IMPOSING SINGULAR POINTS AND MANY NODES TO CURVES ON SURFACES

Edoardo Ballico Luca Chiantini

Dipartimento di Matematica Dipartimento di Matematica Universitá di Trento Universitá di Siena 38050 POVO (TN) - Italy 53100 SIENA - Italy e-mail: ballico@science.unitn.it e-mail: chiantini@unisi.it

Abstract. Let S be a smooth projective surface and consider a very ample linear system |H| on S. Imposing singular points of fixed types τ to the elements of |dH| (supported at general points of S) one gets a subspace of the associated projective space, whose codimension is classically called the number of conditions imposed by the singularities to the linear system. The expected number of conditions is determined by the type τ, but it is achieved in general only for large d. The main problem in this theory is to determine bounds on the number d such that τ imposes the expected number of conditions to |dH|. The paper is concerned with types of singularities τ which contains many points of decreasing type, as nodes for instance. In this case, using a form of the Horace method introduced by Alexander and Hirschowitz, it is shown that one can improve the known general bounds for the minimal d for which the number of imposed conditions is the expected one.

The paper is concerned with the number of conditions imposed by general types of singularity to curves C traced on a smooth surface S (see definition 1.1 for a precise definition of ”type” of singularity or multisingularity). Let Z be a scheme associated to the given multisingularity type and supported at general points of S. If |H| is a very ample polarization, and d is large with respect to the degree of Z, then Z imposes independent conditions to the linear system |dH|; the determination of a minimal integer d for which this happens is however a bold question, still widely open (see e.g. [AH], [BC2], [GL], [GLS1], [GLS2]).

The Horace method, introduced by Hirschowitz (see the paper [AH] for recent powerful applications) allows us to proceed by induction on deg(Z) and d; indeed one can move Z putting some points on a divisor C ∈ |H|, so that C∩Z can be controlled; the problem scales to show that the residue scheme Res_{C}(Z) (defined as in [AH] 2.1) imposes independent conditions to |(d − 1)H|. The reduction step, unfortunately, does not always simplify the problem, inasmuch as the residue scheme is not always suitably simpler than Z. Of course, by repeating several times the procedure, we finally get an empty residue, which can be

Typeset by AMS-TEX

easily handled. This is clearly not enough to guarantee that any inductive step can be performed.

In [BC1] and [BC2] we considered multisingularities on the projective plane corresponding to the union of ”fat points”, with the restriction that many of them are nodes (i.e. ordinary double points) or simple points. The main idea behind the papers consists of showing that the long tail of nodes ”tames” the passage from Z to Res_{C}(Z), for the multiplicity at any node is easily controlled, when passing to the residue. Then the inductive steps can be performed and the procedure works. We found, for these multisingular schemes Z in P^{2}, new bounds for the minimum d such that Z imposes independent conditions to |dH|.

In the present paper, we extend the method to general projective surfaces S, with very ample polarization |H|, and to general types of multisingularities. Even in this general setting, if many singularities of Z are nodes or simple points, then we are able to work by induction taking residues with respect to C ∈ |H| and we find some bounds for the minimum d such that Z imposes independent conditions to |dH|. For the numerical conditions (i.e. the length of the ”tail” of nodes and simple points and the bound on d), we refer to the statement of the Main Theorem.

We also prove that, strengthening the numerical conditions a little bit, one obtains that a general element in |dH| containing Z is smooth outside Z and has exactly the prescribed singularities at the points of Z. As shown by the exceptional case in [AC], Th. 3.2 (degree 6 plane curves with 9 nodes in general position) and several others, this is indeed a subtle question, false even in the plane without some restrictions.

In the Main Theorem we find bounds valid for any type of multisingularity, provided the existence of a suitably long tail of nodes or simple points; this forces us to impose some quite restrictive numerical conditions. For instance, at any step we need to move all bad singularities to C, and this forces us to assume dH^{2} large with respect to Z ∩ C. Of course the bounds can be improved if something more is known about the type τ of multisingularity. Some results are sketched in section 2, where we examine the case in which τ is a set of fat points or a set of singularities of type A_{k}.

