The Wahl map of one-nodal curves on K3 surfaces

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by  Edoardo Sernesi

THE WAHL MAP OF ONE-NODAL CURVES ON K3
SURFACES
7
1
EDOARDO SERNESI
0
2

n
a
J
Abstract. We consider a general primitively polarized K3 sur-

7
face (S,H) of genus g+1 and a 1-nodal curve C ∈|H|. We prove
1
that the normalization C of C has surjective Weahl map provided
] g =40,42 or ≥44.
G e
A
.
h
To L. Ein on the occasion of his 60-th birthday
t
a
m
[
Introduction

1
v In this Note we consider 1-nodal curves lying on a K3 surface and
1
we study the gaussian map, or Wahl map, on their normalization. If
0
8 we consider a primitive linear system |H| on a K3 surface S, then it is
4
well known that every nonsingular C ∈ |H| has a non-surjective Wahl
0
. map
1 e
0 2
7 Φ : H0(C,ω ) −→ H0(C,ω3)
K C C
1
^
:
v (see §1 for the definition) and that, if moreover C ∈ |H| is general
i
X then it is Brill-Noether-Petri general [W2, L]. It is of some interest
e
r to decide whether the same properties hold for the normalization C
a
of a 1-nodal C ∈ |H|, and more generally for the normalization of a
singular C in |H|. The Brill-Noether theory of singular curves on a
e
K3 surface has received quite a lot of attention in recent times, see
e
e.g. [Go, FKP, BFT, CK, Ke]. On the other hand to our knowledge
very little is known on their Wahl map. In [Ke] and [H2] the authors
consider a modified version of the Wahl map, which does not seem to
have a direct and simple relation with the ordinary Wahl map Φ ;
K
in particular their results point towards the non-surjectivity of such
modified map. On the other hand in [FKPS] it is proved that the
normalization C (of genus 10) of a general 1-nodal curve C ∈ |H| on
a general polarized (S,H) of genus 11 has general moduli; then the
e
1
2 EDOARDOSERNESI
main result of [CHM] implies that Φ is surjective for such a curve.
K
This surjectivity result isextended inthe present paper inthefollowing
form:
Theorem 1. Let (S,H) be a general primitively polarized K3 surface
of genus g + 1. Assume that g = 40,42 or ≥ 44. Let C ∈ |H| be a
1-nodal curve and C its normalization. Then the Wahl map
e
2
Φ : H0(C,ω ) −→ H0(C,ω3)
K C C
^
is surjective.
This of course gives yet another proofof themainresult of [CHM] for
the values of g as in the statement because 1-nodal curves are known
to exist in |H| for a general primitively polarized (S,H) of any genus
g +1 ≥ 2 [MM, Ch].
Now a few words about the method of proof. Letting P ∈ S be
the unique singular point of C we consider the blow-up σ : X :=
Bl S −→ S at P and we let E ⊂ X be the exceptional curve. Then
P
e
the normalization of C is the strict transform C = σ∗C − 2E ⊂ X.
The Wahl map Φ on C can be decomposed a
K
e e
Φ = H0(ρ)·Φ
K KX+C
where Φ is a gaussian map on X and H0(ρ) is induced in coho-
KX+C
mology by a restriction homomorphism:
ρ : Ω1 (2K +2C) −→ ω3
X X C
on X. We study these two maps and prove their surjectivity sepa-
rately. This method of proof is analogous to the one adopted in the
work of several authors before, notably [W1, DM, CLM1, CLM2]. The
restriction on the genus depends on the proof: one would expect the
result to hold for g = 10 (as it does indeed, as already remarked)
and for g ≥ 12. In fact the surjectivity of H0(ρ) holds for g = 10 or
g ≥ 12 (see Lemma 2). On the other hand the proof of the surjec-
tivity of Φ , which consists in adapting an analogous proof given
KX+C
in [CLM2] for plane curves, leads to the restrictions on g: in fact this
proof requires that we decompose a certain divisor on X as the sum
of three very ample ones and this decomposition forces the genus to
increase.
