cubo2.dvi


CUBO A Mathematical Journal
Vol.12, No¯ 01, (103–114). March 2010

Brill-Noether Theories for Rank 1 Sheaves on Mg

E. Ballico ∗

Dept. of Mathematics, University of Trento,

38050 Povo (TN), Italy

email : ballico@science.unitn.it

ABSTRACT

Here we discuss some Brill-Noether problems for rank 1 sheaves on stable curves.

RESUMEN

Nosotros discutimos aquí algunos problemas de Brill-Noether para hazes de rango 1 sobre

curvas estables.

Key words and phrases: Stable curve, reducible curve, Brill-Noether theory.

Math. Subj. Class.: 14H10, 14H51.

We stress the plural “Brill-Noether theories” in the title. In section 1 we consider reducible

curves with large genus with degree 1 spanned sheaves (one could say that they have gonality 1).

Since these are our easiest results, we state them in the introduction, before going on to other

possible meanings of the word “gonality”.

Let X be a connected projective curve such that each point of X lying on at least two irre-

ducible components of X is an ordinary node of X. Let Sing(X)′′ be the set of all points of X lying

∗The author was partially supported by MIUR and GNSAGA of INdAM (Italy).



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on at least two irreducible components of X. By assumption every point of Sing(X)′′ is an ordinary

node of X. We assume g := pa(X) ≥ 2. Let B(X) denote the set of all irreducible components of

X. For the elementary properties of depth 1 coherent sheaves on reduced curves, see [21], parts

VII and VIII. A coherent sheaf F on X has depth 1 if and only if no non-zero subsheaf of F is

supported by a finite set. Hence on a reduced curve depth 1 sheaves are often called torsion-free

sheaves. We say that a depth 1 sheaf F on X has pure pure rank 1 if its restriction to Xreg is a

line bundle. Let F be sheaf on X with pure rank 1 and with depth 1. Set Sing(F ) := {P ∈ X : F

is not locally free at P }. Hence Sing(F ) ⊆ Sing(X). The degree deg(F ) of F may be defined by

the Rieman-Roch formula χ(F ) = deg(F ) + χ(OX ), even if the curve is not connected. Let Sk(X)

denote the set of all spanned coherent sheaves on X with depth 1, pure rank 1 and degree k. Set

Dk(X) := Sk(X) ∩ Pic(X).

Theorem 1. Let X be a connected projective curve such that each point of X lying on at least

two irreducible components of X is an ordinary node of X. Assume g := pa(X) ≥ 2. Let S(X)

be the set of all T ∈ B(X) such that T ∼= P1 and X\T has ♯(T ∩ Sing(X)) connected components.

D1(X) 6= ∅ if and only if S(X) 6= ∅. There is a bijection between S(X) and D1(X) constructed

in the following way. Take T ∈ S(X) and an automorphism φ : T → P1. There is a unique

spanned degree 1 line bundle LT on X such that h
0(X, LT ) = 2 and the morphism hLT : X → P

1

induces φ on T , while it sends the connected components Y1, . . . , Ys, s := ♯(T ∩ Sing(X)), of X\T

respectively to the points φ(T ∩ Y1), . . . , φ(T ∩ Ys). If D1(X) 6= ∅, then for every integer d ≥ 2

there is L ∈ Dd(X) such that h
0(X, L) = d + 1.

Theorem 2. Let X be a connected projective curve such that each point of X lying on at least

two irreducible components of X is an ordinary node of X. Assume g := pa(X) ≥ 2. There is

a bijection between the set D(X) of all disconnecting nodes and S1(X)\D1(X) constructed in the

following way. For each P ∈ D(X) let fP : XP → X be the partial normalization of X in which

we only normalize the point P . Then fP ∗(OXP ) ∈ S1(X)\D1(X). Any F ∈ S1(X)\D1(X) has a

unique singular point, this singular point is a disconnecting node, and h0(X, F ) = 2.

In Theorems 1 and 2 we just required that the sheaves are spanned. The universal propety of

projective spaces say that spanned line bundles are essentially equivalent to morphisms to projective

spaces. In section 2 we discuss what happens if we add the condition that the morphism contracts

no irreducible component, i.e. if the spanned line bundle is ample. We only consider P1 as the

target and study the deformation theory of a morphism f : X → P1 which does not contract any

irreducible component of X.