Notice that at any inductive step we just need to know that we can put on C many points of strictly decreasing type (see definition 1.2). Nodes and simple points are the easiest examples of singularities of strictly decreasing type, so we use tails made up with them. One could use the same procedure and prove similar statements substituting the tail of nodes with tails of points of strictly decreasing type and arranging suitably the numerical assumptions.

The paper was written while both the authors were supported by the italian MURST fund.

The authors are grateful to the referee, who found a gap in the proof of corollary 1 in a preliminary version of the paper and indicated the way to complete it.

1. General results

Let S be a smooth projective surface, defined over a field of characteristic 0.

In the paper we use the notation of topological type of singularities and the associated zero-dimensional schemes, as introduced in [GLS2]. Let us recall the basic definitions:

Definition 1.1. Let P ∈ S and consider a germ of curve C ⊂ S in a neighbourhood of P. The tree T of the singularity of C at P is the embedded resolution tree of C at P with the multiplicities of the total transforms of C. Let Q be a point infinitely near to P; let E be the exceptional divisor and let C′ be the strict transform of C in the blowing up defining Q. We say that Q is essential for C if Q ∈ C′ but it is not a node of E ∪ C′. P itself is considered essential, if C is singular at P. We define as singularity type any subtree τ of T which contains all the essential points for C, with the multiplicity given by T. Let τ be a type of singularity. We say that a 0-dimensional subscheme Z of S is associated with τ if Z is isomorphic, as a subscheme of S, to the scheme defined by the ideal:

{f ∈ O_{S,P} : m_{Q}(f) ≥ m_{Q} ∀Q ∈ τ}

where m_{Q} is the multiplicity of Q in τ and m_{Q}(f) is the multiplicity at Q of the strict transform of f. We define accordingly the type of multisingularity and the corresponding associated schemes.

Let us set some notation for a type of multisingularity, arising from the Horace method explained in [AH]:

Definition 1.2. Let C ⊂ S be a reduced curve, belonging to some linear system |H| and let Z be any zero-dimensional scheme. The residue Res_{C}(Z) with respect to C is defined by the ideal I_{Z} : I_{C}, where I_{Z}, I_{C} are the ideals of Z and C (see [AH] 2.1). We have the exact sequence:

(1)  0 → I_{Res_{C}(Z)}(−H) → I_{Z} → I_{Z∩C,C} → 0

Definition 1.3. Let Z be a zero-dimensional scheme such that Z_{red} is a singleton {P }. Call diameter diam(Z) of Z the maximal degree of a scheme D ∩ Z, where D varies in the set of smooth curves. Fix now a smooth curve D containing P. We may define inductively Z_{0} = Z and for i ≥ 0, Z_{i+1} := Res_{D}(Z_{i}). Put for all i z_{i+1} := length(Z_{i} ∩ D); then the sequence of integers (z_{i})_{i≥1} may depend on the choice of D; however for any choice of D we have z_{i} ≥ z_{i+1} for every i and deg(Z) = ∑_{i≥1} z_{i}. Call depth(z_{i}) or depth_{D}(Z) the first positive integer j with z_{j+1} = 0. We say that Z is of strictly decreasing type if there is a smooth curve D through P such that z_{1} = diam(Z) and the associated sequence (z_{i})_{i≥1} satisfies z_{i} > z_{i+1} for every i < diam_{D}(Z).

When Z has many connected components, say (Z_{1}, ..., Z_{s}), then we define the following related concepts: maxdiam(Z) := max_{1≤j≤s}(diam(Z_{j})), sumdiam(Z) := ∑_{1≤j≤s}diam(Z_{j}), depth(Z) := max{depth_{D}(Z_{j}) : 1 ≤ j ≤ s and D varies in the set of smooth curves}.

Similarly, let τ be a type of multisingularity. We define maxdiam(τ) (resp. sumdiam(τ), depth(τ)) as maxdiam(Z) (resp. sumdiam(Z), depth(Z)), where Z is a scheme associated with τ, supported at general points. Also we define length(τ) as the length of any of its associated schemes.

Remark 1.4. Assume that the zero-dimensional subscheme Z ⊂ S with Z_{red} = {P } is associated to the type τ of a singularity. Call (z_{i})_{i≥1} the associated sequence of invariants of Z with respect to a smooth germ D of curves and fix a smooth germ C of a curve at P. Then there exists another zero-dimensional scheme W ⊂ S with W_{red} = {P } and such that W has associated invariants (z_{i})_{i≥1} with respect to C.

Example 1.5. Here we consider a planar singularity of type A_{2k−1}, k ≥ 2, i.e. a (tacnode) singularity of the germ of curve y^{2} = x^{2k} at 0 ∈ K^{2}. The associated zero-dimensional scheme Z has (y^{2}, yx^{k}, x^{2k}) as ideal at (0, 0) ([GLS2], Ex. 2). Take L := (y = 0) as germ of curve. Res_{L}(Z) has equations y = x^{k} = 0. The associated invariants with respect to L are z_{1} = 2k, z_{2} = k and z_{3} = 0. Take D := x = 0 as germ of curve. We have Res_{D}(Z) = (y^{2}, yx^{k−1}, x^{2k−1}). Iterating we see that the the associated sequence has z_{i} = 2 for 1 ≤ i ≤ k, z_{i} = 1 if k + 1 ≤ i ≤ 2k and z_{i} = 0 for i > 2k.

Example 1.6. Here we consider a planar singularity of type A_{2k}, k ≥ 1, i.e. a cuspidal singularity of the germ of curve y^{2} = x^{2k+1} at 0 ∈ K^{2}. The associated zero-dimensional scheme Z has (y^{2}, yx^{k+1}, x^{2k+1}) as ideal at (0, 0) ([GLS2], Ex. 3). Take L := (y = 0) as germ of curve at 0. Res_{L}(Z) has equations y = x^{k+1} = 0. The associated invariants with respect to L are z_{1} = 2k + 1, z_{2} = k + 1 and z_{3} = 0. Take D := (x = 0) as germ of curve. We have Res_{D}(Z) = (y^{2}, yx^{k}, x^{2k}). Iterating we see that the associated sequence has z_{i} = 2 for 1 ≤ i ≤ k + 1, z_{i} = 1 if k + 2 ≤ i ≤ 2k + 1 and z_{i} = 0 for i ≥ 2k + 2.

Example 1.7. Fix a point P on a smooth surface S and an integer m > 0. The fat point mP is the (m − 1)-th infinitesimal neighborhood of P in S and the associated zerodimensional scheme Z of mP has (I_{P,S})^{m} as ideal sheaf. We have length(Z) = m(m+1)/2 and for every germ C at P of a smooth curve the associated invariants (z_{i})_{i≥1} are z_{i} = m + 1 − i if 1 ≤ i ≤ m and z_{i} = 0 if i ≥ m + 1. In particular the sequence (z_{i})_{i≥1} does not depend on the choice of C and mP is of strictly decreasing type for any C. We call node or ordinary double point the singularity type 2P.

The second simple residue trick. We recall here a trick used to check if a zerodimensional scheme Z satisfies H^{1}(I_{Z}(dH)) = 0. Assume that Z = W ∪2P where P ∈ S is in general position with respect to W_{red}. Choose a reduced, irreducible divisor C ∈ |H| with H^{1}(I_{(C∩W)∪Q,C}(dH)) = 0, Q a point of C, general with respect to W_{red}. Consider the union Z′ of Res_{C}(Z) − {P } with a length 2 subscheme Y of C supported at Q. Then H^{1}(I_{Z}(dH)) = 0 if H^{1}(I_{Z′}((d − 1)H)) = 0. Indeed Y is the ”second simple residue” of a node, with respect to C, i.e. Y = D_{2}(2Q) in the notation of [AH]; the claim follows by [AH], Lemma 2.3 (see also [BC], lemma 1.1, to translate from the vanishing of H^{0} to the vanishing of H^{1}).

Observe that one could consider a similar problem on curves: determine how many conditions are imposed by a singular subset of a smooth curve C to linear systems on C. Clearly in this case the type of singularity is determined by the multiplicities. The answer is in fact well known, as soon as the singularities are supported at general points of C. We state the result, for we are going to use it in the proof of our main theorem.

Proposition 1.8. Let V be a (non necessarily complete) linear system on a smooth curve C. Pick general points P_{1}, …, P_{s} of C and positive integers m_{1}, …, m_{s} such that ∑ m_{i} ≤ dim(V). Then the linear system V′ = V (− ∑ m_{i}P_{i}) of divisors of V vanishing on each P_{i} with multiplicity at least m_{i} satisfies

dim(V′) = dim(V) − ∑ m_{i}.

Proof. We make induction on the number s of points. If s = 1, the claim follows by [L], th.15.(i), section 6. The general step follows by replacing V with V (−m_{1}P_{1}) and repeating the argument.

In particular for divisors L and A = ∑ m_{i}P_{i} on C, when deg(A) = h^{0}O_{C}(L) and the P_{i}’s are general, then the evaluation map H^{0}(O_{C}(L)) → H^{0}O_{A} surjects. When L is non special, this implies h^{1}(I_{A}(L)) = 0.

Now we are ready to give our bound on the number d such that multisingularities with many nodes and simple points impose independent condition to a linear system |dH|.

Theorem (Main). Fix a smooth projective surface S and a very ample divisor H ∈ Pic(S). Call g the genus of a general curve of |H|. Let τ be a type of multisingularity on S and set δ := depth(τ). Let θ be the multisingularity type corresponding to the union of τ, m nodes, and s general points on S. Assume m + s ≥ (H^{2} + 1)δ.

Let d be any integer such that the following conditions hold:

(a) h^{1}(O_{S}(d′H)) = h^{2}(O_{S}(d′H)) = 0 for all d′ ≥ d − δ − 1,

(b) (d − δ)H^{2} − g + 1 ≥ sumdiam(τ),

(c) s splits as a sum s = s′ + s” with s′ + m ≥ δ(H^{2} + 1), s” ≤ h^{0}(O_{S}((d − δ)H) − 2 and length(τ) + 3m + s′ ≤ h^{0}(O_{S}(dH)) − h^{0}(O_{S}((d − δ)H)).

Then there is a scheme W ⊂ S associated with θ and such that h^{1}(I_{W} (dH)) = 0.

Proof. Set t = H^{2}. Take a general C ∈ |H|. Thus C is smooth. By assumption, for every n ≥ d − δ we have h^{1}(O_{C}(nH)) = 0, so that by Riemann-Roch: h^{0}(O_{C}((n + 1)H)) − h^{0}(C, O_{C}(nH)) = t. We will apply δ times a kind of Horace Lemma as in [BC1] and [BC2]; in these steps we will consider s′ general points of S. Then we will handle part of H^{0}(O_{S}((d − δ)H)) using s” general points of S, which implies that we will loose essentially h^{0}(O_{S}((d − δ)H)) − s” conditions from the a priori best possible result. We call η the union of τ, m nodes and s′ points of S, i.e. a type of multisingularities in which s′ connected components correspond to smooth points. We start taking some zero-dimensional subscheme Z(1) of S with Z(1)_{red} a general subset of C, length(Z(1) ∩ C) ≤ h^{0}(O_{C}(dH)) and Z(1) corresponding to some connected components of the type η. By Remark 1.4 we may (and will) assume that C computes the diameter of each connected component of Z(1). We choose the type of Z(1) accordingly with the following restrictions (in the given order):

1) Z(1) contains at least t points of strictly decreasing type;

2) the number of points of Z(1) not corresponding to points of the type τ (they are all nodes or simple points) is minimal;

3) the number of simple points of Z(1) not corresponding to points of the type τ is minimal;

4) h^{0}(O_{C}(dH)) − length(Z(1) ∩ C) is minimal. We will handle separately, using the second simple residue trick, the case in which Z(1) contains t + 2 points of strictly decreasing type, h^{0}(O_{C}(dH)) − length(Z(1) ∩ C) = 1 and we still have a node in η, not corresponding to points of Z(1). Assume that h^{0}(O_{C}(dH)) − length(Z(1) ∩ C) is greater than 1 or it is 1 but we are not in the previous situation, so we cannot use the second simple residue trick; then add h^{0}(O_{C}(dH)) − length(Z(1) ∩ C) simple points which will not appear in θ (it just means that in this case the result is not sharp). Call W (1) the scheme obtained from Z(1) adding the described points. By construction we have length(W (1) ∩ C) = h^{0}(O_{C}(dH)). Hence by proposition 1.8 and our assumptions on d, we get h^{1}(I_{W (1)∩C}(dH)) = 0. Observe that condition 1) for Z(1) can always be fulfilled, for nodes and simple points appearing in η are of strictly decreasing type, and we have at least m+s′ ≥ δ(t+1) of them, by our assumptions. So we may assume that we used at most t among nodes and simple points not in τ, if we did not use the second simple residue trick, or at most t+2 nodes if we used the second simple residue trick. Furthermore, since sumdiam(τ) ≤ h^{0}(O_{C}((d−δ)H)), we inserted a layer of all singularities of τ. Call Z(2) the residual scheme Res_{C}(W (1)) with respect to C if we did not use the second simple residue trick; if we used the trick call Z′ the connected components of Res_{C}(W (1)) except the last node N, and Z(2) the union of the residual scheme of Z′ with a degree two Cartier divisor of C supported at a general point. Call η(2) the type of multisingularities which is the disjoint union of the type of Z(2) and of the types of the connected components of η which do not occur in W (1). Observe that by proposition 1.8 and sequence (1) or by the second simple residue trick, we may replace Z with Z(2) and d with d − 1 to prove our statement. Then we continue in a similar way using the same smooth curve C but with the further restriction that among the connected components of η(2) whose support we put on C there is Z(2). We may do this construction for the integer d − 1 if length(Z(2) ∩ C) ≤ h^{0}(O_{S}((d − 1)H)). This is true by condition 1) in the construction of Z(1). We do the same construction δ times. We are never obliged in two consecutive steps to use the second simple residue trick. So all these steps can be done because we assumed m + s′ ≥ δ(t + 1). Observe that at the end, after δ steps, we obtain a scheme E union of the last residual scheme W = W (δ) with W_{red} ⊂ C and general simple points of S. To obtain the claim, by proposition 1.8, it is sufficient to prove the vanishing of h^{1}(I_{W} ((d − δ)H)). The claim follows from the definition of δ and the inequality length(τ) + 3m + s′ ≤ h^{0}(O_{S}(dH)) − h^{0}(O_{S}((d − δ)H)), which guarantees that in these δ steps we get rid of all the singularities in η.♢

Of course condition (c) can be simplified (loosing something in the sharpness of the result) just requiring length(τ) + 3m + s ≤ h^{0}(O_{S}(dH)) − h^{0}(O_{S}((d − δ)H)) (i.e. taking s′ = s and s” = 0).

From the theorem, we do not know whether a general element C ∈ |dH| containing W is irreducible or not, neither we know exactly its singularities. We can get a more precise description of C if we add conditions on d.

Corollary 1. With the same notation as in the statement of the theorem, assume furthermore that (d − δ − 1)H^{2} − g + 1 ≥ sumdiam(τ), s” ≤ h^{0}(O_{S}((d − δ − 1)H) − 2 and length(τ) + 3(m + 1) + s′ ≤ h^{0}(O_{S}((d − 1)H)) − h^{0}(O_{S}((d − δ)H)). Then a general curve C ∈ |dH| containing W is irreducible and for every P ∈ W_{red}, C has at P a singularity of the prescribed type θ and it is smooth outside W.

Proof. As in the theorem, set t = H^{2}. Fix a general zero-dimensional subscheme W of X of type θ and call Π the linear system P(H^{0}(I_{W} (dH))). We need to prove that, under the numerical hypothesis, a general member of Π is irreducible and has singularities exactly of type θ at each point of W_{red}. Fix a general P ∈ S and let A = 2P be the first infinitesimal neighborhood of P in S, a scheme corresponding to a node. By our assumptions, we may apply the theorem replacing m with m + 1, i.e. adding one node to W. Thus h^{1}(I_{W ∪A}(dH)) = 0 and hence h^{0}(I_{W ∪A}(dH))) = h^{0}(I_{W} (dH)) − 3. This implies that the rational map induced by Π has two-dimensional image and hence it is not composed with a pencil of curves. So a general member of Π is the union of the fixed divisor L with an irreducible curve (by Bertini theorem, see [J], Th. 6.3, part 4), p. 67). Thus we get the irreducibility of a general element in |I_{W} (dH)| if we prove that L = ∅. This can be done as follows. Fix any point P outside W. Our assumptions imply h^{1}(I_{W} ((d−1)H) = 0. Taking a smooth curve D′ ∈ |H| through P and applying the Horace method, exploiting D′, we get h^{1}(I_{W ∪{P }}(dH)) = 0, hence h^{0}(I_{W ∪{P }}(dH)) is smaller than h^{0}(I_{W} (dH)) thus P is not a fixed point of Π. Now let us show that a general D has at the points of W_{red} singularities exactly of the assigned type. In order to obtain a contradiction we assume that at some P ∈ W_{red} a general D has singularity of type ω strictly greater than the assigned one, respectively with generic numerical sequence (w_{i}(P))_{i≥1} and (z_{i}(P))_{i≥1}. Thus w_{i}(P) ≥ z_{i}(P) for every i and w_{j}(P) > z_{j}(P) for at least one index j; take the minimal j for which this happens. Let η′ be the singularity of D at P. We repeat the proof of the vanishing part of the theorem taking η′ instead of η. Since length(τ)+3m+s′ +1 ≤ h^{0}(O_{S}(dH))−h^{0}(O_{S}((d−δ)H)), we may repeat the proof at least for the first j steps; we have now a zero-dimensional scheme Ω ⊂ D which gives the same independent conditions as W for the first j − 1 steps, but at step j it gives one more condition. Since h^{0}(I_{W} (dH)) = h^{0}(O_{S}(dH)) − length(W), we obtain h^{0}(I_{Ω}(dH)) < h^{0}(I_{W} (dH)), hence D cannot be general. Now we will finish the proof by checking that W_{red} = Sing(D) for general D. The assumptions are such that we may apply the theorem replacing d with d − 1 and find a zerodimensional scheme W associated to the multisingularity type θ with h^{1}(I_{W} ((d − 1)H)) = 0. Since d − 2 ≥ d − δ − 1, we have h^{2}(I_{W} ((d − 2)H)) = h^{2}(O_{S}((d − 2)H)) = 0. Hence we may apply the Castelnuovo - Mumford Lemma ([Mu], p. 100) and obtain that I_{W} (dH) is spanned. This implies that a general D ∈ |I_{W} (dH)| is smooth outside W_{red} ([J], Th. 6.3). ♢ Corollary 2. Let Hilb(S, θ, d) be the set of all reduced curves in |dH| whose singularities are exactly of type θ. Then under the assumptions of corollary 1, there exists a curve D ∈ Hilb(S, θ, d) such that Hilb(S, θ, d) is smooth at D and of the expected dimension h^{0}(dH) − length(τ) − 3m − s − 1. Furthermore, for every partial smoothing ρ of θ there is a non-empty component F of Hilb(S, ρ, d) which is generically smooth of the expected dimension h^{0}(dH) − 1 − deg(ρ) and such that the closure of F in Hilb(S) contains the irreducible component of Hilb(S, θ, d) containing D.

Proof. The proof follows from the T-property in the sense of [GLS1] or [GLS2] with respect to formal types or equisingular or (over C) topological types for the associated zero-dimensional scheme and vanishing of h^{1}(I_{W} (dH)) and the fact that a general C ∈ |I_{W} (dH)| is reduced and has a singularity scheme exactly of type θ ([GL], 3.7 and 3.12).♢

Remark. It is very easy to obtain results sharper than the main theorem making some restrictions on the type of multisingularity.

In particular, this can be done for two important classes of multisingularities: fat points (i.e. ordinary multiple points considered in Example 1.7) and singularities of type A_{k} considered in Examples 1.5 and 1.6, i.e. tacnodes and planar cusps, perhaps non-ordinary ones.

The most interesting improvements consist of removing or weakening the hypothesis (d − δ)H^{2} − g + 1 ≥ sumdiam(τ), which is quite strong, since sumdiam(τ) can be large. When the singularities are ”good”, one can replace it, at least partially, with assumptions on depth(τ).

For instance, in the case of fat points, one has depth(τ) = maxdiam(τ) and one could replace condition (b) in the theorem with: H^{2}d − g + 1 ≥ (H^{2} + 2) depth(τ), rephrasing accordingly the other conditions. We leave the details to the reader.

Similarly, one can get even more accurate results, splitting the type τ in several subtypes and asking, for each subtype, conditions which imply the desired vanishing. Of course, in all these particular cases, once the main theorem has been settled, one can easily use it to obtain results similar to our Corollaries 1 and 2.

References

[AH] J. Alexander and A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points, Invent. Math. 140-2 (2000), 303-325.

[AC] E. Arbarello and M. Cornalba, Footnotes to a paper of Beniamino Segre, Math. Ann. 256 (1981), 341-362.

[BC1] E. Ballico and L. Chiantini, Nodal curves and postulation of generic fat points on surfaces, preprint math.AG/9804073.

[BC2] E. Ballico and L. Chiantini, Nodal curves and postulation of generic fat points on P^{2}, Arch. der Math. 71 (1998), 501-504.

[GL] G.-M. Greuel and C. Lossen, Equianalytic and equisingular families of curves on surfaces, Manusc. Math. 91 (1996), 323-342.

[GLS1] G.-M. Greuel, C. Lossen and E. Shustin, New asymptotics in the geometry of equisingular families of curves, IMRN (Internat. Math. Res. Notices) (1997), 595-611.

[GLS2] G.-M. Greuel, C. Lossen and E. Shustin, Plane curves of minimal degree with prescribed singularities, Invent. Math. 133 (1998), 539-580.

[J] J.-P. Jouanolou, Théorémes de Bertini et Applications, Progress in Math. 42 (1983), Birkhäuser.

[L] D. Laksov, Wronskians and Plücker formulas for linear systems on curves, Annales Ec. Norm. Sup. 17 (1984), 45-66.

[Mu] D. Mumford, Lectures on curves on an algebraic surface, Annals of Math. Studies 59 (1966), Princeton University Press, Princeton, N. J..

[Re] R. Re, Multiplication of maps and Clifford bounds for stable vector bundles, Comm. Algebra 26 (1998), 1931-1944.