RecentworkbyM.Kemeny[Ke]impliesthatthecurvesC considered
here, i.e. normalizations of 1-nodal curves on a general primitive K3
surface, are generically Brill-Noether-Petri general and fill a locus in
M , the coarse moduli space of curves of genus g, whose closure has
g
THE WAHL MAP OF ONE-NODAL CURVES ON K3 SURFACES 3
dimension19+g. Theorem1and[W2]implythatthisnaturallydefined
locusisnotcontainedintheso-calledK3-locus(i.e. thelocusofsmooth
curves that can be embedded in a K3 surface).
One can ask whether a result analogous to Theorem 1 can be proved
for the normalization of curves on K3 surfaces having a more compli-
cated singular point. We did not consider this case. Note though that,
to our knowledge, such curves are known to exist only in the case of
A -singularities or ordinary triple points (see [GK, Ga]).
k
The paper is organized as follows. In §1 we introduce the gaussian
maps and explain the strategy of proof of the surjectivity of the Wahl
mapofacurve lying inaregularsurface. In§2weprovethesurjectivity
of H0(ρ) and in §3 we prove the surjectivity of Φ . We work over
KX+C
C.
Acknowledgements - We thank M. Halic and M. Kemeny for
helpful e-mail correspondence, and A. L. Knutsen for very useful com-
ments on a preliminary version of this work. The author is member of
GNSAGA-INDAM.
1. Generalities on Gaussian maps
In this section we recall a few definitions and basic facts concerning
gaussian maps. Given line bundles L,M on a nonsingular projective
variety Y we consider:
R(L,M) = ker[H0(Y,L)⊗H0(Y,M) → H0(Y,L⊗M)]
Then we have a canonical map:
Φ : R(L,M) −→ H0(Y,Ω1 ⊗L⊗M)
L,M X
called the gaussian map, or Wahl map, of L,M, which is defined as
follows. Let ∆ ⊂ Y ×Y be the diagonal and p ,p : Y ×Y → Y the
1 2
projections. Then
R(L,M) = H0(Y ×Y,p∗L⊗p∗M ⊗I )
1 2 ∆
Since I ⊗O = Ω1 , the restriction to ∆:
∆ ∆ Y
p∗L⊗p∗M ⊗I −→ p∗L⊗p∗M ⊗I ⊗O
1 2 ∆ 1 2 ∆ ∆
induces Φ on global sections. The exact sequence:
L,M
0 // p∗L⊗p∗M ⊗I2 // p∗L⊗p∗M ⊗I // p∗L⊗p∗M ⊗I ⊗O // 0
1 2 ∆ 1 2 ∆ 1 2 ∆ ∆
shows that the vanishing:
H1(Y ×Y,p∗L⊗p∗M ⊗I2) = 0
1 2 ∆
is a sufficient condition for the surjectivity of Φ .
L,M
4 EDOARDOSERNESI
In case L = M we have R(L,L) = I (Y)⊕ 2H0(Y,L), where
2
I (Y) = ker[S2H0(Y,L) → H0V(Y,L2)]
2
andΦ iszero onI (Y). Therefore Φ isequivalent to itsrestriction
L,L 2 L,L
to 2H0(Y,L), which is denoted by
V 2
Φ : H0(Y,L) −→ H0(Y,Ω1 ⊗L2)
L X
^
In particular, for a non-hyperelliptic curve C we are interested in Φ
K,K
or rather in
2
Φ : H0(C,ω ) −→ H0(C,ω3)
K C C
^
where O (K) = ω is the canonical invertible sheaf.
C C
Suppose that C ⊂ X where X is a nonsingular regular surface.
Then H0(C,ω ) = H0(X,K + C) and Φ can be described as a
C X K
composition:
(1) Φ = H0(ρ)·Φ
K KX+C
where
2 2
Φ : H0(C,ω ) = H0(X,O (K +C)) −→ H0(X,Ω1 (2K +2C))
KX+C C X X X X
^ ^
and ρ : Ω1 (2K +2C) → ω3 is the restriction map. Therefore if both
X X C
H0(ρ) and Φ are surjective, the surjectivity of Φ will follow.
