ {y, x+y}y∈D,

a union of two irreducible rational curves, each being mapped isomorphically by p to _ξ. Then p({y, x+y}y∈D)=D⊂S would be an irreducible rational curve intersecting Suppξ, a contradiction. The same argument works for q⁻¹_η. Therefore, none of the components ofq⁻¹ξ norq⁻¹η are contracted byp. We therefore haveD_ξ⊃ξ andD_η⊃η, viewingξ andηas length-two subschemes of S. (Note thatD_ξ andD_ηare not necessarily distinct.) Moreover,

C∼alg(πB)_∗

q⁻¹ψ⁻_B¹b

=Dξ+Dη+Eξ,η

withE_ξ,η0, and by construction,D_ξ+D_η+E_ξ,ηlies in the border of the family ϕ:C→V of curves on S, and as such, [D_ξ +D_η+E_ξ,η] ∈V, where V is the closure ofV in the component of the Hilbert scheme ofScontainingV. Moreover, as the number of such effective decompositions of[C]is finite (asSis projective), we can find one decomposition_[C] = [D1] + [D2]holding for general ξ, η∈R_V. Since this construction can be repeated for generalξ, η∈R_Vand the set{x∈S|x∈ Supp(ξ )for someξ ∈R_V}is dense inS, as the curves parametrized byV cover the whole surfaceS, the obtained classesD1andD2must move in an algebraic system of dimension at least one. ₂

We now prove an additional, more precise result:

Lemma 3.2. Assume (1) and (6). Then we can find a decomposition as in Proposition3.1satisfying the additional properties

(a) D1.D2pa−pg+2; and

(b) there is a reduced and irreducible component of D₁(resp. D₂) containing ξ (resp.η).

Proof. Letξ and η∈R_V be general and [D₁ +D₂] ∈V such that ξ ⊂D₁ and η⊂D₂as in Proposition 3.1. LetDξ⊆D₁,Dη⊆D₂,ξ⊂RV andη⊂RV be as in the proof of Proposition 3.1, that is,D_ξ:=p(q⁻¹_ξ)andD_η:=p(q⁻¹_η).

Ifq⁻¹_ξwere reducible, it would consist of two rational components, each being mapped isomorphically to_ξ by q. Therefore, ξ, viewed as a length-two scheme onS, would intersect a rational curveγ ⊂S satisfyingγ .CC². As in the proof of Proposition 3.1, forξ general this cannot happen. Henceq⁻¹_ξ is reduced and irreducible and so isD_ξ =p(q⁻¹_ξ)as well. Of course the same reasoning also works to show thatD_η=p(q⁻¹_η)is irreducible. This proves (b).

SetD:=D₁+D₂. We now want to show that there is an effective decomposition D=D1+D2withDξ⊆D1,Dη⊆D2andD1.D2pa−pg+2.

We know that a partial desingularization ofD, say D, which can be obtained by a succession of blowupsf:S→S, is a limit of smooth hyperelliptic curves, as [D] ∈V by Proposition 3.1. LetD_ξ andD_ηdenote the strict transforms ofD_ξ and Dη, respectively. We now claim that there is an effective decomposition

D=D1+D2 withDξ⊆D1,Dη⊆D2andD1.D22.

(9)

To show (9), we first writeD=Dξ+D. We have a short exact sequence 0→ω_D

ξ →ω_D→ω_D(Dξ)→0.

SinceH¹(ω_D

ξ)H¹(ω_D)by Serre duality, the mapH⁰(ω_D)→H⁰(ω_D(D_ξ))is surjective, whence_|ω_D(D_ξ)|is not birational onD_η, since _|ω_D|is 2:1on every nonrational component, asDis a limit of hyperelliptic curves.

Let nowD₁⊆Dbe maximal with respect to the properties thatD_ξ⊆D₁,D₂:=

D−D₁⊇D_ηand|ω_D

2(D₁)|is not birational onD_η.

IfD2=Dη, thenD1.D22by [10, Proposition 2.3], and (9) is proved.

Now assume thatD2D_η.

If there is a reduced and irreducible component D₂₂ ⊆D₂ −D_η such that D₂₂.D₁>0, setD₂₁:=D₂−D₂₂⊇D_η. Then from

0→ω_D

22(D₁)→ω_D

2(D₁)→ω_D

21(D₁+D₂₂)→0 and the fact thath¹(ω_D

22(D₁))=0, we see, as above, that|ω_D

21(D₁+D₂₂)|is not birational onD_η, contradicting the maximality ofD₁.

Therefore D₁.D₂=D₁.D_η and since |ω_D

2(D₁)| is not birational on D_η, and H⁰(ω_D

η(D₁))⊆H⁰(ω_D

2(D₁)), then |ω_D

η(D₁)| is not birational on D_η either. It follows, using [10, Proposition 2.3] again, that

D1.D2=D1.Dη2,

and (9) is proved.

