Degeneration of differentials and moduli of nodal curves on $K3$ surfaces

0.35 MB·

by  C. Ciliberto

DEGENERATION OF DIFFERENTIALS AND MODULI OF NODAL CURVES ON K3
SURFACES
C.CILIBERTO,F.FLAMINI,C.GALATI,ANDA.L.KNUTSEN
Abstract. Weconsider,undersuitableassumptions,thefollowingsituation: Bisacomponentofthemoduli
space of polarized surfaces and Vm,δ is the universal Severi variety over B parametrizing pairs (S,C), with
7 (S,H)∈B and C ∈|mH| irreducible withexactly δ nodes as singularities. The moduli map V →Mg of an
1 irreducible component V of Vm,δ is generically of maximal rank if and only if certain cohomology vanishings
0 hold. AssumingtherearesuitablesemistabledegenerationsofthesurfacesinB,weprovidesufficientconditions
2 for the existence of an irreducible component V where these vanishings are verified. As a test, we apply this
toK3surfacesandgiveanewproofofaresultrecently independently provedbyKemenyandbythepresent
n
authors.
a
J

5
2
1. Introduction

]
G
Let (S,H) be a smooth, projective, polarized, complex surface, with H an ample line bundle such that the
A linear system |H| contains smooth, irreducible curves. We set
. 1
h p:=p (H)= (H2+K ·H)+1,
a S
t 2
a
m the arithmetic genus of any curve in |H|. For any integer m>1, we set
[ ℓ(m):=dim(|mH|) and p(m):=pa(mH)= m(m−1)H2+m(p−1).
2
1
v One has
4 ℓ(m)=χ(O )+m2H2−p(m), for m≫0.
S
2
For any integer δ ∈{0,...,ℓ(m)}, consider the locally closed, functorially defined subscheme of |mH|
2
7 V (S,H)(simplyV (S)orV when H or (S,H) are understood),
m,δ m,δ m,δ
0
which is the parameter space for the universal family of irreducible curves in |mH| having only δ nodes as
.
1
singularities; this is called the (m,δ)–Severi variety of (S,H).
0
We will assume that there exists a Deligne–Mumford moduli stack B parametrizing isomorphism classes of
7
1 polarized surfaces (S,H) as above. Since we will basically deal only with local properties, we can get rid of
: the stack structure. Indeed, up to replacing B with an étale finite type representable cover, we may pretend
v
that B is a fine moduli scheme. Although not necessary, we will assume that B is irreducible (otherwise one
i
X may replace B with one of its components).
r Thenwemay considerthe scheme V ,calledthe (m,δ)–universal Severi variety overB,whichisendowed
m,δ
a
with a morphism
φ :V →B,
m,δ m,δ
whosefiberover(S,H)∈B isV (S,H). ApointinV canbeidentifiedwithapair(S,C),with(S,H)∈B
m,δ m,δ
and C ∈V (S,H).
m,δ
We make the following:
Assumption 1.1. (i) B is smooth;
(ii) for all (S,H) ∈ B, the surface S is regular, i.e., h1(S,O ) = 0, and h0(S,T ) = 0, i.e., S has no positive
S S
dimensional automorphism group;
(iii) for any m > 1 and δ ∈ {0,...,ℓ(m)} and for all (S,H) ∈ B, ℓ(m) is constant and the Severi variety
V (S,H) is smooth, of pure (and expected) dimension ℓ(m)−δ (hence V is smooth, of pure dimension
m,δ m,δ
dim(B)+ℓ(m)−δ, and φ is smooth and surjective).
m,δ
2010 Mathematics Subject Classification. 14H10,14J28, 14B05.
Key words and phrases. Severi varieties,modulimap,nodalcurves, K3surfaces.
1
2 C.CILIBERTO,F.FLAMINI,C.GALATI,ANDA.L.KNUTSEN
Remark 1.2. InAssumption1.1(i)wecouldhaveaskedB tobegenerically smooth,andin(iii)wecouldhave
askedthatforthegeneral (S,H)∈B,theSeverivarietyV (S,H)issmooth,ofpuredimensionℓ(m)−δ. But
m,δ
under these weaker assumptions, (i) and (iii) hold on a Zariski dense open subset. Since we will be interested
only in what happens at the general point of B, we may replace B with this open subset. The hypotheses in
(ii) are technical and not strictly necessary for our purposes, but they make things easier for us.
Conditions (i)–(iii) hold in some important cases, e.g., for polarized K3 surface of genus p (in which case
the moduli stack is usually denoted by K , is of dimension 19, and ℓ(m) = p(m) for any m > 1, cf.,e.g.,[12,
p
15, 16, 5]). Moduli spaces exist also for polarized Enriques surfaces (cf.[14]) and degenerations of (polarized)
Enriques surfaces are also studied (cf.e.g. [8, 18, 20, 22]).
Anotherrelevantclassisthe oneofminimal,regularsurfacesofgeneraltype (S,H)whosemodulispacehas
at least one (generically) smooth component B with points (S,H) verifying, for some m and δ, the conditions
in [3, 10, 11] ensuring smoothness and expected dimension of any component of V (S,H). Particular cases
m,δ
are,forsomem andδ,surfacesin P3 ofdegreed>5(cf. [2])andcomplete intersectionsofgeneraltype in PN.