The main contender for the title of “best Brill-Noether theory” is the one introduced in [6]:

line bundles (or sheaves) satisfying the so-called Basic Inequality. For them see section 3. See [7]

for the case of binary curves, i.e. genus g stable curves with two irreducible components, both of

them smooth and rational. In the introduction we give the combinatorial data which describe all

topological types for stable curves with a fixed genus.

For any nodal curve X let Sing(X)′ (resp. Sing(X)′′) be the set of all singular points of X

lying on exactly one (resp. two) irreducible components of X. To any nodal projective curve X



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Brill-Noether Theories for Rank 1 Sheaves ... 105

we associate the following non-oriented marked graph ‖X‖. There is a bijection between vertices

of ‖X‖ and the set B(X) of all irreducible components of X. For any T ∈ B(X) let [T ] denote

the associated vertex of ‖X‖. For each T ∈ B(X) we give as a marking the non-negative integer

qT , where qT is the geometric genus of T . ‖X‖ contains ♯(Sing(X)
′ ∩ T ) loops with [T ] as their

vertex. For all T, T ′ ∈ B(X), such that T 6= T ′ the vertices [T ] and [T ′] of ‖X‖ are joined by

♯(T ∩ T ′) edges. Call τ the abstract marked graph ‖X‖. The set of all nodal projective curves Y

such that ‖Y ‖ ∼= τ (as marked graphs) is parametrized by an irreducible algebraic variety M[τ ].

If K = C, then the topological type of the complex analytic space X(C) is uniquely determined by

the marked graph τ and two non-isomorphic marked graphs give topologically different complex

analytic spaces. If we forget the marking, i.e. if we forget the integer qT , T ∈ B(X), then ‖X‖

becomes the classical dual graph of the nodal curve X. Fix a topological type τ for nodal connected

curves, say τ = ‖X‖. For all T, J ∈ B(X), T 6= J let qT ≥ 0 be the associated marking, aT the

number of looops based at T and aT J := ♯(T ∩ J). For every T ∈ B(X) fix an integer dT and set

d := (dT )T ∈B(X) and call d a multidegree for τ or for every curve A ∈ M[τ ]. We say that d is

positive and write d > 0 if dT > 0 for all T ∈ B(X). A natural question is to study the Brill-Nother

theory (for all multidegrees, not just for the total degree of the sheaf or line bundle) of a general

element of M[τ ]. See [7] for the case of binary curves

We conclude by giving examples of stable curves X such that ωX contains no spanned line

bundle (Section 4). We don’t touch a very important topic: the limit linear series introduced by

D. Eisenbud and J. Harris for nodal curves of compact type, i.e. such that ‖X‖ has no loop and no

multiple edge ([11]). See [18] for the positive characteristic case and [13] for the case of nodal curves

with two smooth irreducible components. A quick glance at G. Farkas’s survey on the geometry

of Mg and at the references quoted theirin shows the importance of this theory. However, a quick

look at the examples in [11] (resp. [13]) and at Example 2 (resp. Example 3) here shows that

the Brill-Noether theory coming from the limit linear series has nothing in common with the one

considered here in section 1 or the one studied by L. Caporaso in [7].

Theorems 1 and 1 and Section 1 are contained in [2] (in which also degree 2 sheaves are

considered). Section 2 is contained in [3].

1 Curves with Gonality 1

Remark 1. Let X be a reduced and quasi-projective curve, P ∈ X, and F a sheaf on X with pure

rank 1 and depth 1. The germ FP of F at P is a torsion free OX,P -module with rank 1. Hence

there exists an inclusion j : FP →֒ M with M a free OX,P -module with rank 1. The minimal

integer dimK(M/FP ) for all such pairs (j, M ) is an important invariant of the germ FP . Call

ℓ(F, P ) this integer. We have ℓ(F, P ) ≥ 0 and ℓ(F, P ) = 0 if and only if FP is a free OX,P -module.

