A brill - noether theory for k-gonal nodal curves

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RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Tomo LII (2003), pp. 285-296

A BRILL - NOETHER THEORY FOR k-GONAL NODAL CURVES EDOARDO BALLICO – CLAUDIO FONTANARI

Let V (g, x, k, y) be the set of all pairs (X, F), where X is an integral projective nodal curve with pa (X ) = g and card(Sing( X )) = x and F is a rank 1 torsion free sheaf on X with deg(F) = k , card(Sing(F)) = y and h0 (X, F) ≥ 2. Here we study a general (X, F) ∈ V (g, x, k, y) and in particular the Brill-Noether theory of X and the scrollar invariants of F .

1. Introduction. The study of linear series on a general curve of fixed gonality is a classical theme in algebraic geometry, but only recently (see [7] and [8]) a systematic treatment of the question has been successfully pursued. In the present paper we address the singular case, and in particular we point our attention at  0 , the boundary component of M g whose general element corresponds to an irreducible nodal curve of arithmetic genus g. It is easily checked (see section 1 for all the details) that a general element of  0 has gonality [(g + 3)/2], the same as in the smooth case. It seems then natural to fix an integer k < [(g + 3)/2] and define V (g, x, k, y) to be the set of all pairs (X, F), where X is an integral projective nodal curve with p a (X ) = g and card(Sing(X )) = x and F is a rank 1 torsion free sheaf on X with deg(F) = k, card(Sing(F)) = y and h 0 (X, F) ≥ 2. If we let W (g, x, k, y) be the set of all nodal curves X such that there is a sheaf F with (X, F) ∈ V (g, x, k, y), we obtain a locally closed algebraic subset of the moduli space of stable curves. Exactly as for the general k-gonal curve, the main property of a general element of W (g, x, k, y) regards the dimension of multiples of the fixed pencil. Here is the statement in the case y = 0.

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LEMMA 0.1. Fix positive integers g, x , k, a with a ≥ 3, k ≥ 3, (a − 2)(k − 1) < g ≤ (a − 1)(k − 1) and 0 ≤ x ≤ g. Let (X, L) a general element of V (g, x, k, 0). Then for every integer r with 0 ≤ r ≤ a − 2 we have h 0 (X, L ⊗r ) = r + 1 and for every integer s ≥ a − 1 we have h 0 (X, L ⊗s ) = ks − g + 1, i.e. h 1 (X, L ⊗s ) = 0. As an application, we will deduce the following Brill - Noether statement, which is the analogous of [8], Corollary 2.3.2 (recall the Brill - Noether number ρ(g, r, d): = g −(r +1)(g −d +r ) and the Brill - Noether loci W dr (C): = {L: L is a locally free sheaf on C , h 0 (X, L) = r + 1, deg(L) = d}). THEOREM 0.2. Let C a general element of W (g, x, k, y). Then if d − g < r ≤ k − y − 2 and ρ(g, r, d) ≥ 0 there exists an irreducible component V of Wdr (C) with dim(V ) = ρ(g, r, d). In particular, W dr (C) = ∅. Here it is the plan of the paper. First of all, in section 1 we discuss background and motivations, make precise our set-up and give a proof of Theorem 0.2, assuming for a moment Lemma 0.1. Next, in section 2, we turn to Lemma 0.1 and after its proof we describe an extension to the case y > 0. Finally, section 3 is devoted to Brill - Noether theory for pencils, which is essentially related to the gonality of the curve. We work over an algebrically closed field K with char(K ) = 2. This research was partially supported by MIUR (Italy).