Consider now the moduli map
ψ :V →M , where g =p(m)−δ,
m,δ m,δ g
and where M denotes the moduli space of smooth, genus–g curves: ψ sends a curve to the isomorphism
g m,δ
class ofits normalization. In this set–up,one is interestedin the following generalproblem: find conditions on
m and δ ensuring the existence of a component V of V such that ψ | is either generically finite onto its
m,δ m,δ V
image or dominant onto M . By taking into account Assumption 1.1, in principle, one may expect
g
dominance if dim(B)+ℓ(m)−δ >dim(M ),
g
generic finiteness onto its image if dim(B)+ℓ(m)−δ 6dim(M ).
g
The typicalexampleis thecaseofpolarizedK3surfacesstudiedbyvariousauthors(cf.,e.g.,[19,12,15,16,
17, 5]). In particular, [5] and Kemeny in [17] independently show that, as expected, ψ is generically finite
m,δ
on some component for all g = p(m)−δ > 11 with only a few finite possible exceptions (m,g) with m 6 4,
for fixed p; moreover [5] shows that ψ is dominant, as expected, for g 6 11 with only a few finite possible
m,δ
exceptions (m,g) with m64, for fixed p. The precise result for m=1 is the following:
Theorem 1.3. Let V be the universal Severi variety over the moduli space K of polarized K3 surfaces
m,δ p
(S,H) of genus p. For m=1 and g =p−δ one has:
(a) [5, 17] if g >15 there is a component V of V such that ψ is generically finite onto its image;
1,δ 1,δ|V
(b) [5] if g 67 there is a component V of V such that ψ is dominant onto M .
1,δ 1,δ|V g
In case (a) Kemeny’s result is stronger in the sense that he may weaken the assumptions on g for infinitely
many p’s.
The proofs in [5, 17], although different, both rely on studying the fibers of the moduli map on curves
on special K3 surfaces. Kemeny’s proof is inspired by ideas of [19], and uses appropriate curves on K3
surfaces with high rank Picard group. The approach in [5] is by specialization to a reducible K3 surface in
a partial compactification of K and therefore uses an extension of the moduli map to an appropriate partial
p
compactification of the Severi variety containing reducible curves, with target space M .
g
In the present paper we want to present a different approach to the aforementioned general problem. This
approach relies on two different techniques. Firstly, it is based on the analysis of first order deformations
of pairs (S,C) ∈ V as in [12, §4]. The strategy in [12], which requires Assumption 1.1, was originally
m,δ
introducedforpolarizedK3surfacesandform=1,but canbe easilyadaptedto m>1andtothe casewhere
the canonical bundle is not necessarily trivial (cf. also [15, Thm. 1.1(ii)]). The upshot is the following. In the
above setting, take (S,C) ∈ V and set Z := Sing(C). Arguing as in [12, §§4-5], the differential dψ of
m,δ (S,C)
ψ :=ψ at(S,C)canbeidentifiedwithasuitablecohomologymap(theH1(τ)in[12,(4.21)]). Inparticular,
m,δ
if µ : S˜ → S is the blowing-up of S at Z, C˜ is the strict transform of C and T and T are the tangent
Z S C˜
bundles of S and C˜ respectively, then
coker(dψ )≃H2(µ∗(T )(−C˜)) and ker(dψ )≃H1(µ∗(T )(−C˜))/H0(T ).
(S,C) Z S (S,C) Z S C˜
Moreover,by the Serre duality theorem and Leray isomorphism, as in [12, Proof of Thm (5.1)]), one has that
H2(µ∗(T )(−C˜))≃H0(Ω (mH +K )⊗J ) and H1(µ∗(T )(−C˜))≃H1(Ω (mH +K )⊗J ).
Z S S S Z/S Z S S S Z/S
DEGENERATION OF DIFFERENTIALS AND MODULI OF NODAL CURVES ON K3 SURFACES 3
By summarizing, if (S,C) belongs to a component V of V , the map ψ is
m,δ m,δ|V
dominant if and only if h0(Ω (mH +K )⊗J )=0,
(1) S S Z/S
generically finite if h1(Ω (mH +K )⊗J )=0.
S S Z/S
Finally, the vanishing of H1(Ω (mH +K )⊗J ) is equivalent to generic finiteness of ψ if g ≥2.
S S Z/S m,δ|V
Unfortunately, the vanishings (1) are, in general, not so easy to be proved, even if one assumes δ = 0 (cf.
[1]). This is where the second tool of our approach enters the scene (see §2). In order to prove the above
vanishings, we propose to use degenerations. We assume in fact that the surfaces in B and nodal curves on
them possess good semistable degenerations with limiting surfaces that are reducible in two components (the
more general case of reducibility in more components could be treated, but, for simplicity, we do not dwell
on this here). Then we look at the limits of the relevant cohomology spaces. The latter are driven by the
so–called abstract log complex (see [13, §3]). Using this we arrive at sufficient conditions for the vanishings in
(1)to hold,expressedintermsofcohomologicalpropertiesofsuitable sheavesofforms onthe twocomponents
of the limit surface (cf. §2.2). These properties are hopefully easier to prove than the vanishings in (1), since
these componentsare simplerthan S. The resultsare summarizedin Proposition2.7, whichis the mainresult
of this note.