This invariant may be computed on the formal completion of OX,P . Let mX,P be the maximal

ideal of the local ring OX,P . Notice that mX,P is a free OX,P -module if and only if P ∈ Xreg.

Hence if P ∈ Sing(X) and FP ∼= mX,P , then ℓ(F, P ) = 1. Now assume that X is projective.



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Fix a finite set S ⊆ Sing(X) and let f : C → X be the partial normalization of X in which we

normalize only the points of S. The torsion of f ∗(F ) is supported on the finite set f −1(S). Set

G := f ∗(F )/Tors(f ∗(F )). G is a coherent sheaf on C with depth 1 and pure rank 1. Since X and

C are projective, the integers deg(F ) and deg(G) are well-defined and satisfy the Riemmann-Roch

formulas χ(F ) = deg(F ) + χ(OX ), χ(G) = deg(G) + χ(OC ) even if X or C are not connected. We

have

deg(G) = deg(F ) −
∑

P ∈S

ℓ(F, P ) (1)

We need this formula only when each point of S is an ordinary node of X. In this case we may

decompose f into ♯(S) partial normalizations of a single node. Hence for nodes it is sufficient to

prove it when ♯(S) = 1, say S = {P }. In this case (1) is obviously true if FP is free. If F = IP ,

then (1) holds, because ℓ(IP , P ) = 1 and G is the ideal sheaf of the two points f
−1(P ). For an

arbitrary FP use the next result.

Remark 2. Take the set-up of the first part of Remark 1. Assume that P is either an ordinary

node or an ordinary cusp of X. Assume P ∈ Sing(F ). By the classification of torsion free modules

on simple curves singularities ([15], or, for nodes, [21], pp. 163–166) the germ of F at each P

is formally equivalent to the maximal ideal mX,P of the local ring OX,P . Hence Remark 1 gives

ℓ(F, P ) = 1.

Remarks 1 and 2 immediately give the following result.

Corollary 1. Let X be a reduced projective curve and F a coherent sheaf on X with depth 1

and pure rank 1. Fix S ⊆ Sing(F ). Assume that each point of S is an ordinary node or an

ordinary cusp of X. Let h : D → X (resp. f : C → X) be the partial normalization of X

in which we normalize only the points of S (resp. the points of S and the singular points of X

at which F is locally free). Set L := f ∗(F )/Tors(f ∗(F )) and R := h∗(F )/Tors(h∗(F )). Then

deg(L) = deg(R) = deg(F ) − ♯(S).

Lemma 1. Let X, Y reduced, projective curves and f : Y → X a finite surjective morphism. Let

A be a coherent sheaf on Y . Then deg(f∗(A)) = deg(A) + χ(OX ) − χ(OY ).

Proof. Obviously, h0(X, f∗(A)) = h
0(Y, A). Since f is finite, R1f∗(A) = 0. Hence the Leray

spectral sequence of f gives h1(X, f∗(A)) = h
1(Y, A). Thus χ(A) = χ(f∗(A)). Since χ(A) =

deg(A) + χ(OY ) and χ(f∗(A)) = deg(f∗(A)) + χ(OX ), we are done.

Remark 3. For any X (even not connected) and any L ∈ Pic(X) we have

∑

T ∈B(X)

deg(L|T ) = deg(L) (2)

Notice that (2) is true for non-locally free L if we only assume that L is locally free at each point

of X lying on at least two irreducible components.



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Brill-Noether Theories for Rank 1 Sheaves ... 107

Proof of Theorem 1. Fix any L ∈ D1(X). Since deg(L|C) ≥ 0 for all C ∈ B(X), there

is TL ∈ B(X) such that deg(L|TL) = 1, while the morphism hL : X → P
r, r := h0(X, L) − 1,

contracts to points all other components. Let Y1, . . . , Ys be the closures in X of the connected

components of X\TL. Since L|TL is spanned, TL ∼= P
1, and hL|TL is bijective. This implies that

hL(Yi) = hL(Yi ∩ T ) for all i. The second part of the statement of Theorem 1 shows how to

construct from any T ∈ S(X) a morphism hLT : X → P
1 such that the spanned line bundle

h∗LT (OP1 (1)) has degree 1. Obviously, the last (resp. first) construction is the inverse of the first

(resp. last) one. To check the last assertion take L constructed in a similar way by taking the

degree d Veronese embedding T →֒ Pd of T ∼= P1, instead of the isomorphism φ.