1. PRELIMINARIES AND PROOF OF THEOREM 0.2. Let X be an integral projective curve. Even though we are mainly interested in the Brill - Noether theory of line bundles, we have also to consider torsion free sheaves: in fact, if L ∈ Pic(X ), h 0 (X, L) ≥ 2 but L is not spanned by its global sections at some point of Sing(X ), the subsheaf of L generated by H 0 (X, L) may be not locally free but only torsion free. If X is nodal, however, we have a very clear picture of the torsion free sheaves which may arise on X . Remark 1.1. Let X be an integral nodal curve, F a rank 1 torsion free sheaf on X , P ∈ Sing(X ), R the completion of the local ring O X,P and M the completion of the germ of F at P . Hence R ∼ = K [[x, y]]/(x y). Call m the maximal ideal of the local ring R. Assume that F is not locally free at P , i.e. assume M not isomorphic to R. Since X is integral, M has rank 1 at each of the two branches of R. By the classification of torsion free modules over R

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(see [5] p. 24, [13] or, in positive characteristic, [14]) we have M ∼ = m. Here we use that M has rank 1 on each branch of R: the difference is discussed in [16], last two lines of p. 165 and Prop. 3 at p. 166, [15] or, if char(k) = 2, [4] p. 60. Remark 1.2. Let X be an irreducible projective nodal curve. Fix Z ⊆ Sing(X ) and let f : Y → X be the partial normalization in which we normalize exactly the set Z of nodes of X . Set z: = card(Z ). For any R ∈ Pic d (Y ) f ∗ (R) is a rank 1 torsion free sheaf with Sing( f ∗(R)) = Z , h 0 (X, f ∗ (R)) = h 0 (X, R) and deg( f ∗ (R)) = d + z. Conversely, take a rank 1 torsion free sheaf F with Sing(F) = Z and set M: = f ∗ (M) / Tors( f ∗ (M)). Then deg(M) = deg(F) − z, F ∼ = f ∗ (M) and hence h 0 (X, F) = h 0 (Y, M). Summing up, if for every subset (possibly empty) S ⊆ Sing(X ) we define W dr (X, S) = {F: F is a rank 1 torsion free sheaf on X , h 0 (X, F) = r + 1, deg(F) = d and Sing(F) = S}, we have a natural isomorphism between Wdr (X, Z ) and r (Y, ∅). Wd−z If we assume, moreover, that X is a general irreducible nodal curve, it is worth noticing that line bundles on X are governed by the classical Brill - Noether theory. Indeed, by [12], Prop. 1.2, there is a rational curve with g nodes such that Petri’s condition holds for it and even for any its partial desingularization. It follows that the general irreducible curve with x ≤ g nodes X has G rd (X ) = ∅ if ρ(g, r, d) < 0 and dim(G rd (X )) ≤ ρ(g, r, d) if ρ(g, r, d) ≥ 0. It remains to verify that G rd (X ) = ∅ if ρ(g, r, d) ≥ 0. We may do it by induction on g. If we define Hdr (X ) = {F: F is a rank 1 torsion free sheaf on X , h 0 (X, F) = r + 1, deg(F) = d}, since X has only smoothable singularities and is a flat limit of smooth curves of genus g, we have dim(Hdr (X )) ≥ ρ(g, r, d). If G rd (X ) = ∅ it should be dim(Hdr (X ) \ G rd (X )) ≥ ρ(g, r, d), but, by Remark 1.2, Hdr (X )\G rd (X ) = ∪0 0. We do the same construction for all such pairs (X, L) and we obtain an equidimensional locally closed subset W (g, x, k, y) with dim(W (g, x, k, y)) = 3(g − x) − 3 + (x − y) + 2y. We want to check that

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(at least if char(K ) = 0)W (g, x, k, y) is irreducible, not just equidimensional. This fact follows from the irreducibility of G 1g−x,k−y proved in [10], Theorem 1’. We stress that we need only the simple transitivity results for r = 1 and this is probably true for any K with char(K ) = 2; anyway, in the case ρ(g − x, 1, k − y) = 0 we may work with any irreducible component of W (g, x, k, y). Proof of 0.2. If y = 0, we may copy almost verbatim the proof given in [8], (2.3) for the smooth case. Indeed, the only nontrivial ingredients in the proofs of [8] Prop. (2.3.1) and Cor. (2.3.2) are Prop. (2.1.1), Lemma (2.2.1) and Prop. (A.1). We remark that (2.1.1) is nothing but the case x = 0 of Lemma 0.1, while (2.2.1) is a straightforward application of [9], Cor. 5.2, which stands for an arbitrary reduced and irreducible projective variety; finally, the construction performed in (A.1) following [1] does not involve the smoothness property in any essential way. If y > 0, we notice that W (g, x, k − y, 0) ⊂ Z (g, x, k, y). Since by the case y = 0 we already know that a (general) curve in W (g, x, k − y, 0) satisfies the desired property, the thesis now follows immediately from [8], (A.1).