In the restof the paper (i.e. §§3 and 4) we test our approachin the (known)case of K3surfaces for m=1,
giving a new proof of Theorem 1.3. We do not claim that this is easier than the proofs in [5, 17], but it works
quite nicely and gives good hopes to fruitfully apply the same method in other unexploredcases, like the ones
mentioned at the end of Remark 1.2. We also mention that the approach of this paper can be applied to the
m>1 case(using aslightly differentdegeneration,namely the one in [4]), butwe leavethis outas the bounds
we obtain depend linearly on m and are thus considerably weaker than the bounds in [5, 17]. Furthermore,
althoughthe presentapproachgives the same resultas [5] for m=1,the analysisof this casein [5] is finer (as
it studies the family ofdegenerateK3sonwhichthe curvesinthe fibersofthe moduli maplive)andisneeded
in the proof of the m > 1 case. Thus, the approach in this paper cannot replace the proof of the m = 1 case
in [5].
Terminology and conventions. We work over C. For X any Gorenstein, projective variety, we denote by
O and ω ≃ O (K ) the structural and the canonical line bundle, respectively, where K is a canonical
X X X X X
divisor. We denote by T and Ω the tangent sheaf and the sheaf of 1-forms on X, respectively. For Y ⊂X
X X
any closed subscheme, J (or simply J if X is intended) will denote its ideal sheaf whereas N (same
Y/X Y Y/X
as above) its normal sheaf. We use ∼ to denote linear equivalence of divisors. We often abuse notation
and identify divisors with the corresponding line bundles, using the additive and the multiplicative notation
interchangeably. Finally, we usethe conventionthat ifF is a sheafona schemeX andY ⊂X is a subscheme,
then H0(F) is the image of the restriction map H0(F)→H0(F ⊗O ).
|Y Y
Acknowledgements. The first three authors have been supported by the GNSAGA of Indam and by the
PRIN project “Geometry of projective varieties”, funded by the Italian MIUR.
2. Semistable degenerations and the abstract log complex
In this section we will provide a tool for proving the vanishings of the cohomology groups occurring in (1)
by degenerationof the surface andsemicontinuity. The main results aresummarizedin Proposition2.7 below.
2.1. Semistable degenerations. We recallsomebasic facts concerningsemistable degenerationsofcompact
complex surfaces and the associated abstract log complex (see [13, §3]). This complex allows to define flat
limits of the sheaves occurring in (1).
Definition 2.1. (i) Let R and R be connected, complex analytic varieties. Let ∆ = {t ∈ C| | t |< 1}. A
proper, flat morphism α:R→∆ is said to be a deformationofR if R is the scheme theoretical fibre of α over
0. Accordingly, R is said to be a flat limit of R , the scheme theoretical fibre of α over t6=0.
t
(ii) If there is a line bundle H on R, set H :=H and H :=H for t6=0. Then the pair (R,H) is said to
|R t |Rt
be a limit of (R ,H ) for t6=0.
t t
(iii) The deformation (or, equivalently, the degeneration) is semistable if R is smooth (so that we may assume
that R is smooth for t6=0) and R has at most normal crossing singularities.
t
(iv) Assumethatthereareδ disjoint sectionss ,...,s ofαandthattheirimages Z ,...,Z aresmoothcurves
1 δ 1 δ
in R, disjoint from Sing(R). Set Z := δ Z . Then we say that Z :=Z is a limit of Z :=Z for t6=0.
i=1 i |R t |Rt
S
4 C.CILIBERTO,F.FLAMINI,C.GALATI,ANDA.L.KNUTSEN
(v) If conditions (i)-(iv) are satisfied, we will say that (R,H,Z) is a semistable degeneration of (R ,H ,Z ),
t t t
for t6=0, or that (R ,H ,Z ), for t6=0, admits the semistable degeneration (R,H,Z).
t t t
α
Let R −→∆ be a semistable degeneration of surfaces as in Definition 2.1. Assume that all components of
R are smooth and R has no triple point. Then Sing(R) consists of the transversalintersection points of pairs
of components of R.
Consider the sheaf Ω (logR) on R defined by the exact sequence (cf. [23] and [7, §3.3])
R/∆
(2) 0 // α∗(Ω (0)) ι //Ω (logR) //Ω (logR) // 0.
∆ R R/∆
The map ι in (2) has rank one at every point, whence Ω (logR) is locally free of rank 2, as recalled in the
R/∆
following remark.
Remark 2.2. Away from R, one has Ω (logR)≃Ω and Ω (logR)≃Ω . In particular
R R R/∆ R/∆
Ω (logR)⊗O ≃Ω , for any t6=0.
R/∆ Rt Rt
Let P ∈R−Sing(R), and let x, y, z be local coordinates on R around P. Let t be the coordinate on ∆ and
assume that α is locally defined by t = x around p. Then Ω (logR) is locally free generated by dx, dy, dz,
R x
the map ι is defined by
dt dx
−→
t x
hence Ω (logR) is locally free generated by dy, dz (cf.[9, Prop.2.2.c]).
R/∆
Let now P ∈ Sing(R). From our assumptions, we may assume that α is locally defined by t = xy. Then
Ω (logR) is locally free generated by dx, dy, dz, the map ι is defined by
R x y
dt dx dy
−→ +
t x y
and so Ω (logR) is locally free generated by dx =−dy, dz.
R/∆ x y
Following [13, §3], we set
Λ1 :=Ω (logR)|
R R/∆ R
which is locally free on R.