KX+C K
This is how we will prove it in our situation.
2. The surjectivity of H0(ρ)
As in the Introduction, we let (S,H) be a K3 surface with a polar-
ization of genus g +1 ≥ 3 and let C ∈ |H| be a curve with one node,
i.e. an ordinary double point, at P ∈ S and no other singularities.
e
Consider the blow-up σ : X := Bl S −→ S of S at P, let E ⊂ X be
P
the exceptional curve and C = σ∗C −2E ⊂ X the strict transform of
C. Then we have a homomorphism of sheaves
e
e ρ : Ω1X(2KX +2C) −→ ωC3
induced by the natural restriction Ω1 → ω . We prove:
X C
Lemma 2. Suppose that (S,H) is a general primitively polarized K3
surface of genus g+1, with g = 10 or g ≥ 12. Then, with the notations
introduced above, the map:
H0(ρ) : H0(X,Ω1 (2K +2C)) −→ H0(C,ω3)
X X C
is surjective.
THE WAHL MAP OF ONE-NODAL CURVES ON K3 SURFACES 5
Proof. We have an exact sequence on X:
0 // Ω1 (log C)(−C) // Ω1 // ω // 0
X X C
where Ω1 (log C) is the sheaf of 1-forms with logarithmic poles along
X
C [EV]. Tensoring with O(2K +2C) we obtain:
X
0 // Ω1 (log C)(2K +C) // Ω1 (2K +2C) ρ // ω3 // 0
X X X X C
Since K = E we have O (2K + C) = σ∗H and therefore in order
X X X
to prove the Lemma it suffices to prove that
(2) H1(X,Ω1 (log C)⊗σ∗H) = 0
X
Consider the relative cotangent sequence of σ:
0 // σ∗Ω1 // Ω1 η // ω // 0
S X E
and tensor it by σ∗H:
0 // σ∗[Ω1 ⊗H] // Ω1 ⊗σ∗H ηH // ω // 0
S X E
We have:
H2(X,σ∗[Ω1 ⊗H]) = H2(S,Ω1 ⊗H) = H0(S,T ⊗H−1)∨ = 0
S S S
Fromtheassumption aboutthegenus andfromthegenerality of(S,H)
it follows that we also have:
H1(X,σ∗[Ω1 ⊗H]) = H1(S,Ω1 ⊗H) = 0
S S
(see [B], 5.2). Therefore η induces an isomorphism:
H
H1(η ) : H1(X,Ω1 ⊗σ∗H) ∼= H1(E,ω ) ∼= C
H X E
Then from the exact sequence:
0 // Ω1 ⊗σ∗H // Ω1 (log C)⊗σ∗H // σ∗H ⊗O // 0
X X C
ω (E)
C
we see that, since H1(C,ω (E)) = 0, in order to prove (2) it suffices
C
to show that the coboundary map:
(3) ∂ : H0(C,ω (E)) −→ H1(X,Ω1 ⊗σ∗H)
H C X
is non-zero.
6 EDOARDOSERNESI
We can argue as follows. Consider the commutative diagram:
0 // Ω1 // Ω1 (log C) // O // 0
X X C
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // Ω1 ⊗σ∗H // Ω1 (log C)⊗σ∗H // ω (E) // 0
X X C
where the vertical maps are induced by multiplication with a section
0 6= s ∈ H0(X,σ∗H). Then we have the following commutative dia-
gram involving the coboundary maps of both sequences:
H0(C,O ) ∂ // H1(X,Ω1 )
C X❘❘❘❘❘❘H❘❘1❘(η❘)❘❘❘((
s s H1(E,ω )
(cid:15)(cid:15) (cid:15)(cid:15) ❧❧❧❧❧H❧❧1❧(❧η❧H❧)❧❧66 E
H0(C,ω (E)) ∂H // H1(X,Ω1 ⊗σ∗H)
C X
and it suffices to prove that
H1(η)·∂ : H0(C,O ) → H1(E,ω )
C E
is non-zero.