2. PROOF OF LEMMA 0.1 AND GENERALIZATIONS. The proof of [3], Prop. 1, gives verbatim the following remark. Remark 2.1. Fix positive integers g, k, a, with a ≥ 3, k ≥ 3, and (a −2)(k −1) < g ≤ (a −1)(k −1). Fix an integral curve C ⊂ P 1 × P 1 of type (k, a). Fix S ⊆ Sing(C) with card(S) = (a − 1)(k − 1) − g and assume that C has only ordinary nodes or ordinary cusps at every point of S. Assume that S imposes independent conditions to forms of type (k − 2, 0), i. e. that the points of S have different images under the projection P 1 × P 1 → P 1 on the second factor. Let π : X → C be the partial normalization of C at the points of S; hence pa (X ) = g. Let L ∈ Pick (X ) be the degree k linear series on X induced by the composition of π and the projection P 1 × P 1 → P 1 on the second factor. Then for every integer r with 0 ≤ r ≤ a−2 we have h 0 (X, L ⊗r ) = r +1 and for every integer s ≥ a − 1 we have h 0 (X, L ⊗s ) = ks + 1 − g, i. e. h 1 (X, L ⊗s ) = 0. Notice that the values for h 0 (X, L ⊗r ) and h 0 (X, L ⊗s ) are the minimal ones compatible with Riemann - Roch and the assumptions h 0 (X, L) ≥ 2, deg(L) = k and pa (X ) = g.

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Proof of 0.1. By [17], proof of 3.2 and 3.3, there is an irreducible nodal curve D ⊂ P 1 × P 1 of type (k, a) and whose normalization is rational, i. e. with exactly ka − k − a + 1 ordinary nodes as only singularities. Moreover we may find such a curve D with the further property that there is Z ⊆ Sing(D) with card(Z ) = (a − 1)(k − 1) − g and such that Z imposes card(Z ) independent conditions to forms of type (k − 2, 0), i. e. such that no two different points of Z are on the same line of type (1,0) on P 1 × P 1 ; for instance the existence of such a pair (D, Z ) follows by smoothing a suitable set of a + k − 1 nodes of the union D ⊂ P 1 × P 1 of k lines of type (1,0) and a lines of type (0,1) and then applying [17], 2.11 and 2.13. Take a subset Z of Sing(D) \ Z with card(Z ) = x and call Z ∪ Z the set of assigned nodes of D and Sing(D) \ (Z ∪ Z ) the set of unassigned nodes of D. Since the anticanonical bundle of P 1 × P 1 is ample, we may apply [17], Prop. 2.11, and obtain the existence of a flat family {Dt }t∈T of nodal curves of type (k, a) on P 1 × P 1 with T irreducible affine curve, o ∈ T , Do = D, Dt with exactly x+ card(Z ) nodes for every t ∈ (T \ {o}) and such that the family {Dt }t∈(T \{o}) has Z ∪ Z as flat limit as t goes to o. In particular, for a general t ∈ T the set of nodes of Dt may be divided into two disjoint subsets, Z t and Z t , with Z t going to Z and Z t going to Z . By semicontinuity for general t the set Z t imposes independent conditions to forms of type (k − 2, 0). Hence we may apply Remark 2.1 with C = D t and S = Z t for general t and conclude. Fix (X, L) ∈ V (g, x, k, y) and set S: = Sing(L). Let π : C → X be the partial normalization in which we normalize exactly the subset S of the nodes of X . Hence there is R ∈ Pick−y (C) with L ∼ = π∗ (R), h 0 (C, R) = 2 and R {t} spanned. For every integer t ≥ 1 set L : = π∗ (R ⊗t ). Hence L {t} is a rank 1 torsion free sheaf on X with deg(L {t} ) = t (k − y) + y, L {t} spanned and h 0 (X, L {t} ) = h 0 (C, R ⊗t ). For every integer t ≥ 1 let L [t] ∈ Pict (k−y)+2y (X ) be a line bundle such that L {t} ⊂ L [t] , and L [t] /L {t} is a skyscraper sheaf of lenght y with a 1-dimensional vector space over every point of S. Hence h 0 (X, L {t} ) ≤ h 0 (X, L [t] ) ≤ h 0 (X, L {t} ) + y. PROPOSITION 2.2. Fix integers g, x , y, k, t with g − y ≥ 2(k − y), k ≥ y + 2, g ≥ x ≥ y ≥ 2, t ≥ 1, g − y ≥ t (k − y) + 2y. Then for a general X ∈ W (g, x, k, y) we have h 0 (X, L [t] ) = h 0 (X, L {t} ) = t + 1. Proof. Take π , C , R as above. By the generality of X , C is a general element of W (g − y, x − y, k − y, 0) and X is obtained from C by gluing together y general pairs of points of C , say {Pi , Q i }1≤i≤y , to create y new nodes. By Lemma 0.1 the inequality g − y ≥ 2(k − y) implies h 0 (C, R ⊗t ) =