Lemma 2. Let T be an integral projective curve. There is no spanned rank 1 torsion-free sheaf F

on T such that Sing(F ) 6= ∅ and deg(F ) = 1.

Proof. Assume the existence of such a sheaf F . Since Sing(F ) 6= ∅, T is singular. Hence pa(T ) ≥

1 = deg(F ). Since F 6= OX and F is spanned, h
0(X, F ) ≥ 2. The contradiction comes from

Clifford’s inequality ([12], Theorem A at p. 532).

Proof of Theorem 2. Assume the existence of F ∈ S1(X)\D1(X) and set c := ♯(Sing(F ))

and b := ♯(Sing(F ) ∩ Sing(X)′′). By assumption c > 0. If b = 0, then Lemma 2 and the last

sentence of Remark 3 gives a contradiction. Hence we may assume b > 0. Let h : D → X be

the partial normalization of X in which we normalize only the points of Sing(F ) ∩ Sing(X)′′. Set

R := h∗(F )/Tors(h∗(F )). Corollary 1 gives deg(R) = 1−b ≤ 0. R is a spanned sheaf with pure rank

1 and depth 1. Hence b = 1 and R ∼= OD. Hence c = b and F has a unique singular point. Since

F has no torsion, the natural map h∗ : H0(X, F ) → H0(D, R) is injective. Hence h0(X, R) ≥ 2.

Thus the only singular point, P , of Sing(F ) is a disconnecting node of X. Hence D = XP and

h = fP . Since XP has two connected components, we get h
0(X, F ) = 2. Hence to prove all the

assertions of Theorem 2 it is sufficient to check that A := fP ∗(OXP ) ∈ S1(X)\D1(X). Lemma 1

gives deg(A) = 1. Obviously, A is not locally free at P . We have h0(X, A) = h0(XP , OXP ) = 2.

Let A′ be the subsheaf of A spanned by H0(X, A). If A′ = A, then we are done. Assume

A′ 6= A. Hence deg(A′) ≤ deg(A) − 1 ≤ 0. Since X is connected, we get h0(X, A′) ≤ 1. Since

h0(X, A′) = h0(X, A) = 2, we get a contradiction.

Example 1. Fix integers g > q > 0. Let X be a genus g stable curve with 2 irreducible components

X1 and X2 such that pa(X1) = q and pa(X2) = g − q. Since pa(X) = 1, we have ♯(X1 ∩ X2) = 1.

The only point P of X1 ∩ X2 is a disconnecting node of X. It is easy to check that the degree 1

spanned sheaf associated by Theorem 2 to this disconnecting node satisfies the Basic Inequality

(5) (see section 3) if and only if g = 2q.



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2 The Deformation Theory of Maps to P1

Fix a positive multidegree d for τ . Set δ(d) :
∑

T ∈B(X) dT (the total degree of d). For any positive

multidegree d = {dT } let G
1(τ, d) denote the set of all pairs (X, f ) such that X ∈ M[τ ] and

f : X → P1 is a morphism with multidegree d, i.e. such that deg(f ∗(OP1 (1)) = dT for every

T ∈ B(X). In this section we prove the following result.

Theorem 3. Fix a topological type τ for nodal and connected projective curves and a positive

multidegree d for τ . Either G1(τ, d) = ∅ or G1(τ, d) is smooth and of pure dimension 2δ(d) + 2g −

2 − s, where X is any element of M[τ ], g := pa(X), and s := ♯(Sing(X)).

Remark 4. Let X be a connected projective curve with only ordinary nodes as singularities. Let

ΘX denote the dual of the cotangent sheaf Ω
1
X . Since ΘX is the dual of a generically rank 1 coherent

sheaf, ΘX has no torsion. It is easy to check that ΘX = (Ω
1
X /Tors(Ω

1
X ))

∗. Fix P ∈ Sing(X). It

is well-known that the connected component of Tors(Ω1X ) supported by P has length 1 and that

Ω1X /Tors(Ω
1
X ) is not locally free at P (see [10], formula (4.1), or [16], p. 33). Since X is Gorenstein,

every depth 1 sheaf on X is reflexive. Thus Ω1X /Tors(Ω
1
X )

∼= Θ∗X . Hence Sing(ΘX ) = Sing(X).