Now let E1, . . . , E_n be the (total transforms of the) exceptional divisors of f:S→S, so that K_S =f^∗K_S+

E_i,D₁=f^∗D₁−

α_iE_i andD₂=f^∗D₂− β_iE_i, forα_i, β_i0, whereD₁⊇ξ andD₂⊇η. We compute

2p_g−2=(D1+D2).(D1+D2+K_S)

=(D₁+D₂).(D₁+D₂+K_S)

−2

α_i.β_i+

α_i(1−α_i)+β_i(1−β_i)

D.(D+K_S)−2

α_i.β_i=2pa−2−2 α_i.β_i,

whenceαi.βipa−pg. Inserting this into D₁.D₂=

f^∗D₁− α_iE_i .

f^∗D₂−

β_iE_i =D₁.D₂− α_i.β_i

and using (9), we obtain the desired resultD1.D2pa−pg+2. 2

A consequence is the following Reider-like result. Note that whenp_a=p_g, that is, the family consists of smooth hyperelliptic curves, we retrieve the results of Reider [34].

We make the following notation: ifC⊂San irreducible curve with hyperelliptic normalization and f:S→S a birational morphism inducing the normalization ν:C→C, then we define

W_[2](C):=

ξ⊂Csmooth|ξ=f_∗(Z)withZ∈g¹₂(C)

⊂Hilb²(S).

Theorem 3.3. Assume(1)and(6). Then there is an effective divisorDonSsuch thath⁰(D)2,h⁰(C−D)2,D²(C−D)²and

02D²

(i)D.CD²+p_a−p_g+2

(ii)2(pa−p_g+2), (10)

with equalities in(i)or(ii)if and only ifC≡2D.

Furthermore there is a flat family parametrized by a reduced and irreducible complete subschemeV_D of the component of the Hilbert scheme ofS containing [D]with the following property: for general[C] ∈V there is a complete rational curveV_D(C)⊆V_Dsuch that for generalξ∈W_[2](C), there is a[D_ξ] ∈V_D(C)such thatξ⊂Dξ.

Proof. We have, by Proposition 3.1 and Lemma 3.2, a decomposition into alge- braically moving classesC∼algD1+D2withD1.D2pa−pg+2. Without loss of generality we can assume thatD²₁D₂², or equivalentlyD1.CD2.C. We first show that we can assume thatD₁²0.

Indeed, ifD²₁<0, then the algebraic system in whichD₁moves must have a base component >0. We can writeD₁∼algD₀+, whereD₀moves in an algebraic system of dimension at least one, without base components. In particular0D²₀=

D²₁−2D1.+²< ²−2D1., so that²>2D1.. Moreover, we have.D2

−.D1asCis nef. Hence

D0.(D2+)=D1.D2−.D2+.D1−² D₁.D₂+2.D1−²< D₁.D₂,

and we can substituteD₁ with D₀⊆D₁, as clearly Supp(ξ )∩= ∅ for general ξ∈R_V. Therefore, we can assumeD₁²0.

Combining with the Hodge index theorem, we get (2D1.C)·D₁²C²·D²₁ (D1.C)², so that 2D²₁D1.C, with equality if and only if C ≡2D1. Moreover, D²₁¹₂D1.C=¹₂(D²₁+D1.D2)¹₂(D²₁+pa−pg+2)yieldsD₁²pa−pg+2, again with equality if and only ifC≡2D1.

Finally, from Proposition 3.1 and the fact that we have at most removed base components of the obtained family, it is clear that there is a reduced and irreducible complete schemeVD₁parametrizing curves algebraically equivalent toD1with the property that for general_[C] ∈V and generalξ∈W_[2](C), there is a_[Dξ] ∈VD₁

such that ξ ⊂D_ξ. For fixed C this yields a complete curveV_D1(C)⊆V_D1 such that all_[D_ξ] ∈V_D1(C) for generalξ ∈W_[2](C). This gives a natural rational map C− →V_D1(C)inducing a morphism between the normalizationsC→V_D1(C)that is composed with the hyperelliptic double coverC→P¹. HenceV_D1(C)admits a surjective map from_P¹and is therefore rational.

Ifh⁰(D1)=1, then the variety parametrizing curves algebraically equivalent to D₁is abelian and therefore cannot contain rational curves, cf. e.g. [7, Proposition 4.9.5]. Henceh⁰(D₁)2.

SubstitutingD₁withD₂we also obtainh⁰(D₂)2, thus finishing the proof. 2 In particular, we have a slight improvement of [15, Corollary 4.7]:

Corollary 3.4. Assume(1)and in addition that there is no decompositionC∼alg

C₁+C₂such thath⁰(OS(C_i))2fori=1,2.

ThendimV 2ifp_g(S) >0anddimV 4otherwise.

The conditions in Corollary 3.4 are for instance satisfied if NS(S)Z[C]. Theorem 3.3 gives additional restrictions on the existence of such a family as in (1) and (6). In particular it shows that when the differenceδ:=p_a−p_gis “small”, then such a family cannot exist unless there are some quite special divisors on the surface (cf. also [16, Theorem 1], where we in fact show the nonexistence of curves with hyperelliptic normalizations in the primitive linear system |H| with δ^pa₂⁻³ on a general primitively polarizedK3surface(S, H )). Unlike the results of Reider, Theorem 3.3 cannot be used to say that if[C] ≡mLfor somem0, then families as in (6) do not occur, asp_agrows quadratically withm, but it shows that the differenceδ:=p_a−p_gmust get bigger asmgrows. This was already seen in Examples 2.7 and 2.9(iv) above.

Note that the families in Examples 2.7 and 2.9(iv) forn2and in Example 2.8 satisfy the conditions (1) and (6) and that the conditions in Theorem 3.3 are satisfied forD=H in Examples 2.7 and 2.9(iv) and forD=2E1in Example 2.8.