Lemma 2.3. Let(R,H,Z)beasemistable degeneration of(R ,H ,Z )for t6=0, as in Definition 2.1. Assume
t t t
furthermore that all components of R are smooth and R has no triple points. Then for all m ∈ Z, i ∈ N and
t6=0, one has
hi(Λ1 ⊗O (mH +K )⊗J )>hi(Ω ⊗O (mH +K )⊗J ).
R R R Z/R Rt Rt t Rt Zt/Rt
Proof. The statement follows by semicontinuity as Ω (logR) is flat over ∆ and the ideal sheaf J is flat
R/∆ Z/R
over ∆ by [21, Prop.4.2.1(ii)]. (cid:3)
2.2. Degenerations of differentials. From now on we assume that R := R ∪R , with R ,R smooth,
1 2 1 2
projective surfaces, with transversalintersection along a smooth, irreducible curve E :=R ∩R . Then
1 2
(3) K =(K +R) =(K +R +R ) =K +E, i=1,2.
R|Ri R |Ri R 1 2 |Ri Ri
In this situation there are exact sequences involving Λ1 ⊗O (mH+K )⊗J , which allow us to compute
R R R Z/R
its cohomology by conducting computations on R and R .
1 2
Consider the exact sequences
(4) 0 // Ω //Ω (logE) ρi //O //0, i=1,2,
Ri Ri E
and
(5) 0 //Ω (logE)⊗O (−E) //Ω ri //ω // 0, i=1,2,
Ri Ri Ri E
where ρ is the residue map and r is the trace map of differential forms, cf. [9, §2].
i i
For the reader’s convenience, we recall how ρ and r are defined locally around a point of E. We may
i i
assume that locally, in an open subset of C4 with coordinates x,y,z,t, the equationof R is xy−t=0. We let
R be given by x=t=0 and R by y =t=0, so that E is given by x=y =t=0. In this chart
1 2
O ≃C[[y,z]], Ω ≃C[[y,z]]dy⊕C[[y,z]]dz,
R1 R1
DEGENERATION OF DIFFERENTIALS AND MODULI OF NODAL CURVES ON K3 SURFACES 5
dy
Ω (logE)≃C[[y,z]] ⊕C[[y,z]]dz, O ≃C[[z]]andω ≃C[[z]]dz,
R1 y E E
dy
ρ f(y,z) +g(y,z)dz :=f(0,z), r f(y,z)dy+g(y,z)dz :=g(0,z)dz
1 1
y
(cid:16) (cid:17) (cid:16) (cid:17)
and similarly for R .
2
Let a:R ⊔R →R be the desingularization of R; consider the exact sequence
1 2
(6) 0 // A //a Ω (logE)⊕Ω (logE) ρ //O // 0,
∗ R1 R2 E
(cid:16) (cid:17)
defining A, where ρ:=ρ +ρ . As in [13, Pf. of Lemma3.1,p.94-95], Λ1 fits in the exact sequence
1 2 R
(7) 0 // Λ1 //A r // ω //0,
R E
where r is locally defined as follows: if
dy dx
(ω ,ω )= f (y,z) +g (y,z)dz, f (x,z) +g (x,z)dz
1 2 1 1 2 2
y x
(cid:16) (cid:17)
is a local section of A, then
r(ω ,ω ):= g (0,z)−g (0,z) dz.
1 2 1 2
Remark 2.4. Notice that (cid:0) (cid:1)
(8) a (Ω ⊕Ω )֒→A and r| =r −r ,
∗ R1 R2 a∗(ΩR1⊕ΩR2) 1 2
where r and r are the maps in (5). One may verify that the map r does not extend to a map on A.
1 2 i
Havinginmind(1)andLemma2.3,wearelookingforconditionsensuringthevanishingofhi(Λ1⊗O (mH+
R R
K )⊗J ), for i=0,1. To this end, consider the maps
R Z/R
F:=ρ⊗O (mH +K )⊗J , q :=r⊗O (mH +K )⊗J ,
R R Z/R R R Z/R
F :=ρ ⊗O (mH +K +E)⊗J and q :=r ⊗O (mH +K +E)⊗J .
i i Ri Ri Zi i i Ri Ri Zi
Then by (4), (5) and the fact that Z avoids E, for i=1,2 we have
i
(9) 0
(cid:15)(cid:15)
0 //ΩRi(logE)(mH+KRi)⊗JZi //ΩRi(mH+KRi+E)⊗JZi qi ////ωE(mH+KRi+E) //0
(cid:15)(cid:15)
ΩRi(logE)(mH+KRi+E)⊗JZi
Fi
(cid:15)(cid:15)(cid:15)(cid:15)
OE(mH+KRi+E)
(cid:15)(cid:15)
0
Moreover,by (6)–(8), we have
(10)
0 //A⊗OR(mH+KR)⊗JZ/R //a∗(cid:16)⊕2i=1ΩRi(logE)(mHOO +KRi+E)⊗JZi(cid:17) F=F1+F2//OE(mH+KR) //0
(cid:31)?
a∗(cid:16)⊕2i=1ΩRi(mH(cid:127)+_(cid:15)(cid:15) KRi+E❱)❱⊗❱❱J❱Z❱i❱q(cid:17)❱1❱−❱q2❱❱❱❱❱❱❱❱❱++
0 //Λ1R(mH+KR)⊗JZ/R //A⊗OR(mH+KR)⊗JZ/R q //ωE(mH+KR) //0.