Observe that ∂ associates to 1 ∈ H0(C,O ) the Atiyah-Chern class
C
ofO (C)andH1(η)(∂(1))isitsrestrictiontoE. Wehavethatker(H1(η)) ∼=
X
H1(X,σ∗Ω1) is generated by the Atiyah-Chern classes of curves com-
S
ing from S, which are trivial when restricted to E. Since E·C = 2 we
see that ∂(1) ∈/ ker(H1(η)) and this proves the Lemma. (cid:3)
3. The gaussian map on X
We keep the notations of §2. We will prove the following:
Proposition 3. Suppose that (S,H) is a general primitively polarized
K3 surface of genus g +1, with g = 40,42 or ≥ 44; let C ∈ |H| be a
1-nodal curve and C is normalization. Let X = Bl (S) where P ∈ C
P e
is the node. Then the gaussian map
e
2
Φ : H0(X,O (K +C)) −→ H0(X,Ω1 (2K +2C))
KX+C X X X X
^
is surjective.
We will need the following result and its corollary:
THE WAHL MAP OF ONE-NODAL CURVES ON K3 SURFACES 7
Proposition 4. a) For every d ≥ 10 there is a K3 surface S containing
two very ample nonsingular curves A,B such that Pic(S) = Z[A] ⊕
Z[B], with intersection matrix:
A2 A·B 8 d
=
(cid:18)B ·A B2 (cid:19) (cid:18)d 8(cid:19)
and A and B non-trigonal.
b) For every d ≥ 12 there is a K3 surface S containing two very
ample nonsingular curves A,B such that Pic(S) = Z[A] ⊕ Z[B], with
intersection matrix:
A2 A·B 10 d
=
(cid:18)B ·A B2 (cid:19) (cid:18)d 8(cid:19)
and A and B non-trigonal.
In both cases (a) and (b) the surface S does not contain rational
nonsingular curves R such that A·R = 1 or B ·R = 1.
Proof. The Proposition is a special case of [Kn], Theorem 4.6. We
obtainthePropositionbytaking (withthenotationsused there) (n,g−
1) = (4,4) and (n,g − 1) = (5,4) respectively. The restriction on d
is forced by the requirement that the hypotheses of the theorem apply
symmetrically w.r. to A an B so that both are very ample. The non-
trigonality follows from the fact that S is embedded by both |A| and
|B| so to be an intersection of quadrics. The last assertion is proved as
done in loc. cit. for (−2) curves, by comparing discriminants. (cid:3)
Corollary 5. If S is as in Prop. 4(a) then H = A + 2B defines a
primitive very ample divisor class of genus g +1 = 21+ 2d for every
d ≥ 10. If S is as in Prop. 4(b) then H = A+2B defines a primitive
very ample divisor class of genus g +1 = 22+2d for every d ≥ 12.
Proof. ClearlyH isveryampleanditisprimitivebecausethegenerator
A of Pic(S) appears with coefficient 1. The genus in either case is
(cid:3)
readily computed using the intersection matrix.
Proof. of Proposition 3. By semicontinuity it suffices to prove the
Proposition for just one primitively polarized K3 surface for each value
of g as in the statement. We take (S,H) as in Corollary 5, distinguish-
ing cases (a) and (b) according to the parity of g. Letting C,C,P and
X as in the statement, consider the product X ×X and the blow-up
e
π : Y = Bl (X × X) → X × X along the diagonal ∆. Let Λ ⊂ Y
∆
be the exceptional divisor. For any coherent sheaf F on X we define
F = (p ·π)∗F, i = 1,2. It suffices to prove that
i i
H1(X ×X,p∗O (K +C)⊗p∗O (K +C)⊗I2) = 0
1 X X 2 X X ∆
8 EDOARDOSERNESI
which is equivalent to:
H1(Y,(K +C) +(K +C) −2Λ) = 0
X 1 X 2
Note that we have:
(K +C) +(K +C) −2Λ = K +C +C −3Λ
X 1 X 2 Y 1 2
and therefore we want to prove that:
(4) H1(Y,L) = 0
where we set L = O (K +C +C −3Λ). The proof is an adaptation
Y Y 1 2
of the proofs of Lemmas (3.1) and (3.10) of [CLM2]. One uses the
following:
Lemma 6. Assume that D is a very ample divisor on X. Then D +
1
D −Λ is big and nef on Y.