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t + 1 and hence h 0 (X, L {t} ) = h 0 (C, R ⊗t ) = t + 1. From g − y ≥ t (k − y) +2y it follows that h 1 (C, R ⊗t ) ≥ 2y and then, if D: = 1≤x≤y Pi + Q i is a general sum of 2y points of C , we have h 0 (C, R ⊗t (D)) = t + 1. Thus h 0 (X, π∗ (R ⊗t (D))) = t + 1. Notice that π∗ (R ⊗t (D)) is a rank 1 torsion free sheaf which contains L {t} because the functor π∗ is left exact. T : = π∗ (R ⊗t (D))/L {t} is a skyscraper sheaf and over every point P ∈ S the connected component of T supported by P is 2-dimensional. By the classification of rank 1 torsion free modules on nodes, there is M ∈ Pic(X ) with deg(M) = t (k − y) + 2y, M contained in π ∗ (R ⊗t (D)) and containing L {t} . We may take M as L [t] . Since L {t} is a subsheaf of M and M is a subsheaf of π∗ (R ⊗t (D)), we have t + 1 = h 0 (X, L {t} ) ≤ h 0 (X, M) ≤ h 0 (X, π∗ (R ⊗t (D))) = t + 1 and hence the thesis. 3. BRILL - NOETHER THEORY FOR PENCILS. PROPOSITION 3.1. Fix integers g, k, x , d with g ≥ 2k −1 ≥ 3, g ≥ x ≥ 0, and ρ(g, 1, d): = 2d − 2 − g < 0. Let X be a general element of W (g, x, k, 0) and L ∈ Pick (X ) the associated pencil. Then for every M ∈ Picd (X ) with h 0 (X, M) ≥ 2 there is an effective Cartier divisor D with deg(D) = d −k ≥ 0 and such that M ∼ = L(D). Taking d = k (resp. d < k) in Proposition 3.1 we obtain the following corollaries. COROLLARY 3.2. Fix integers g, k, x , d with g ≥ 2k − 1 ≥ 3, g ≥ x ≥ 0. Then for a general X ∈ W (g, x, k, 0) we have card(Wk1 (X, ∅) = 1. COROLLARY 3.3. Fix integers g, k, x , d with g ≥ 2k + 1 ≥ 5, d < k and g ≥ x ≥ 0. Then for a general X ∈ W (g, x, k, 0) we have Wd1 (X, ∅) = ∅. We obtain also the following result. COROLLARY 3.4. Fix integers g, k, x , d with g ≥ 2k − 1 ≥ 3, d ≤ k, g ≥ x > 0 and a non-empty subset Z of Sing(X ). Then for general X we have Wd1 (X, Z ) = ∅. Proof. Set z: = card(Z ). Assume the existence of F ∈ Wd1 (X, Z ). Let f : C → X be the partial normalization of X in which we normalize exactly the nodes of Z . Hence pa (C) = g − z and card(Sing(C)) = x − z. By the generality of X , C is a general element of W (g−z, x −z, k, 0) if g−z ≥ 2k−1