The next two lemmas provide sufficient conditions for the vanishings of the two first cohomology groups of
Λ1(mH +K )⊗J .
R R Z/R
6 C.CILIBERTO,F.FLAMINI,C.GALATI,ANDA.L.KNUTSEN
Lemma 2.5. Assume that
(11) im(H0(F ))∩im(H0(F ))={0}.
1 2
Then h0(Λ1(mH +K )⊗J )=0 if and only if
R R Z/R
(12) im(H0(q ))∩im(H0(q ))={0}
1 2
and
(13) h0(Ω (logE)(mH +K )⊗J )=0 for i=1,2.
Ri Ri Zi
Proof. By (10), we have that h0(Λ1(mH +K )⊗J ) = 0 if and only if H0(q) is injective. Moreover, as
R R Z/R
H0(F)=H0(F )+H0(F ), we get from (11), (10) and the vertical sequence in (9) that
1 2
2 2
H0(A⊗O (mH +K )⊗J )=ker(H0(F))= ker(H0(F ))= H0(Ω (mH +K +E)⊗J ).
R R Z/R i Ri Ri Zi
i=1 i=1
M M
Hence, by (10) again, we have H0(q) = H0(q )+H0(q ). The statement now follows by the isomorphism
1 2
ker(H0(q))≃ ⊕ ker(H0(q ) im(H0(q ))∩im(H0(q )) and the horizontal sequence in (9). (cid:3)
i i 1 2
(cid:16) (cid:17) (cid:16) (cid:17)
Lemma 2.6. Assume that L
2
(14) H0(q ) is surjective,
i
i=1
X
2
(15) H0(F ) is surjective
i
i=1
X
and
(16) h1(Ω (logE)(mH +K +E)⊗J )=0, for i=1,2.
Ri Ri Zi
Then h1(Λ1(mH +K )⊗J )=0.
R R Z/R
Proof. This follows from (10). (cid:3)
We summarize our main results as follows:
Proposition 2.7. Let S′ be a smooth, projective surface, H′ a line bundle on S′ and Z′ ⊂S′ a reduced zero-
dimensional scheme. Assume that (S′,H′,Z′) admits a semistable degeneration (R,H,Z) with R = R ∪R ,
1 2
R ,R smooth, with transversal intersection E =R ∩R , and Z =Z ∪Z , with Z ⊂R −E, i=1,2.
1 2 1 2 1 2 i i
If (11), (12) and (13) hold, one has h0(ΩS′(mH′+KS′)⊗JZ′/S′)=0.
If (14), (15) and (16) hold, one has h1(ΩS′(mH′+KS′)⊗JZ′/S′)=0.
Proof. This follows from the two preceding lemmas and Lemma 2.3. (cid:3)
3. The K3 case
In the rest of the note, we will show how to apply Proposition 2.7 to semistable degenerations of smooth,
primitively polarized K3 surfaces, thus giving, via (1), a new proof of Theorem 1.3.
3.1. A semistable degeneration of K3 surfaces. This degeneration is well known (see [6]) and we recall
it to fix notation.
Let p = 2n+ε >3 be an integer, with n > 1 and ε ∈{0,1}, and let E′ ⊂ Pp be a smooth, elliptic normal
curve of degree p+1. Consider two general line bundles L ,L ∈Pic2(E′) with L 6=L . In particular there
1 2 1 2
is no relation between L1,L2 and OE′(1) in Pic(E′).
We denote by R′ and R′ the rational normal scrolls of degree p−1 in Pp described by the secant lines of
1 2
E′ generated by the divisors in |L | and |L |, respectively. We have
1 2
P1×P1 if p=2n+1,
R′ ≃F =
i 1−ε (F1 if p=2n
(F is called the type of the scrolls R′ and R′) and R′ and R′ transversely intersect along E′, which is
1−ε 1 2 1 2
anticanonical on both (cf.[6]).
DEGENERATION OF DIFFERENTIALS AND MODULI OF NODAL CURVES ON K3 SURFACES 7
Denotingbyp :R′ →P1 thestructuralmorphismandbyσ andF asectionwithminimalself–intersection
i i i i
and a fiber of p , respectively, we have σ2 =ε−1, σ ·F =1, F2 =0 and Pic(R′)≃Z[σ ]⊕Z[F ]. One has
i i i i i i i i
(17) OR′(1)≃OR′(σi+nFi) and KR′ ∼−2σi−(3−ε)Fi ∼−E′,
i i i
and ΩR′ fits in the exact sequence
i
(18) 0 // p∗i(ωP1)≃OR′i(−2Fi) // ΩR′i //ΩR′i/P1 ≃OR′i(−E′+2Fi) // 0,
which splits if ε=1.
Set R′ :=R′ ∪R′. The first cotangent sheaf T1 (cf. [13, §1]) is the degree 16 line bundle on E′
1 2 R′
(19) TR1′ ≃NE′/R′1 ⊗NE′/R′2 ≃OE′(4)⊗(L1⊗L2)⊗(3−p),
the last isomorphism coming from (17).
The Hilbert point of R′ sits in the smooth locus of the component X of the Hilbert scheme whose general
p
point represents a smooth K3 surface of degree 2p−2 in Pp having Picard groupgeneratedby the hyperplane
section (cf.[6, Thms.1,2]).