2
(cid:3)
Proof. See [BEL], Claim 3.3.
Let M := σ∗H(−3E) on X. We have:
M = σ∗A(−E)+2(σ∗B(−E))
Since A and B are non-trigonal S has no trisecant lines through P and
does not contain a line through P whether it is embedded by |A| or
by |B| (Proposition 4). Therefore every curvilinear subscheme of S of
length 3 containing P imposes independent conditions to both |A| and
|B|. Then it follows from [Co], Prop. 1.3.4, that both σ∗A(−E) and
σ∗B(−E) are very ample on X. Therefore by the Lemma we have that
both σ∗A(−E) +σ∗A(−E) −Λ and σ∗B(−E) +σ∗B(−E) −Λ are
1 2 1 2
big and nef. It follows that the divisor M +M −3Λ is big and nef,
1 2
being the sum of three big and nef divisors.
Since C ∼ M + E we have C + C ∼ M + M + E + E and
1 2 1 2 1 2
therefore we have an exact sequence on Y:
0 → O (K +M +M −3Λ) → L → O (L) → 0
Y Y 1 2 E1+E2
By Kawamata-Vieweg we have H1(Y,K +M +M −3Λ) = 0: there-
Y 1 2
fore in order to prove (4) it suffices to show that
(5) H1(E +E ,O (L)) = 0
1 2 E1+E2
Letting W := E ∩E we have an exact sequence:
1 2
0 → O (L−W) → O (L) → O (L) → 0
E1 E1+E2 E2
and, by symmetry, it suffices to prove that:
(6) H1(E ,O (L−W)) = 0
1 E1
and
H1(E ,O (L)) = 0
1 E1
THE WAHL MAP OF ONE-NODAL CURVES ON K3 SURFACES 9
We can then consider the exact sequence on E :
1
0 → O (L−W)) → O (L) → O (L) → 0
E1 E1 W
and finally we are reduced to prove (6) and
(7) H1(W,O (L)) = 0
W
∼
Let U = E × E be the proper transform of E × E in Y. Then in
E we have W = U + Λ . As in [CLM2], proof of Lemma (3.1),
1 |E1
one shows that Λ|E1 ∼= PE, where E = OP1(1)⊕OP1(−2), and L|PE =
OPE(2C0 +2f) where C0 ∈ |OPE(1)| and f is a fibre of PE → P1. Now
theproofproceedsasin[CLM2], Lemma(3.1), afterhavingprovedthat
O (L−W)) = O (K +M +M −3Λ) is big and nef. This last fact
E1 E1 Y 1 2
is obtained exactly as in the proof of Lemma (3.10) of [CLM2], using
the fact that both σ∗A(−E) and σ∗B(−E) are very ample on X. (cid:3)
Proof. of Theorem 1. Recalling (1), the theorem follows immediately
(cid:3)
from Lemma 2 and from Proposition 3.
References
[BFT] Ballico E., Fontanari C., Tasin L.: Singular curves on K3 surfaces.
Sarajevo J. of Math. 6 (2010), 165-168.
[B] BeauvilleA.: Fano threefolds andK3surfaces. Proceedings of the Fano
Conference-Torino2002.EditedbyA.Collino,A.Conte,M.Marchisio
(2004).
[BEL] Bertram A., Ein L., Lazarsfeld R.: Vanishing theorems, a theorem of
Severi,andtheequationsdefiningprojectivevarieties.J.AMS 4(1991),
587-602.