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and a general curve of genus g − z with x − z nodes if g − z ≤ 2k − 2. In both cases respectively by Proposition 3.1 and Brill - Noether for a general nodal 1 (C, ∅) = ∅. By Remark 1.2, F induces curve ([12], Prop. 1.2) we have Wd−z

1 F ∈ Wd−z (C, ∅), contradiction. Now we turn to the proof of Proposition 3.1. Let f : C → X be the normalization. Hence C has genus g −x . Take M ∈ Picd (X ) with h 0 (X, M) ≥ 2. Hence f ∗ (M) ∈ Wd1 (C). If g − x ≤ 2k − 1, then C is a general smooth curve of genus g − x ; if g − x ≥ 2k − 1, then C is a general k-gonal curve. In both cases we have Wd1 (C) = ∅ if ρ(g − x, 1, d) < 0 i. e. if ρ(g, 1, d) < −x , respectively by Brill - Noether for curves with general moduli and by [2], Th. 2.6. Hence from now on we may assume 0 ≤ ρ(g − x, 1, d) < x . DEFINITION 3.5. Let V ⊆ H 0 (C, M) with dim V = 2; we say that the pair (M, V ) ∈ G 1m (C) has P , Q ∈ C with P = Q as a neutral pair if either V spans M neither at P nor at Q or if V spans M both at P and at Q and the rational function u (which is regular both at P and at Q) has u(P) = u(Q). For every integer n ≥ 1 we set G(P1 , Q 1 , ..., Pn , Q n ): = {(M, V ) ∈ G 1d (C): all the pairs Pi , Q i , 1 ≤ i ≤ n, are neutral for (M, V )}. If g − x ≤ 2k − 2, then dim(G 1k (C)) = ρ(g − x, 1, k) and we fix a general R ∈ Wk1 (C), so R is spanned and h 0 (C, R) = 2. If g − x ≥ 2k − 1, then C has a unique gk1 ([2], Th. 2.6) and we let R be the associated spanned line bundle with h 0 (C, R) = 2. In both cases, call u: C → P 1 the induced morphism. We point out the following fact. LEMMA 3.6. Let C be a smooth curve of genus g − x and let R ∈ Pic k (C) be a line bundle spanned with h 0 (C, R) = 2. Let G 1d (C)∗ : = {(M, V ) ∈ G 1d (C) such that there is no effective divisor D of degree d − k with M ∼ = R(D)}. Then G 1d (C)∗ has the minimal possible dimension ρ(g − x, 1, d). Assuming for a moment Lemma 3.6, we are ready to prove Proposition 3.1. Proof of 3.1. Assume that for a certain integer z, 1 ≤ z ≤ x , we have shown that for general A 1 , . . . , A z ∈ P 1 there are {Pi , Q i } ⊆ u −1 (Ai ), 1 ≤ i ≤ z, with Pi = Q i for every i such that the set G(P1 , Q 1 , . . . , Pz , Q z ) ∩ G 1d (C)∗ has codimension at least z in G 1d (C)∗ (if z > dim(G 1d (C)) = g − x − 2d − 2, this means G(P1 , Q 1 , . . . , Pz , Q z ) ∩ G 1d (C)∗ = ∅). We claim that for a general A z+1 ∈ P 1 there is {Pz+1 , Q z+1 } ⊆ u −1 (A z+1) with Pz+1 = Q z+1 such that G(P1 , Q 1 , . . . , Pz+1 , Q z+1 ) has codimension at

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least one in G(P1 , Q 1 , . . . , Pz , Q z ). Call Z 1 , . . . , Z s the irreducible components of G(P1 , Q 1 , . . . , Pz , Q z ) ∩ G 1d (C)∗ of dimension ρ(g − x, 1, d) − z. Fix (Mi , Vi ) ∈ Z i and assume that for all A z+1 ∈ P 1 \ {A1 , . . . , A z } with card(u −1 (A z+1 )) ≥ 2 and every {Pz+1 , Q z+1 } ⊆ u −1 (A z+1 ) with Pz+1 = Q z+1 Pz+1 , Q z+1 is neutral for (M, V ). Then the rational map from C to P 1 induced by (Mi , Vi ) factors through u and this implies M i ∼ = R(D) for some effective divisor D, contradicting the fact that (M, V ) ∈ G 1d (C)∗ . Hence there is a sufficiently general A z+1 ∈ P 1 and {Pz+1 , Q z+1 } ⊆ u −1 (A z+1) such that {Pz+1 , Q z+1 } is not neutral for (M i , Vi ); we may choose A z+1 , Pz+ 1, Q z+1 which work for all (M i , Vi ), 1 ≤ i ≤ s because the algebraic set G(P1 , Q 1 , . . . , Pz , Q z ) has a finite number of irreducible components. Hence we obtain the claim. Applying x times the claim (starting with the case z = 0) gives Proposition 3.1, since a pair (M, V ) ∈ W m1 (C) spanned at each point of f −1 (Sing (X )) descends to a pair (M , V ) ∈ Wm1 (X, ∅) spanned at each point of Sing (X ) if and only if for every A ∈ Sing(X ) the pair f −1 (A) is neutral for (M, V ). Proof of 3.6. If 2k ≥ g − x + 2, C has general moduli and by Gieseker - Petri G 1d (C) is equidimensional. Hence we have only to consider the case 2k ≤ g − x + 1. First assume d ≤ g − x (notice that this is the case if d ≤ k). By [6], Th. 2.2.1, every irreducible component of W d1 (C) different from Wk1 (C)+Wd−k (C) has dimension ρ(g − x, 1, d). Let T be an irreducible family of G 1d (C) which is not an irreducible component of W d1 (C) and whose generic element is not of the form R(D) with D an effective divisor of degree d − k. Then a general M ∈ T has h 0 (C, M) ≥ 3 and h 0 (C, M ⊗ R ∗ ) = 0. We assume dim(T ) ≥ ρ(g − x, 1, d) and we look for a contradiction. Fix P ∈ C , P general so that it is not a base point of all M ∈ T . For a general M ∈ T we have h 0 (C, M(−P)) = h 0 (C, M) − 1, deg(M(−P)) = d − 1; notice that ρ(g − x, 1, d − 1) = ρ(g − x, 1, d) − 2 and that the Grassmannian G(2, H 0 (C, M)) (resp. G(2, H 0 (C, M(−P)))) of 2-dimensional linear subspaces of H 0 (C, M) (resp. H 0 (C, M(−P))) has dimension 2(h 0 (C, M) − 2) (resp. 2(h 0 (C, M(−P)) − 2), i. e. dim(G(2, H 0 (C, M(−P)))) = dim (G(2, H 0 (C, M))) − 2. Hence from T we obtain an irreducible family, , of G 1d−1 (C) with dim( ) ≥ dim(T )−2 > ρ(g−x, 1, d −1); a general element of is not of the form R(D), because otherwise a general element of T would be of the form R(D + P). Hence we conclude after finitely many steps, at each of them decreasing by 1 deg and generic h 0 : we stress that the final contradiction 1 comes from [6], Th. 2.2.1, i. e. from a result for W d−n (n: = h 0 (C, R) − 2).

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Now assume d > g − x . We assume the existence of an irreducible family T of G 1d (C) with dim(T ) > ρ(g − x, 1, d) and such that for a general M ∈ T we have h 0 (C, M ⊗ R ∗ ) = 0. We make the same trick as before decreasing one by one the degree until we arrive at an integer m ≤ d such that there is an irreducible family, , of G 1d−1 (C) with dim( ) > ρ(g − x, 1, m) whose general element, M , has h 0 (C, M) = 2 and is not of the form R(D). By Riemann - Roch we have h 1 (C, M) = g − x + 1 − m. If m = g − x + 1, then generically consists of non-special line bundles and the corresponding component of G 1m (C) has dimension ρ(g − x, 1, m), contradiction. If m ≤ g − x , then the contradiction comes from [6], Th. 2.2.1, because the morphism → Wm1 (C) is birational, since the general element, M , of has h 0 (C, M) = 2. PROPOSITION 3.7. Fix integers g, k, y, y1 , d with g ≥ 2k − 1 ≥ 3, g ≥ x ≥ 0, x ≥ y ≥ 0, y = y1 , and d < k. Let X be a general element of W (g, x, k, y). Then X ∈ W (g, x, k, y1 ), for every S ⊆ Sing(X ) with card(S) = y we have Wk1 (X, S) = ∅ and for every rank 1 torsion free sheaf F on X with deg(F) = d we have h 0 (X, F) ≤ 1. Furthermore, there is a unique rank 1 torsion free sheaf, L , on X with deg(L) = k, card(Sing(L)) = y and h 0 (X, L) ≥ 2; L is spanned and h 0 (X, L) = 2. Proof. Let f : C → X be the partial normalization of X in which we normalize exactly Sing(L). Hence C is a general element of W (g − y, x − y, k − y, 0). Let M be a rank 1 torsion free sheaf on X with m: = deg(M) ≤ k. Set z: = card(Sing(M)) and w: = card(Sing(M)∩ Sing(L)). Consider A: = f ∗ (M)/ Tors( f ∗ (M)), with deg(A) = m − w and card(Sing(A)) = z − w, and R: = f ∗ (L)/ Tors( f ∗ (L)): we have A ∼ = R if and only if M ∼ = L . Hence 1 we obtain the uniqueness part for W k (X ,Sing(L)) by applying Corollary 3.2 to C ; Corollaries 3.2, 3.3 and 3.4 cover also the cases Sing(L) ⊆ Sing(M). Hence from now on we may assume m < k and w < y. Let u: Y → X be the partial normalization of X in which we normalize exactly the nodes in Sing(M)∩ Sing(L), B: = u ∗ (M)/ Tors(u ∗ (M)) and D: = u ∗ (L)/ Tors(u ∗ (L)). The pair (Y, D) corresponds to a general element of W (g − w, x − w, k − w, y − w). We have deg(D) = m − w ≤ k − w. Hence if w > 0 we may apply induction and conclude. Thus it is sufficient to handle the case w = 0. Let v: W → X be the partial normalization of X in which we normalize exactly the nodes in Sing(M). Set B : = v ∗ (M)/ Tors(v ∗ (M)) and D : = v ∗ (L). Notice that D is torsion-free and W is a general element of W (g −

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z, x − z, k, y) with D as associated pencil; if g − z ≤ 2k − 2 we mean as usual that W is a general nodal curve with arithmetic genus g − z and x − z nodes. We have deg(B ) = k − z, so if z > 0 we may conclude again by induction. Thus we are reduced to find a contradiction in the case M ∈Pic m (X ) is locally free; moreover, taking M(−P), P ∈ X reg , instead of M and using also all the other cases with M non locally free, we may assume that M is spanned and h 0 (X, M) = 2. Let f : C → Y be the normalization of X . Set B

: = f ∗ (M) and D

: = ∗ f (L)/Tors( f ∗ (L)). The pair (C, D

) shows that C is a general element of W (g − x, 0, k − y, 0) with the convention that C has just general moduli if g−x ≤ 2(x −y)−2. Hence we obtain dim(G 1m (C)) = 2m−2−g+x < x either by Brill - Noether for curves with general moduli or by Lemma 3.6 (notice that in this case G 1m (C) = G 1m (C)∗ since we are assuming m < k). Now it is sufficient to show that each time we create a node to pass from C to X we decrease by at least one dim(G 1m (C)). For the first x − y nodes (corresponding to the nodes of Sing(X )\ Sing(Y )) we may apply the proof of Prop. 3.1, getting at the end a general element of W (g− y, x − y, k − y, 0), say C . The remaining y nodes are easier: we do the same proof without even speaking about neutral pairs, because X is obtained from C just gluing together y general pairs of points of C .

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Department of Mathematics University of Trento 38050 Povo (TN) Italy e-mail: [email protected], [email protected]