The fact that T1 is non-trivial on E′ implies that R′ does not admit any semistable deformation (cf. [13,
R′
Prop.1.11]). Indeed, the total space of a generalflat deformation of R′ in Pp is singular along 16 points on E′
that are the zeros of a global section of T1 (cf.[13, §2]). More precisely if R′ −α→′ ∆ is a (general) embedded
R′
deformation of R′ in Pp corresponding to a (general) section τ ∈H0(R′,NR′/Pp), then the total space R′ has
double points at the 16 (distinct) points of the divisor W ∈|T1 |, with
R′
W :={sτ =0} where τ ∈H0(NR′/Pp)−κ→sτ :=κ(τ)∈H0(TR1′),
themapκbeingsurjective(see[6,Cor.1]). ByblowingupR′ alongthesesingularpointsandcontractingevery
exceptional divisor on one of the two irreducible components of the strict transformof R′, one obtains a small
resolution of singularities Π : R → R′ and a semistable degeneration R −α→ ∆ of K3 surfaces, with central
fiber R=R ∪R , where R =Π−1(R′), i=1,2. Then E :=R ∩R =Sing(R) is such that E′ =Π(E)≃E
1 2 i i 1 2
and T1 ≃ O . The curve E′ [resp. E] is anticanonical on R′ [resp. on R ], for i = 1,2, hence both R′ and R
R E i i
have trivial dualizing sheaf (see [13, Rem.2.11]). On R′ there is a line bundle H′ restricting to the hyperplane
bundle on each fiber. We set H=Π∗(H′).
Themapπ :=Π :R →R′ isthecontractionofk disjoint(−1)-curvese ,...,e ,suchthate ·E =1,
i |Ri i i i i,1 i,ki i,j
to distinct points x ,...,x on E′. We set W = x +···+x and e := ki e , for i = 1,2. Then
i,1 i,ki i i,1 i,ki i j=1 i,j
W =W +W ∈|T1 | is general, hence reduced.
1 2 R′ P
If R′ −α→′ ∆ is generalin the above sense, we will accordingly say that R−α→∆ is general and (R ,H ), for
t t
t6=0, can be thought of as the general point of K .
p
3.2. Technical lemmas. We willnowdeveloptoolstoverifytheconditions(11)-(13)and(14)-(16)inPropo-
sition 2.7.
Consider the relative cotangent sequence of the map π : R → R′, together with the dual of the exact
i i i
sequence defining its normal sheaf Nπi (c.f. e.g. [21, Ex.3.4.13(iv)]); we have ΩRi/R′i ≃ Ext1(Nπi,ORi) ≃
Ext1(Oei(−ei),ORi). One easily verifies that Ext1(Oei(−ei),ORi)≃⊕kj=i1ωei,j, whence
(20) 0 // πi∗(ΩR′i) //ΩRi //ΩRi/R′i ≃⊕kj=i1ωei,j //0.
On each (−1)–curve e on R , with i=1,2 and j =1,...,k , we can consider the two points Y and Y′
i,j i i i,j i,j
respectively cut out on e by the strict transformon R of the ruling of R′ through x and by E. Note that
i,j i i i,j
Y = Y′ if and only if x is a ramification point on E of the linear series |L |. This will not be the case
i,j i,j i,j i
if W ∈ |T1 | is general. We will consider the 0–dimensional scheme Y = kj Y on R , for i = 1,2. Since
R′ i j=1 i,j i
π (Y )=π (Y′ )=x , we have W =π (Y ), for i=1,2. Then:
i i,j i i,j i,j i i i P
Lemma 3.1. Let W ∈|T1 | be general. We have an exact sequence
R′
(21) 0 //O (−2π∗(F )) // Ω //O (−E+2π∗(F ))⊗J //0.
Ri i i Ri Ri i i Yi
Proof. To simplify notation, we set S = R ,S′ = R′,π = π , F = F , L = L , for i = 1,2. By the local
i i i i i
nature of the claim, we may and will assume that π :S →S′ is the blow–up at only one point w∈E′, which,
by the generality assumption, is not any of the four ramification points of the pencil |L|. Denote by e the
π–exceptional divisor.
8 C.CILIBERTO,F.FLAMINI,C.GALATI,ANDA.L.KNUTSEN
The cokernel of the injective map π∗(OS′(−2F)) → π∗(ΩS′) → ΩS obtained from (18) and (20) is torsion
freewithsecondChernclassc =1. Fordeterminantalreasons,itmustbeequaltoO (−E+2π∗(F))⊗J for
2 S Y
some Y ∈e. We will prove, with a local computation, that Y is the intersection of e with the strict transform
on S of the ruling of S′ passing through w. This will prove (21).
Choose a chartU ⊂S′ centered at w with coordinates (x,z), such that the structuralmap S →P1 is given
on U by (x,z) 7→ x and E′ has equation z = 0 (we can do this because w is not a ramification point of |L|).
Then ΩS′|U is generatedby dx,dz, the sheaf π∗(OS′(−2F))|U is generatedby dx and dz is the local generator
for the quotient line bundle.
Consider U ⊂ U ×P1 the blow-up of U at w = (0,0). If [ǫ,η] are homogeneous coordinates on P1, an
equation for U in U ×P1 is xη = ǫz. The open subset U where η 6= 0, sits in U ×A1 and has equation
0
x = tz there,ewhere t = ǫ is the coordinate on A1. So we have coordinates (t,z) on U and the equation for
η 0
e =e∩(U ×eA1) is z =0, whereas t=0 is the equation ofethe strict transform of the ruling on S′ though w.