Here we present an alternative proof. We claim that the germ at P of the sheaf Ω1X /Tors(Ω
1
X ) is

isomorphic to a colength 1 module F of the trivial OX,P -module of ωX,P . It would be sufficient to

prove the claim, because no such F is locally free by the classification of torsion free modules over

an ordinary node ([21], huitième partie, propositions 2 and 3). The claim is just part (2) of [5],

Lemma 6.1.2, in which the following notation is used: λ is the colength which we want compute,

µ is the Milnor number of X at P (µ = 1 for an ordinary node), δ is the genus of the singularity

(δ = 1 for an ordinary node) κ (the cuspidal number) is equal to the multiplicity minus the number

of branches by part (1) of [5], Lemma 6.1.2 (hence κ = 0 for an ordinary node); the formula says

that µ ≥ λ ≥ δ + κ, i.e. 1 ≥ λ ≥ 1 + 0.

Lemma 3. Let X be a connected projective curve with only ordinary nodes as singularities. Then

deg(ΘX ) = 2 · χ(OX ) + ♯(Sing(X)).

Proof. Since X is Gorenstein, every torsion free coherent sheaf F on X is reflexive and deg(F ∗) =

− deg(F ). Since ΘX ∼= (Ω
1
X /Tors(Ω

1
X ))

∗, it is sufficient to prove that

deg((Ω1X /Tors(Ω
1
X )) = −2 · χ(OX ) − ♯(Sing(X)).

Since deg(ωX ) = −2·χ(OX ), it is sufficient to prove the existence of an inclusion j : Ω
1
X /Tors(Ω

1
X ) →֒

ωX whose cokernel is a torsion sheaf with length ♯(Sing(X)). This assertion is the last part of Re-

mark 4, i.e. [5], Lemma 6.1.2.

Let X be a nodal and connected projective curve, M a smooth and projective variety and

f : X → M be a morphism whose restriction to each irreducible component of X is non-constant.

The latter condition implies TX/M = 0, where TX/M (also denoted with TX/f /Y or Tf ) is the

subsheaf of ΘX defined in [21], p. 387. Our assumption on f is called “non-degenerate ” in [21],



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Brill-Noether Theories for Rank 1 Sheaves ... 109

Definition 3.4.5. Let N ′f denote the cokernel of the natural map ΘX → ΘM ; the same sheaf is

denoted with Nf in [21], Definition 3.4.5. Since TX/M = 0, we have an exact sequence of coherent

sheaves on X ([21], p. 162).

0 → ΘX
f̃
→ f ∗(ΘM ) → N

′
f → 0 (3)

We are interested in the functor of locally trivial deformations of the map f with M fixed, mainly

when M = P1. Since f is non-degenerate, the vector space H0(X, N ′f ) is the tangent space to the

functor of locally trivial deformations of the map f with M fixed, while the vector space H1(X, N ′f )

is an obstruction space for the same functor ([21], Lemma 3.4.7 (iii) and Theorem 3.4.8). Since X

is nodal, saying “locally trivial ” means that we only look at nearby pairs (X′, f ′) with X′ a nodal

curve with ‖X′‖ = ‖X‖, i.e. in which no node is smoothed, i.e. with the topological type of X.

Proposition 1. Let X be a nodal and connected projective curve. Let f : X → P1 be a morphism

such that f|T is not constant for every irreducible component T of X. Set g := pa(X), s :=

♯(Sing(X)) and d := deg(f ). The the functor of locally trivial deformations of f is smooth at f

and of dimension 2d + 2g − 2 − s.

Proof. Both ΘX and f
∗(ΘP1) are sheaves with depth 1 and pure rank 1. Hence the injectivity

of the map f̃ in the exact sequence (3) gives that N ′f is supported by finitely many points of X.

Hence h1(X, N ′f ) = 0 and h
0(X, N ′f ) = deg(f

∗(ΘP1 )) − deg(ΘX ) = 2d + 2g − 2 − s (Lemma 3).

Proof of Theorem 3. The theorem is just a restatement of Proposition 1.

Let X be a nodal and connected projective curve. Let gon1(X) denote the minimal integer

d such that there is a degree d morphism X → P1. Obviously, gon1(X) ≥
∑

T ∈B(X) gon1(T ).

Thus gon1(X) ≥ ♯(B(X)) and the inequality is strict if at least one of the components of X is

not isomorphic to P1. There are topological types τ such that h0(X, f ∗(OP1 (1)) ≥ 3 for any nodal

curve X with topological type τ and f computing gon1(X). A stupid example is given by the

topological type of a reducible conic. A more interesting example is given by the graph curves X

of genus g ≥ 4 ([4]) for which gon1(X) ≥ ♯(B(X)) = 3g − 3 and hence h
0(X, f ∗(OP1 (1)) ≥ g − 1

(Riemann-Roch). Let τ be a topological type for nodal and connected projective curves. Let

gon1,−(τ ) (resp. gon1,+(τ )) denote the minimal (resp. maximal) integer d such that there is X ∈ τ

such that gon1(X) = d.

Question 1. Compute gon1,−(τ ) and gon1,+(τ ) for every topological type τ .

3 Caporaso’s Compactification: The Basic Inequality

Fix an integer g ≥ 2. In [6] L. Caporaso constructed a compactification over Mg of the set of

all line bundles with fixed degree on smooth genus g curves. Recall that for every stable genus g

curve X the canonical sheaf ωX is an ample line bundle. R. Pandharipande proved that Caporaso’s



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compactification is equivalent to the moduli scheme of equivalence classes of all slope-semistable

(with respect to the polarization ωX ) coherent sheaves with depth 1, pure rank 1 and degree d

([19], Theorem 10.3.1; see [21] for the construction of this moduli space). This is a very nice result,

because it shows that, at least if we take the canonical polarization, the Brill-Noether theory of

elements of Caporaso’s compactification or of semistable sheaves are the same. We will use the

set-up of [6], because it has a very important feature: if X is reducible, then we may refine the

degree, prescribing (at least for line bundles) the multidegree, i.e. the degree of the restriction to

each irreducible components. Easy examples show that for certain multidegrees a Brill-Noether

locus may be empty, while for other multidegrees with the same total degree the corresponding

moduli space is non-empty ([6], Proposition 12). This is not an exceptional situation: it happens

very frequently.

We recall the definition of semibalanced or balanced line bundle and of Basic Inequality ([17],

Definition 1.1). A semistable curve X is called quasistable if any two exceptional irreducible

components of X (i.e. smooth rational component intersecting the other components at two points)

are disjoint. X is quasistable if and only if either X is stable or its stable reduction u : X → Y

have the following property: there is S ⊆ Sing(Y ) such that u|u−1(X\S) : u−1(X\S) → X\S is

an isomorphism and u−1(P ) ∼= P1 for all P ∈ S. If X is quasistable, but not stable, then (u, Y, S)

are uniquely determined by X, while X is uniquely determined by the pair (Y, S). Let X be a

quasistable curve of genus g ≥ 2. For any proper subcurve A of X set kA := ♯(A ∩ X\A) and

wA := deg(ωX|A) = −2 · χ(OA) + kA. Fix L ∈ Pic(X). L is said to be semibalanced if the following

inequality (called the Basic Inequality)

deg(L) · wZ /(2g − 2) − kZ /2 ≤ deg(L|Z) ≤ deg(L) · wZ /(2g − 2) + kZ /2 (4)

holds for every proper connected subcurve Z of X. L is said to be balanced if it is balanced and

deg(L|E) = 1 for every exceptional component E of X. Let X be a stable curve. Fix S ⊆ Sing(X).

Let uS : X[S] → X be the partial normalization of X in which we normalize exactly the point of

S. Thus X[S] is nodal ♯(Sing(X[S]) = ♯(Sing(X)) − ♯(S) and χ(O[S]) = ♯(S) + χ(OX ). X[S] may

be disconnected. Let vS : XS → X denote the stable reduction of the quasistable model of the

pair (X, S), i.e. XS is quasistable with ♯(S) exceptional components, each of them mapped to a

different point of S and vS|v
−1
S (X\S) : v

−1
S (X\S) → X\S is an isomorphism. XS is connected

and pa(XS ) = pa(X). X[S] is a subcurve of XS and uS = vS|X[S].

Remark 5. Fix a pure rank 1 torsion free sheaf F on X. Write uF , vF , X[F ] and XF instead of

XS , vF , X[S] and XS if S := Sing(F ). Set F̃ := u
∗
F (F )/Tors(u

∗
F (F )). Since F̃ has no torsion and it

is localy free at each singular point of X[F ], F̃ is a line bundle. Fix P ∈ Sing(F ). The classification

of all pure rank 1 torsion free sheaves on a nodal singularity gives that the germ of F at P is

isomorphic to the maximal ideal of the local ring OX,P ([21], huitième partie, Propositions 2 and

3). Hence deg(F̃ ) = deg(F ) − ♯(S). There is a unique line bundle F on XS such that F |X[F ] = F̃

and deg(F |E) = 1 for every exceptional curve E of XF . We have deg(F ) = deg(F̃ )+♯(S) = deg(F ).

Since F has no torsion, the pull-back map f ∗ induces an inclusion f ∗ : H0(X, F ) → H0(X[S], F̃ ).

Hence h0(X[S], F̃ ) ≥ h
0(X, F ). Now assume that F is spanned by a linear subspace V ⊆ H0(X, F ).



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Brill-Noether Theories for Rank 1 Sheaves ... 111

Since the tensor product is a right exact functor, f ∗(V ) spans F̃ . Recall that XF is obtained from

X[F ] adding ♯(S) disjoint exceptional curves. Since deg(F |E) = 1 for every exceptional curve E of

XF , ♯(S) Mayer-Vietoris exact sequences give h
0(XS , F ) = h

0(X[S], F̃ ) and that F is spanned if

and only if F̃ is spanned (see Lemmas 4 and 5)).

Lemma 4. Fix a nodal curve X and S ⊆ Sing(X). Fix L ∈ Pic(X[S]) and let M be the only line

bundle on XS such that M|X[S] ∼= L and deg(M|E) for every exceptional curve E of vS. We have

h0(X[S], L) = h
0(XS , M ). If L is spanned, then h

0(XS , M ) ≤ h
0(X[S], L) + ♯(S). M is spanned if

and only if L is spanned and for any exceptional curve E of vS there is f ∈ H
0(X[S], L) vanishing

at one of the point of E ∩ X[S], but not vanishing at the other point of E ∩ X[S].

Proof. By induction on S we reduce to the case ♯(S) = 1. This inductive procedure is the reason for

not requiring the stability of X (if ♯(S) ≥ 2), since this assumption would be lost in the inductive

step. Set {P } := S. Let E be the only new exceptional curve of X[S] and P1, P2 the points of

E ∪ X[S]. We have a Mayer-Vietoris exact sequence on XS :

0 → M → M|X[S] ⊕ M|E → M|{P1, P2} → 0 (5)

Since the restriction map H0(E, M|E) → H0({P1, P2}, M|{P1, P2}) is bijective, (5) gives that the

restriction map ρ : H0(XS , M ) → H
0(X[S], L) is bijective. Hence h

0(X[S], L) = h
0(XS , M ). The

bijectivity of ρ gives that L is spanned if and only if M is spanned at each point of X[S]. Since

E ∼= P1 and deg(M|E) = 1, M is spanned at every point of E if and only if the restriction map

η : H0(XS , M ) → H
0(E, M|E) is surjective. The cohomology exact sequence of (5) gives that η is

surjective if and only if the restriction map β : H0(X[S], L) → L|{P1, P2} is surjective. Since L is

spanned, the surjectivity of β is equivalent to require the existence of f ∈ H0(X[S], L) vanishing

at P1, but not at P2.

Lemma 5. Fix a nodal curve X and a sheaf F on X with depth 1 and pure rank 1. Assume that

F is spanned. Then F̃ and F are spanned and h0(XF , F ) = h
0(X[F ], F̃ ).

Proof. We know that F̃ is spanned. Apply Lemma 4 to S := Sing(F ) and get equality h0(XF , F ) =

h0(X[F ], F̃ ). To check the spannedness of F we need to check the last condition of Lemma 4. Fix

P ∈ Sing(F ) and let E be the corresponding exceptional curve. Set {P1, P2} := E ∩ X[F ]. Since

the germ of F at P is isomorphic to the maximal ideal of OX,P and OX,P is an ordinary node, the

fiber F |{P } of F at P is a 2-dimensional vector space. Since F is spanned at P and the natural

map u∗F : H
0(X, F ) → H0(X[F ], L) is injective, we get h

0(X[F ], I{P1,P2} ⊗ L) ≤ h
0(X[F ], L). Hence

the last part of Lemma 4 gives the spannedness of F .

SUMMARY OF THE SECTION: We hope to have convinced the reader that in many

cases we may separately study the Basic Inequality and the geometric properties (spannedness,

very ampleness and so on) of a potential element of a Brill-Noether theory.



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4 Stable Curves with ωX without Spanned Locally Free Sub-

sheaves

Let X be a stable curve of genus g. The dualizing sheaf ωX is spanned if and only if X has no

disconnecting node, i.e. there is no P ∈ Sing(X) such that X\{P } is not connected ([8], Theorem

D, or [1], part (a) of Theorem 1.2, or [9], part (b) of Theorem 3.3). Since X is nodal, X\{P } has

two connected comnponents if P is a disconnecting node of X. Here we give two examples of genus

g stable curves X such that there is no locally free spanned subsheaf L of ωX with h
0(X, L) ≥ 2

(i.e. L 6= OX ). Since ωX is locally free, in any such example ωX is not spanned, i.e. X has a

disconnecting node. The first example works for any genus g ≥ 2. For the second example we need

to assume g large, say g ≥ 9. In the second example we have no injective map OX →֒ ωX .

Example 2. Let X be a chain of g curves of genus 1, i.e. assume X = ∪
g
i=1Ti, pa(Ti) = 1 for all

i, Ti ∩ Tj 6= ∅ if and only if |i − j| ≤ 1 and ♯(Ti ∩ Ti+1) = 1 for all i ∈ {1, . . . , g − 1}. Notice that

X has g − 1 disconnecting nodes. The proofs of [8], Theorem D, or of of [1], Theorem 1.2, or an

easy exercise left to the reader show that the subsheaf F of ωX spanned by H
0(X, ωX ) has the

property that ωX /F = ⊕P ∈Sing(X)′′ KP , where KP denote the skyscraper sheaf supported by P

and such that h0(X, KP ) = 1. Hence deg(F ) = 2g − 2 − ♯(Sing(X)
′′) = g − 1 and F is not locally

free at any point of Sing(X)′′. Every spanned subsheaf of ωX is contained in F . If L is a locally

free subsheaf of F , then the torsion sheaf F/L must have every point of Sing(X)′′ in its support.

Thus deg(L) ≤ deg(F ) − (g − 1) ≤ 0. If L is also spanned, we get L ∼= OX .

Example 3. Let X be a genus g stable curve such that there is an irreducible component T ∼= P1.

Set k := ♯(X ∩ X\T ). Notice that ωX|T has degree k − 2. Assume that at least k − 1 of

the points of X ∩ X\T are disconnecting node of X. Let F be the subsheaf of ωX spanned

by H0(X, ωX ) and let L be any locally free subsheaf of F . Since ωX is not spanned at any

disconnecting node, the inclusion map j : L →֒ ωX drops rank at each disconnecting node of X.

Hence deg(L|T ) ≤ deg(ωX|T )− (k − 1) ≤ −1. Thus L is not spanned and any section of L vanishes

identically on T . In particular L 6= OX .

Question 2. Is it possible to describe all genus g stable curves X such that there is no locally free

spanned subsheaf L of ωX with h
0(X, L) ≥ 2 and/or the ones for which there is no injective map

OX →֒ ωX ? Is it true that any example with the latter property has a smooth rational component

T such that every section of ωX vanishes identically on T ?

Received: October, 2008. Revised: October, 2009.

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