[Ch] Chen X.: Nodal curves on K3 surfaces. arXiv:1611.07423.
[CHM] Ciliberto C, Harris J., Miranda R.: On the surjectivity of the Wahl
map. Duke Math. J. 57 (1988), 829-858.
[CK] Ciliberto C., Knutsen A. L.: On k-gonal loci in Severi varieties on
general K3 surfaces and rational curves on hyperk¨ahler manifolds. J.
Math. Pures Appl. (9) 101 (2014), no. 4, 473?494.
[CLM1] Ciliberto C, Lopez A.F., Miranda R.: On the corank of gaussian maps
for general embedded K3 surfaces. In Israel Mathematical Conference
Proceedings.PapersinhonorofHirzebruch’s65thbirthday,vol.9,AMS
Publications (1996), 141-157.
[CLM2] Ciliberto C, Lopez A.F., Miranda R.: On the Wahl map of plane
nodal curves. In Complex Analysis and Algebraic Geometry, a volume
in Memory of Michael Schneider. Thomas Peternell and Frank-Olaf
Schreyer, editors. Walter de Gruyter, Berlin/New York 2000. p. 155-
163.
[Co] Coppens M.: Very ample linear systems on blowings-up at general
points of smooth projectivevarieties. Pacific J. Math. 202 (2002),313-
327.
10 EDOARDOSERNESI
[DM] Duflot J., Miranda R.: The gaussian map for rational ruled surfaces.
Trans. AMS 330 (1992), 447-459.
[EV] H. Esnault, E. Viehweg: Lectures on Vanishing Theorems, DMV Sem-
inar, vol. 20, Birkhauser Verlag (1992).
[FKP] FlaminiF.,KnutsenA.L.,PacienzaG.: SingularcurvesonaK3surface
and linear series on their normalizations. International J. of Math. 18
(2007), 671-693.
[FKPS] Flamini F., Knutsen A.L., Pacienza G., Sernesi E.: Nodal curves with
generalmodulionK3surfaces.Comm.inAlgebra 36(2008),3955-3971.
[Ga] Galati, C.: On the existence of curves with a triple point on a K3
surface.AttiAccad. Naz. Lincei Rend. Lincei Mat. Appl.23(2012),no.
3, 295?317.
[GK] Galati, C., Knutsen, A. L.: On the existence of curves with Ak-
singularities on K3 surfaces. Math. Res. Lett. 21 (2014), no. 5,
1069?1109.
[Go] G´omez, T. L.: Brill-Noether theory on singular curves and torsion-free
sheaves on surfaces. Comm. Anal. Geom. 9 (2001), no. 4, 725?756.
[H1] Halic M.: Modular propertiesof nodalcurvesonK3 surfaces.Math. Z.
270 (2012), 871-887.
[H2] Halic M.: Erratum to: Modular properties of nodal curves on K3 sur-
faces. Math. Z. 280 (2015), 1203-1211.
[Ke] KemenyM.: ThemoduliofsingularcurvesonK3surfaces.J. de Math.
Pures et appliqu´ees 104 (2015), 882-920.
[Kn] SmoothcurvesonprojectiveK3surfaces.Mathematica Scandinavica 90
(2002) 215-231.
[L] Lazarsfeld R.: Brill-Noether-Petri without degenerations. J. Diff.
Geom. 23 (1986) 299- 307.
[MM] Mori S., Mukai S.: The uniruledness of the mduli space of curves
of genus 11. Algebraic Geometry. Proceedings Tokyo/Kyoto, 334-353.
Springer LNM 1016 (1983).
[W1] Wahl J.: Gaussian maps on algebraic curves. J. Diff. Geometry 32
(1990), 77-98.
[W2] Wahl J.: The cohomology of the square of an ideal sheaf. J. Algebraic
Geom. 6 (1997), 481-511.
E-mail address: [email protected]
Dipartimento di Matematica e Fisica, Universita` Roma Tre, L.go
S.L. Murialdo 1, 00146 Roma.