0
In U we have dx=tdz+zdt so the injection in (18) gives e
0
e z
0 // O (cid:18) t (cid:19)//O⊕2(−t z)////O ,
U0 U0 U0
where: e e e
• O⊕2 ≃Ω is generated by dt, dz;
U0 U0
• thee incluseion OS(−2π∗(F))֒→ΩS is locally given in U0 by 17→zdt+tdz, and
• the image of the map (−t z) is the ideal generated by t and z in O ,
U0
hence Y has equation t=z =0, as wanted. e (cid:3)
e
For i=1,2, we denote by X ∈|2L | the ramification divisor of |L | on E′, or, by abuse of notation, on E.
i i i
Lemma 3.2. Same assumptions as in Lemma 3.1. Then the composed map
O (−2π∗(F ))֒→Ω −r→i ω ,
Ri i i Ri E
given by (5) and (21), is non–zero, for i=1, 2. Its image is the line bundle ω (−X ).
E i
Proof. We use the same simplified notationas in the proofofLemma 3.1. Accordinglywe will writeX for X .
i
Take a point P in X on E. By the generality assumption on W, R and R′ are isomorphic around P. So, if
we worklocally,we may do it onR′. Choosea chartU onR′ centeredatP, with coordinates (x,z), suchthat
the structural map S →P1 is given on U by (x,z)7→x and E′ has equation x=z2. So we may take z as the
coordinate on E′.
On U the sheaf injection in (21) is the same as the one in (18) which, by the proof of Lemma 3.1, is given
by 1 7→ dx. Composing this injection with the trace map on E′ gives 1 7→ dx = 2zdz, which shows that the
map is non–zero and its image is a differential form on E vanishing at P.
The proof is accomplished by making similar local computation at points P ∈E which are not on X. This
can be left to the reader. (cid:3)
By Lemmas 3.1 and 3.2, we have the following commutative diagram
(22)
0 0 0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //ORi(mH−2πi∗(Fi)−E)⊗JZi //ΩRi(logE)(mH−E)⊗JZi //ORi(mH−E+2πi∗(Fi))⊗JXi∪Yi∪Zi //0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //ORi(mH−2πi∗(Fi))⊗JZi //ΩRi(mH)⊗JZi //ORi(mH−E+2πi∗(Fi))⊗JYi∪Zi //0
qi
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
// // // //
0 ωE(mH)(−Xi) ωE(mH) OXi 0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 0 0
DEGENERATION OF DIFFERENTIALS AND MODULI OF NODAL CURVES ON K3 SURFACES 9
wherethesecondverticalandhorizontalexactsequencesarerespectively(5),(21)tensoredbyO (mH)⊗J ,
Ri Zi
and where we used the isomorphism O ≃ω .
E E
The composed injection O (mH −2π∗(F ))⊗J // Ω (mH)⊗J // Ω (logE)(mH)⊗J
Ri i i Zi Ri Zi Ri Zi
obtained from the above diagram and (4) tensored by O (mH)⊗J fits in the diagram
Ri Zi
(23) 0 0
(cid:15)(cid:15) (cid:15)(cid:15)
0 //ORi(mH−2πi∗(Fi))⊗JZi //ΩRi(mH)⊗JZi //ORi(mH−E+2πi∗(Fi))⊗JYi∪Zi //0
(cid:15)(cid:15) (cid:15)(cid:15)
0 //ORi(mH−2πi∗(Fi))⊗JZi //ΩRi(logE)(mH)⊗JZi //ORi(mH+2πi∗(Fi))⊗JXi∪Yi∪Zi //0
Fi
(cid:15)(cid:15) (cid:15)(cid:15)
OE(mH) OE(mH)
(cid:15)(cid:15) (cid:15)(cid:15)
0 0
We wantto describeim(H0(F ))andim(H0(q ))inthe casem=1. Tothis endwefirstdefine the following
i i
subspaces of H0(O (H))≃H0(ω (H)). Recalling conventionand notation as at the end of the introduction,
E E
from the right-most vertical sequence in (23), we set
(24) V :=H0(O (H +2π∗(F ))⊗J ) ⊆H0(O (H)), i=1, 2.
i Ri i i Xi∪Yi∪Zi |E E
The inclusion works as follows: take a (non–zero) section s ∈ V (which vanishes along a curve C containing
i
X ), restrict it to E, then divide by fixed local equations of the points in X (i.e., remove X from the divisor
i i i
cut out by C on E).
Similarly, from the left-most vertical sequence in (22), we define
(25) U :=H0(O (H −2π∗(F ))⊗J ) ⊆H0(ω (H)(−X ))⊂H0(ω (H)), for i=1,2.
i Ri i i Zi |E E i E
At divisor level, the inclusion is given by taking a divisor in |O (H −2π∗F )⊗J |, restricting it to E, and
Ri i i Zi
then adding the points X .
i
Lemma 3.3. If m=1, then
(26) im(H0(q ))=U ≃H0(O (H −2π∗(F ))⊗J )
i i Ri i i Zi
and
(27) im(H0(F ))⊆V ≃H0(O (H +2π∗(F ))⊗J ),
i i Ri i i Xi∪Yi∪Zi
with equality ifh1(O (H −2π∗(F ))⊗J )=0.
Ri i i Zi
Proof. This follows from (22) and (23) with m=1, and h0(O (H−E+aπ∗(F )))=0 for any integer a. (cid:3)
Ri i i
3.3. A new proof of Theorem 1.3. We consider V (R) the locally closed subscheme of the linear system
δ
|H| on R parametrizing the universalfamily of curves C ∈|H| having only nodes as singularities, exactly δ of
them (called the marked nodes) off the singular locus E of R, and such that the partial normalization of C at
the δ marked nodes is connected, i.e., the marked nodes are non-disconnecting nodes (cf. [4, §1.1]). Under a
semistabledeformationα:R→∆ofRasinDefinition2.1,itispossibletodeformsuchacurveC toaδ-nodal
curveonthe fibresR ofα:R→∆, fort6=0,preservingits markednodes andsmoothingthe remainingg+1
t
nodes of C located at its intersection with E (see [4, Lemma1.4]).
Usually we will assume the deformation α : R → ∆ to be general. Then, given C ∈ V (R), we may find a
δ
pair (C′,S′) general in some component V ⊆V , where (S′,H′) is general in K and C′ ∈V (S′,H′), such
1,δ p 1,δ
that C is a flat limit of C′. Let Z [resp. Z′] be the scheme of the δ marked nodes of C [resp. of C′]. Then
(possiblyaftershrinking∆further)thetriple(R,H,Z)isasemistabledegenerationof(S′,H′,Z′)andwemay
apply Proposition2.7to showthe desiredvanishingsin(1)neededtoproveTheorem1.3. Forthis weneedthe
following result, whose proof we postpone until the next section.
Proposition 3.4. There exist W ∈|T1 | and C ∈V (R), with Z its scheme of δ marked nodes, such that:
R′ δ
(i) if g :=p−δ >15, then the maps 2 H0(q ) and 2 H0(F ) are surjective and
i=1 i i=1 i
h1(ORi(H +2πi∗(Fi))⊗JXPi∪Yi∪Zi)=h1(ORPi(H −2πi∗(Fi))⊗JZi)=0, for i=1,2;
10 C.CILIBERTO,F.FLAMINI,C.GALATI,ANDA.L.KNUTSEN
(ii) if g :=p−δ 67, then
im(H0(F ))∩im(H0(F ))=im(H0(q ))∩im(H0(q ))={0}.
1 2 1 2
Proof of Theorem 1.3. We will prove the desired vanishings in (1) using Proposition 2.7.
If g >15, conditions (14) and (15) in Proposition 2.7 are satisfied by Proposition 3.4(i), whereas condition
(16) is satisfied by the middle horizontalsequence in (23) and the vanishings of h1 in Proposition 3.4(i).
If g 6 7, conditions (11) and (12) in Proposition 2.7 are satisfied by Proposition 3.4(ii), whereas condition
(13) is satisfiedby the upper horizontalsequence in (22) and the fact that h0(O (H−E±2π∗(F )))=0, for
Ri i i
i=1,2. (cid:3)
4. Proof of Proposition 3.4
With a slight abuse of terminology, we will call lines the curves on R in the pencil |π∗(F )|, for i = 1,2.
i i
The following component of V (R) has been introduced in [5].
δ
Definition 4.1. For any 06δ 6p−1, we define W (R) to be the set of curves C in V (R) such that:
δ δ
(i) C does not contain any of the exceptional curves e of the contractions π :R →R′, i=1,2;
i,j i i i
(ii) C has exactly δ := ⌊δ⌋ nodes on R −E and δ := ⌈δ⌉ nodes on R −E, hence it splits off δ lines on
1 2 1 2 2 2 i
R , for i=1,2;
i
(iii) the union of these δ =δ +δ lines is connected.
1 2
For any curve C in W (R), we denote by C the connected union of δ lines as in (iii), called the line chain of
δ
length δ of C, and by γ the irreducible component of the residual curve to C on R , for i=1,2.
i i
It has been provedin [5, Prop. 4.2] that W (R) is a smoothopen subset of a component of V (R), with the
δ δ
nodes described in (ii) as the marked nodes of any of its members. We write W (R′) for the set of images in
δ
R′ of the curves in W (R). Without further notice, we will denote the image of C ∈W (R) and its line chain
δ δ
C and components γ by C′, C′ and γ′, respectively.
i i
Members ofW (R)areshownin Figure1 below. The points P andQin the picture (the starting point and
δ
P
P
(δ−1)/2nodes δ/2nodes
(δ+1)/2n odes δ/2 nodes
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
Q
Q
E E
}
}
g pts
g pts
Figure 1. Members of W (R) when δ is odd (left) and even (right)
δ
the end point of C ) satisfy the following relation on E:
δ+1 δ−1
(28) P +Q ∼ L − L , when δ is odd;
2 1
2 2
δ
(29) P −Q ∼ (L −L ), when δ is even.
2 1
2
We denote by c the intersection of C with E, considered as a reduced divisor on E. This consists of δ+1
points, i.e., P +Q plus the δ−1 double points of C which are all located on E. One has:
δ−1
(30) c ∼ P +Q+ L when δ is odd;
1
2
δ δ
(31) c ∼ P + L ∼Q+ L when δ is even.
1 2
2 2
Recalling (24), we have: