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Halphen conditions and postulation of nodes

We give sharp lower bounds for the postulation of the nodes of a general plane projection of a smooth connected curve C in P^r and we study the relationships with the geometry of the embedding. Strict connections with Castelnuovo's theory and Halphen's theory are shown.


Introduction
This paper deals with smooth projective non-degenerate curves C ⊂ P r , of degree d and genus g, and their general projections C → C 0 ⊂ P 2 . If N is the set of nodes of C 0 , it is classically known that the postulation of N determines many intrinsic and extrinsic geometric properties of C, such as, for example, the existence of special linear series, the classical relation among d, g and the degree δ of N and the equality e = d − 3 − α, where e is the speciality of C and α is the least degree of a curve containing N (see Section 1).
Much subtler results show how the existence of special linear series g n k depends on the existence of surfaces of low degree s containing C, via complicate but sometimes sharp numerical relations between r, n, k, s, d and g, that one can find in the literature (see [1], [2], [5], [8], [9], or [4], in which our principal procedure is settled).
The main goal of this paper is to prove some sharp lower bounds for ∂h N , the first difference of the Hilbert function of N, and their relationships with the geometry of the embedding C ֒→ P r . Since α is also the maximum of ∂h N , this approach brings into play, in a natural way, the speciality e. Indeed our bounds will be given in terms of d, r and α (rather than e) and, when r = 3, we consider also an integer s such that C is contained in no surface of degree s − 1. These bounds are obviously related to upper bounds for g in terms of d, r, α (hence e) and s. So our theory is strictly related to Castelnuovo's theory and to Halphen's theory of maximal genus, on which we hope to bring some new point of view.
The paper is organized as follows. After the preliminary Section 1, in the second section, we deal with non-degenerate curves in P r , for arbitrary r ≥ 3 and we give a sharp lower bound for ∂h N , depending only on degree and speciality, using the main result in [7]. The result is a Castelnuovo-like theory on the relationship between postulation of nodes and genus of curves.
In the rest of the paper we work in P 3 and we bring s into play. In Section 3 we give some general results which show how s affects ∂h N , the most used in the sequel being Theorem 3.5. For this we use two main tools: first we translate the Hilbert function of N into the dimensions of some linear series on C, using the classical approach and the results contained in [7]; next we bound the dimensions of the linear series, using the procedure introduced in [4] and the results of [6].
In Section 4 we prove the key Theorem 4.2, which produces our main lower bound for ∂h N , depending on d, s and α. As immediate but useful Corollaries we get an upper bound for g (in terms of d, s and α) and a lower bound for α (in terms of d and s).
Then we study the sharpness of our bounds, with the extra assumption d > s(s − 1). In this case our lower bound for α can be made explicit: thus we get a purely elementary arithmetical proof of the celebrated Halphen-Gruson-Peskine bound for e (Proposition 4.8) and we can show that the curves with α minimal and minimal node function are exactly the Halphen curves. Indeed our approach allows an elementary proof of some known characterizations of such curves (Theorem 4.10).
The lower bounds for ∂h N is obtained so far when the value of α is minimal and the curves attaining the bound turn out to be aCM. Our next step, developed in Section 5, is to consider minimal nodal functions with α non-minimal. Here the geometry becomes much more difficult, and we are able to give some results only for s ≤ 4. The case s = 2 is easy and we can give a complete picture. For s = 3 and s = 4 we show that our bounds are sharp for every d > s(s − 1), by producing explicitly curves on the smooth cubic surface or in the so-called Mori surfaces (see [15]). Moreover, using Theorem 3.5, we show that, under some mild numerical restriction, every curve attaining the bound must lie on a surface of degree s. Surprisigly the curves we produce, except for some sporadic cases, are not aCM: this makes us believe that the study of curves with minimal nodal postulation but non-minimal α, and of their cohomology, is a challanging problem, even for low values of s.

Preliminaries and known results
In this section we fix some notation and we collect some known results. We work over an algebraically closed field of characteristic zero. Let C ⊂ P r , r ≥ 3, be a smooth connected non-degenerate curve of degree d and genus g.
We put n := h 0 (O C (1)) − 1, whence C is a projection of a linearly normal curve of degree d in P n (i.e. of a curve in P n which is not a projection of a curve of the same degree spanning a higher dimensional projective space). In particular C is linearly normal if and only if n = r.
We denote by e the index of speciality of C, namely Recall that C is said to be special if and only if e > 0.
Let C → C 0 ⊂ P 2 be a general plane projection, so that C 0 has only nodes as singularities.
Let N ⊂ P 2 be the reduced subscheme of the nodes of C 0 and denote by ∆ the pull-back of N to C, viewed as a divisor on C.
If δ is the length of N, we have the classical relation: Let h N be the Hilbert function of N and let ∂h N be its first difference. Finally let α := min{j | h 0 (I N (j)) = 0}.
The classical theory of adjoint curves relates the linear systems of curves passing through N to the geometry of C. The starting points of this theory can be summarized in the following remark (for details see e.g. [7]): Remark 1.1. Let D be a hyperplane divisor of C. Then we have: (i) the canonical map H 0 (I N (j)) → H 0 (O C (jD − ∆)) is surjective for every j ∈ Z and also injective for (iii) from (i) and (ii) it follows, by Riemann-Roch and standard calculations: (iv) by (ii) the the map of (i) can be viewed as H 0 (I N (j)) → H 0 (ω C (d − 3 − j)), hence from (iii) it follows: (v) from the above we get α = d − 3 − e. It follows α ≤ d − 2, with equality if and only if g = 0. Moreover α ≤ d − 4 if and only if C is special.
By general properties of the Hilbert function of a zero-dimensional subscheme of P 2 and by Remark 1.1 we have the following result: Proposition 1.2. The difference function ∂h N satisfies: (a) ∂h N (j) = 0 for j < 0 and for j > d − 3; Proposition 1.2 above gives an accurate description of the function ∂h N , except for the interval [α, d − 4].
The following result ( [7], Theorem 5.1), which holds in arbitrary characteristic, provides a lower bound for the descent of ∂h N (j) at any step between α and d − 3. More refined bounds for r = 3 will be given in the next sections.
In particular ∂h N is strictly decreasing in the interval [α, d − 2].
It is easy to derive some (well known) speciality bounds from the previous result.
Corollary 1.4. The following inequalities hold: Proof. α is the maximum for ∂h N . On the other hand by Theorem 1.3 ∂h N decreases of at least n − 2 at any step from j = α to j = d − 3; furthermore ∂h N (d − 3) = n − 2 by Proposition 1.2(d) and (a) follows.
Clearly (b) is equivalent to (a). Finally (c) follows from (b) and Remark 1.1(iii).
In P 3 we get back the well known speciality bound e ≤ (d−4) 2 , attained for instance by complete intersections on quadrics. A complete classification of the curves in P 3 achieving this speciality bound will be given in Proposition 5.3 (see also Remark 5.4).
Remark 1.5. Concerning the relations between the values ∂h N (α − 1) = α and ∂h N (α) we can say the following: (a) by Theorem 1.
Indeed C is subcanonical if and only if αd = 2δ. Moreover in this case there is a double structure on N which is a complete intersection of type (α, δ) (see [7]); (c) if α is minimal with respect to d and r, namely . This is an easy consequence of Corollary 1.4 (details to the reader).
We end this section by showing that when C is arithmetically Cohen-Macaulay (aCM) ∂h N can be computed from the Hilbert function of a general hyperplane section of C (and conversely).
We recall first an easy fact.
Remark 1.6. Let Γ be a general hyperplane section of C. Then it is easy to see that with equality whenever h 1 (I C (j)) = 0: moreover equality holds for all j ∈ Z if and only if C is aCM.
Hence by Remark 1.1(iii) we have with equality whenever h 1 (I C (j)) = 0; moreover equality holds for every j ≤ d − 1 if and only if C is aCM.
Example 1.7. Assume that C is aCM and let Γ be a general hyperplane section of C. Then by Remark 1.6 we have It is also easy to see that the index of speciality of C is For example if r = 3 and C is a complete intersection of type (s, s + u), with u ≥ 0, we have if j ≥ 2s + u − 1 It follows (as expected): e = 2s + u − 4 whence α = d + 1 − (2s + u) and, with some calculations, Remark 1.8. The Hilbert function of N is strictly related with the spectrum ℓ C of C, defined as (see [16]): ℓ C (j) := ∂ 2 h 0 (O C (j)) for all j ∈ Z Indeed from Remark 1.1(iii) it follows easily that For example if C is aCM and Γ is a general hyperplane section of C the spectrum of C is ℓ C (j) = ∂h Γ (j) for j ∈ Z. as follows easily by Example 1.7.

Postulation of nodes and Castelnuovo's curves
In this section we use Theorem 1.3 to produce a lower bound for ∂h N in terms of d and r and we show that this bound is sharp for curves of maximal genus, that is Castelnuovo's curves. Moreover we give some lower and upper bounds for the length δ of N depending on d, r and α, hence lower and upper bounds for g depending on d, r and e.
Remark 2.2. Proposition 2.1 shows that the lower bound for ∂h N provided by Theorem 1.3 and the lower bound for α given by Corollary 1.4 are sharp, since they are attained by Castelnuovo's curves. However we will see that these bounds can be achieved also by other curves in P 3 , where the situation is quite clear (see Section 5). In higher-dimensional spaces several things are still unclear. For example even considering complete intersections of quadrics we have the following situation.
On the other hand, the reader can easily realize (e.g. by using Example 1.7) that complete intersections of quadrics in higher dimensional projective spaces, which are not Castenuovo's curves, do not have minimal descent of r − 2 at any step after α + 1.
Also in P 4 more general complete intersections do not have the minimal descent, as the following example shows. Example 2.3. Let C be a complete intersection of type (2, 2, 3) in P 4 . Then d = 12 and g = 13, whence δ = 55 − 13 = 42. Since Then ∂h N (j) = j + 1 for j < 7 while ∂h N (j) = 0 for j > d − 3 = 9; furthermore the sum of ∂h N is 42.
Observe that there is an intermediate step in which the descent is by 3, i.e. more than the minimum allowed r − 2 = 2, whence, in particular, C is not a Castelnuovo curve. However α reaches the minimal value established by Corollary 5.3.
Then (a) follows by a straightforward calculation. To The conclusion follows by a direct computation.
Remark 2.5. The bounds of Proposition 2.4 are equivalent to bounds for the genus in terms of d, r and e. The easy transformation is left to the reader.

Postulation of nodes and flag conditions
The sharpness of the previous results is rather flabby because as soon as one takes curves which do not lie on surfaces of minimal degree, then stronger restrictions seem to apply (consider for example complete intersections of surfaces of degree at least 3 in P 3 , see 4.7).
So one is led to ask whether more information about the flags of subvarieties of P r containing C could give more restrictive constraints to the behavior of ∂h N .
To obtain sharper information on the shape of ∂h N one may start with the formula ). This formula can be read as where D is a hyperplane divisor and L : . Then one is led to study the growth of the linear series |L + jD| with respect to j, that is the growth Hilbert function h L of L, defined as: which was introduced in [2] and studied intensively by many authors.
With this notation (5) can be rewritten as and any result on the step-by step growth of h L can be translated into results on the descent of ∂h N from α to 0, and conversely. Equivalently, one may start from equation in Remark 1.1(iii), which can be rewritten as Here information on the growth of h L are translated, reversing the orientation, into information on the decrease of ∂h N , starting from d − 3 and going backward to α. We will take both points of view in order to relate geometric properties of the embedding C ⊂ P r to arithmetic properties of the functions h L above.
Remark 3.1. In order to show how this method works we give an alternative short proof (in characteristic zero) of Theorem 1.3.
Let M be a divisor corresponding to ω C (j − d + 3) and let D be a hyperplane divisor of C. Then by assumption one has h 0 (O C (M − D)) > 0 for all j ≥ α + 1. Let Γ be a general hyperplane section of C and look at the exact sequences: ) be the dimensions of the images in H 0 (O Γ ) of the two left hand side maps above. Then Γ is in uniform position with respect to the linear series |M| and |D|, so one may apply Castelnuovo's lemma of §1 in [5]. It turns outthat v ′ ≥ v + r − 1. A direct computation shows that the same inequality holds also when j = α. Now by equality (5) above, we obtain: and the claim follows. Now we go back to our main problem, namely to take into account some flag conditions in order to get sharper information on ∂h N . As far as we know, such a theory is available only for curves in P 3 , where the knowledge of a number s such that C is contained in no surfaces of degree smaller than s influences the growth of the dimensions of linear series on the curve, under the addition of multiples of D. Indeed let us recall a corollary of the main technical result of [6], appearing in [4] (see also [16] for generalizations to non-integral curves): Theorem 3.2. Let C ⊆ P 3 be a smooth connected curve and let L be a divisor on C. Let t and j be integers such that: Then: Proof. See [4], Theorem 2.3.
Notice that the assumption of the above theorem implies t−1 2 < d − t 2 , hence also d > (t − 1) 2 + 1, whence by Laudal's Lemma also the general hyperplane section of C is not contained in any curve of degree t − 1.
We point out the consequences of the previous theorem for the Hilbert function of N, in the following: Theorem 3.3. Let C ⊆ P 3 be a smooth connected curve not lying on any surface of degree smaller than s. Then for all j with α ≤ j ≤ d − 2, if t is any integer satisfying: one has: Proof. The claim follows easily by (6) and Theorem 3.2.
We want to explain roughly how one can use Theorem 3.3 to understand the behavior of ∂h N (j), for j ∈ [α + 1, d − 3]. Indeed α is the maximum for the function ∂h N , achieved for j = α − 1; in the next few steps, when j is a little bit bigger than α, then the difference j + 1 − ∂h N (j) can be small and one may apply theorem 3.3 only for small (but nevertheless increasing) t; so the rate of the descent of ∂h N is small, but increasing. Eventually j + 1 − ∂h N (j) becomes bigger than s−1 2 , and one is allowed to apply the theorem taking s = t (of course provided that j + 1 − ∂h N (j) is still smaller than d − s 2 ): here ∂h N (j) is forced to decrease by s−1 (at least) at any steps. Finally when j approaches d − 2, then j + 1 − ∂h N (j) becomes bigger than d − s 2 and once again one must take t < s, in order to apply the theorem: in other words the rate of the descent of ∂h N is allowed to become smoother.
The following picture synthesizes the previous discussion on the shape of ∂h N : Let us see the minimal descent of ∂h N in the following numerical case: d = 24, s = 4, g = 73, α = 15 (here δ = 180). The function ∂h N increases by 1 at any step from 0 to α − 1 = 14. Then: -for j = 15, we get 15 = α ≥ ∂h N (15). Assume the maximum is attained; then for t = 2, one sees that the inequalities (a),(b) of the previous theorem are satisfied (in fact t = 2 is the maximum which satisfies the left hand side) so ∂h N is forced to decrease at least by 1; -for j = 16, we get then 14 ≥ ∂h N (16). Assume the maximum is attained; then for t = 3, one sees that the inequalities (a),(b) of the previous theorem are satisfied (in fact t = 3 is the maximum which satisfies the left hand side) so ∂h N is forced to decrease at least by 2; -for j = 17, we get then 12 ≥ ∂h N (17). Assume the maximum is attained; then for t = 4, one sees that the inequalities (a),(b) of the previous theorem are satisfied (in fact t = 4 is the maximum which satisfies the left hand side) so ∂h N is forced to decrease at least by 3; -for j = 18, we get then 9 ≥ ∂h N (18). Assume the maximum is attained; then the inequality (b) of the previous theorem holds for t = 5, but 5 > s; we may just conclude that ∂h N is forced to decrease at least by 3; -for j = 19, we get then 6 ≥ ∂h N (19). Assume the maximum is attained; then for t = 4, one sees that the inequalities (a),(b) of the previous theorem are satisfied (in fact, now t = 4 is the maximum which satisfies the right hand side) so ∂h N is forced to decrease at least by 3; -for j = 20, we get then 3 ≥ ∂h N (20). Assume the maximum is attained; then for t = 3, one sees that the inequalities (a), (b) of the previous theorem are satisfied (in fact t = 3 is the maximum which satisfies the right hand side) so ∂h N is forced to decrease at least by 2; -for j = 21, we get then 1 ≥ ∂h N (21). Here the maximum must be attained, for 21 = d − 3 and we know that ∂h N (d − 3) ≥ 1 for all curves in P 3 .
A similar conclusion holds taking the reverse point of view and using (8) instead of (6).
Theorem 3.5. Let C ⊆ P 3 be a smooth connected curve not lying on any surface of degree smaller than s. Then for all j with α ≤ j ≤ d − 2, if t is any integer satisfying: Proof. The claim follows from (8) and Theorem 3.2.
-for j = 21 = d −3, we must have ∂h N (21) ≥ 1. Assume that the minimum is attained; then for t = 3, one sees that the inequalities (a),(b) of Theorem 3.5 are satisfied (in fact t = 3 is the maximum which satisfies the left hand side) so ∂h N is forced to increase at least by t − 1 = 2, going one step back; -for j = 20, we get then ∂h N (20) ≥ 3. Assume the minimum is attained; then for t = 4, one sees that the inequalities (a),(b) of the previous theorem are satisfied (in fact t = 4 is the maximum which satisfies the left hand side) so ∂h N is forced to increase at least by 3, going one step back; -for j = 19, we get then ∂h N (19) ≥ 6. Assume the minimum is attained; then for t = 5, one sees that condition (b) of the previous theorem is satisfied, but condition (a) fails. All we can say, taking again t = 4, is that ∂h N is forced to increase at least by 3, going one step back; -for j = 18, we get then ∂h N (18) ≥ 9. Assume the minimum is attained; then for t = 5, one sees that condition (b) of the previous theorem is satisfied, but condition (a) fails. All we can say, taking again t = 4, is that ∂h N is forced to increase at least by 3, going one step back; -for j = 17, we get then ∂h N (18) ≥ 12. Assume the minimum is attained; then for t = 3, one sees that the inequalities (a),(b) of the previous theorem are satisfied (in fact, now t = 3 is the maximum which satisfies the right hand side) so ∂h N is forced to increase at least by 2, going one step back; -for j = 16, we get then ∂h N (16) ≥ 14. Assume the minimum is attained; then for t = 2, one sees that the inequalities (a),(b) of the previous theorem are satisfied (in fact, now t = 3 is the maximum which satisfies the right hand side) so ∂h N is forced to increase at least by 1, going one step back; -for j = 15 = α, finally we get then ∂h N (21) ≥ 15 = α. Here the minimum must be attained.
which are exactly the same values as before.
Notice that the numbers d = 24, s = 4, g = 73, α = 15 above are not random: they are the numbers of a complete intersection C of type (4, 6), see Example 1.7.

Minimal node functions and Halphen curves
In this section we use the previous results to construct a new function which bounds ∂h N from below, and we show that this bound is sharp for the so-called Halphen's curves (see Theorem 4.10). Further examples will be given in the next section.
we agree that the maximum of the empty set is −∞).
(b) Given integers d, s, n, α with d ≥ 3, s ≥ 2, n ≥ 3 and α ≥ 0 we define the minimal node function ∆ with respect to d, s, n, α as follows: We set The next theorem shows that minimal node functions provide lower bounds for ∂h N , which are sharp in some significant cases, as we shall see later.
Theorem 4.2. Let C ⊆ P 3 be a smooth connected curve of degree d, lying on no surface of degree smaller than s (s ≥ 2). Let ∆ be the minimal node function, with respect to d, s, n, α. Then ∂h N (j) ≥ ∆(j) for all j ∈ Z.
Proof. Notice that ∂h N (j) = ∆(j) for all j outside of the interval [α, d − 4], and in particular ∂h N (d − 3) = ∆(d − 3) = n − 2. We prove the non-trivial part of our statement by descending induction on j, starting with j = d − 3; namely we show that, for α and since ∂h N (j − 1) ≥ ∂h N (j) + n − 2 by Theorem 1.3, the conclusion is clear.
In view of Theorem 4.2 it is natural to give the following Definition.
Definition 4.3. Let C ⊆ P 3 be a curve of degree d not lying on any surfaces of degree s (s ≥ 2). We say that C has minimal nodal postulation when ∂h N coincides with the minimal node function with respect to d, s, n, α.
From the above theorem we get the following two Corollaries which give an upper bound for the genus and a lower bound for α (whence an upper bound for the speciality). These bounds are sharp in some important cases (see Proposition 4.8, Theorem 4.10 and section 5).  Proof. Let ∆ be the minimal node function with respect to d, s, n and α. Then by Theorem 4.2 and Proposition 1.2 we have ∆(α) ≤ ∂h N (α) ≤ α, whence the conclusion by the definition of ∆.
Remark 4.6. (i) Let us observe that once we fix d, s n and α, the minimal nodal function ∆ with respect to these values is completely determined, so is its sum δ(∆). Hence the upper bound for g given by Corollary 4.4 depends only on these numbers, or, equivalently, on s, d n and e.
Likewise if we fix d, s and n the lower boundᾱ for α given by Corollary 4.5 is completely determined by d and s.
(ii) We will be able to compute explicitlyᾱ under the assumption d > s(s − 1) and n = 3 (see Proposition 4.8), and from this we will be able to classify the curves with minimal nodal postulation and minimal α in the given range: they are, of course, the well-known Halphen curves (see Theorem 4.10).
The study of curves with minimal nodal postulation but non-minimal α seems to be much harder; in particular it is not clear whether the above bound for the genus is sharp. We will give some partial results for d > s(s − 1) and s ≤ 4 (see Section 5).
Our results and examples seem to indicate that the classification of curves with minimal nodal postulation is not straightforward and could be a real challenge.
Before coming to the main results of this section we want to say something on complete intersections. Observe that our previous examples 3.4 and 3.6 show that complete intersection curves of type (4, 6) have minimal nodal postulation. More generally the following example shows that this is true for all complete intersections.
Example 4.7. Let C ⊆ P 3 be a complete intersection of type (s, s + u), with s ≥ 2 and u ≥ 0. Then d = s 2 + su, s has the usual meaning, α = d − 2s − u + 1 (see Example 1.7). Moreover ∂h N (d − 3) = 1, whence n = 3. The minimal nodal function ∆ with respect to d, α, s as above and n = 3 can be computed easily from the definition and is: By Example 1.7 we have ∆ = ∂h N , that is C has minimal nodal postulation.
Now we want to show that minimal nodal postulation is achieved not only by complete intersection curves but also by Halphen's curves. We begin by showing a lower bound for α. This bound is well-known, but we give a new proof as a consequence of our theory of minimal nodal functions. Proposition 4.8. Let C ⊆ P 3 be a curve of degree d not lying on any surface of degree < s. Assume s ≥ 2 and d > s(s − 1). Let (k, u) be the unique pair of integers such that d = s 2 + us − k, 0 ≤ k ≤ s − 1 and u ≥ 0. Assume further that C is not a complete intersection (i.e. k ≥ 1). Then we have: then the minimal nodal function with respect to d, s, α and n = 3 is the following: Proof. To prove (i) we use Corollary 4.5. A lengthy and tedious but elementary calculation allows to compute the function ∆ (see Definition 4.1) with respect to d, s and n = 3. Going backwards step by step starting with d − 3 and using repeatedly the assumption d > s(s − 1) we get: If k ≥ 2 we have ∆(d−2s−u+2) = d−2s−u+2, and if k = 1 we have ∆(d−2s−u+2) = d − 2s − u + 1. Then (i) follows from Corollary 4.5.
The proof of (ii) is another tedious but elementary calculation using (i) and the definition of ∆. Remark 4.9. Since e = d − 3 − α it is immediate to see that the bound of Proposition 4.5 is equivalent to the Halphen-Gruson-Peskine bound e ≤ d s + s − 4. The first modern proof of this bound for e is in [11].
Another proof of the bound for e can be found in [16], where the Halphen-Gruson-Peskine bound is generalized to non-integral curves, using a fine study of the spectrum. (ii) if k = 0 the function ∂h N is as in Example 4.7. If k > 0 we have α = d−2s−u+2 and (iii) C has minimal nodal postulation with respect to d, s, n = 3 and In other words: if ∆ is the minimal node function with respect to d s α and n = 3 we have: ∆(j) = ∂h N (j) for every j ∈ Z; (iv) C has maximal genus in the set of all curves of degree d not lying on any surface of degree < s.
Proof. (i) ⇒ (ii). If k = 0 follows immediately by Example 4.7. Assume now k > 0 and let Y be the plane curve linked to C on S. Then α = d−2s−u+2, since ω C is cut by surfaces of degree 2s + u − 4 passing through Y (outside the intersection Y ∩ C). Moreover C is aCM, and hence to compute ∂h N we can use Example 1.7. Now the general hyperplane section Γ of C is directly linked in a complete intersection (s, s+u) with the general hyperplane section of Y , which is collinear of degree k. Then by a well known relation between the Hilbert functions of linked zero-dimensional schemes (see [10]) we have (12) ∂h On the other hand by Example 1.7 we have and a straightforward calculation shows (ii).
(ii) ⇒ (i). Assume first that k > 0 and let C ′ be a curve of degree d linked to a plane curve of degree k by a complete intersection of type (s, s + u). Let N ′ be the set of nodes of a general plane projection of C ′ . Then ∂h N = ∂h N ′ , whence by Remark 1.1(iii) h 0 (O C (j)) = h 0 (O C ′ (j)) for all j ∈ Z. This easily implies, since C ′ is aCM, that h 0 (I C (j)) − h 0 (I C ′ (j)) = h 1 (I C (j)), whence, in particular, h 0 (I C (j)) ≥ h 0 (I C ′ (j)). By easy degree considerations this implies that h 0 (I C (j)) = h 0 (I C ′ (j)) for j ≤ s + u − 1, whence h 1 (I C (j)) = 0 for j ≤ s + u − 1. In particular C is contained in an irreducible surface S of degree s, and a surface T of degree s + u not containing S.
Let now Γ and Γ ′ be the general hyperplane sections of C and C ′ respectively. Then by Remark 1.6 it follows that h Γ ′ (j) ≥ h Γ (j) for every j. We want to show that h Γ ′ = h Γ . Assume this is not true and set t := min{j ∈ Z|h Γ ′ (j) > h Γ (j)}. By the above considerations we have that t ≥ s + u and that ∂h Γ is strictly decreasing for j ≥ t until it vanishes. Now ∂h Γ ′ is explicitly given in (12) (with Γ replaced by Γ ′ ), and since ∂h Γ ′ (j) = d = ∂h Γ (j) an easy calculation leads to a contradiction. Then h Γ ′ = h Γ , whence C is aCM by Remark 1.6 and Γ is linked to a collinear scheme in a complete intersection of type (s, s + u). Then it is easy to see that S ∩ T = C ∪ C ′′ , where C ′′ is a plane curve of degree k.
If k = 0 the proof is similar to (and easier than) the previous one and is left to the reader.
Example 4.11. There are some (few) examples in which the minimal node function is the effective Hilbert function of the nodes of a curve coming from P r , r ≥ 4. This is the case, for instance, of a curve C ⊆ P 3 which is a general projection of a Castelnuovo curve in P r , at least when d > (r − 2)(r − 1).
Indeed every Castelnuovo curve C ′ ⊆ P r with d > 2r lies on a surface of (minimal) degree r − 1 (see [3] or [12]), which is also the minimal degree of a surface containing the projection C of C ′ to P 3 , if d > (r − 2)(r − 1). Then in the definition of minimal node function, we get here t ≤ s = r −1 so that max{t−1, r −2} = r −2 hence, by construction (4.1 (2)), the minimal node function satisfies: Now just observe that Castelnuovo's curves reach exactly this bound by Proposition 2.1.
In order to get more refined bounds, similar to the ones in P 3 , it seems reasonable to take into account some flag conditions, such as, for example, the ones used in [6] to bound the genus. Observe also that for these curves we have α = ⌈ (d−2)(r−2) r−1 ⌉, whence α is not minimal with respect to d and s = r − 1 > 2.
We will see in the next section that if α is not minimal there are also linearly normal non-aCM curves having minimal node postulation.

Curves on surfaces of small degree
In this section we study a class of curves having minimal node postulation, with n = 3, s ≤ 4, d > s(s − 1) and α not minimal. This shows, in particular, that the Halphen type bound for the genus in terms of d, s and e given in Corollary 4.4) is sharp in some non-trivial cases. We leave the easy translation to the reader. 5.1. The case s = 2. If s = 2 the inequality d > s(s − 1) is always satisfied, the minimal node function ∆ coincides with the lower bound given by Theorem 1.3 and is defined as follows: It follows that Proposition 5.1. Assume C lies on a quadric surface Q. Then: (c) C has minimal node postulation (4.3).
Proof. (a) We have e = b − 2 (e.g. by the adjunction formula) and the conclusion follows by Remark 1.1.
(b) If d is even C is a complete intersection of type (2, d 2 ) and then e = d 2 − 2, whence α = d 2 − 1. If d is odd then C is linked to a line ℓ in a complete intersection of type (2, d+1 2 ). Then the complete canonical series of C is cut out by the linear system of all surfaces of degree d+1 2 − 2 containing ℓ, outside the divisor ℓ · C. It follows that e = d+1 2 − 3, whence α = d−1 2 . (c) Set p := d − 2 − α. Then p ≥ 0 and a straightforward calculation, using (a), (b) and (14) shows that The conclusion follows by Theorem 1.3 and an easy computation.
The following Proposition gives a characterization of curves lying on a quadric surfaces, which completes Proposition 5.1.
As an application of the above Proposition we give the following characterization of the curves in P 3 having minimal α. (i) (d, g) ∈ {(3, 0), (4, 1), (5, 1), (5, 2)} (in particular C is non-special); (ii) C is a special Castelnuovo curve; (iii) d = 9 and C is a complete intersection of two cubics; (iv) d = 7 and C is linked to a conic in a complete intersection of type (3,3).
Assume If C is special and lies on a quadric then C is a Castelnuovo curve by Proposition 5.1.
Finally assume that C is special and does not lie on a quadric. Then d ≥ 6 and by [7], Proposition 5.5 d must be odd. Then by Proposition 4.8 (with s = 3) we must have either d = 9 or d = 7. Moreover by Proposition 5.2 we must have (15) ∂h Remark 5.4. (i) Proposition 5.1(c) shows that every curve in P 3 lying on a quadric, has minimal node postulation. This is true in particular also by non-Castelnuovo curves (that is curves of type (a, b) on a smooth quadric, with |a − b| ≥ 2). Moreover it is easy to check that every function (13) occurs on a smooth quadric, provided α satisfies the obvious requirement α ≥ d − 2 − α (just take curves of type (α + 1, d − α − 1) on the quadric).
(ii) On the other hand Proposition 5.2 shows that every special curve with minimal node postulation (13) must lie on a quadric, whence by Proposition 5.3 every special curve in P 3 achieving both the bound of Theorem 1.3 and the lower bound for α is necessarily a Castelnuovo curve.
(iii) It follows easily by Remark 1.1(v) that a non-special curve C ⊆ P 3 has α ≥ d − 3 hence it trivially attains the lower bound of Theorem 1.3, by Riemann-Roch and by 1.1 (iii). On the other hand such a C does not necessarily lie on a quadric surface (e.g. general rational curves of degree d ≥ 5 and elliptic curves of degree d ≥ 5). Compare with Proposition 5.2.
(iv) We feel it would be interesting to generalize the above results to higher-dimensional spaces, replacing quadric surfaces with surfaces of minimal degree. In particular we don't know if (ii) holds for such curves.
Proposition 5.5. (a) Let S ⊆ P 3 be a smooth cubic surface. Then for every pair of integers (v, z) with v ≥ 0 and z ≥ −1 there is a smooth connected curve C ⊆ S having minimal node postulation (16). (b) Conversely any smooth curve C ⊆ P 3 having minimal node postulation (16) with z ≥ 2 and d > 12 lies on a cubic surface.
Proof. We use [13], Ch. V, §4 as a basic reference for the geometry of the cubic surface. We begin by showing that S contains a smooth rational curve C ′ of degree v + 1. Indeed (notation as in [13]) it is sufficient to consider a general element of the linear system if v is odd This shows our claim for z = −1. Assume now z ≥ 0 and let H be a hyperplane divisor on S. Then the linear system |C ′ + (z + 1)H| is very ample by [13], V.4.12, whence a general curve C ∈ |C ′ + (z + 1)H| is smooth and irreducible.
We claim that C has the required properties. Indeed one computes easily that C has degree v + 3z + 4 = d and genus g = z(z+1) 2 + (z + 1)(v + 3), whence δ = δ(∆) by (17). Then by Theorem 4.2 C has minimal node postulation provided it has the right speciality, namely e(C) = z. Now since −H is a canonical divisor for S we have that (C ′ + zH) · C is a canonical divisor of C, whence e(C) ≥ z. Then we need to show that (C ′ − H) · C is non-effective.
From the exact sequence recalling that C ′ is rational and that (C ′ − H) · C ′ is a canonical divisor of C ′ , we get H 0 (O S (C ′ − H)) = 0. Hence from the exact sequence This concludes the proof of (a).
Remark 5.6. The curves with minimal node postulation we produce in the proof of Proposition 5.5 are, by construction, bilinked to the rational curve C ′ , whence they have, up to shift, the same first cohomology of C ′ (see e.g. [14]). In particular if v + 1 = deg(C ′ ) ≥ 4 all our curves are not aCM. It would be interesting to know if there are aCM curves of the same degrees and specialities, having minimal node postulation. 5.3. The case s = 4. Now we consider the case s = 4. The case where α is minimal and C has the minimal node function is yet covered by Theorem 4.10(ii). Hence we may assume that α is not minimal, namely α > ⌈ 3d 4 ⌉ − 3. We shall also assume that α ≤ d − 5, otherwise the meaningful part of the node function, from α to d − 3, is too short and everything trivializes. Then the minimal node function can be written as follows, for suitable integers v ≥ 1 and z ≥ 1: (18) ∆ : . . . , 0, 1, . . . , v + 3z + 3, 3z + 3, . . . , 6, 3, 1, 0, . . .
It remains to show that C has the required speciality, namely e(C) = z + 1. It is clear that e(C) ≥ z +1 and then we have to show that the linear system |C ′ −H| is non-effective (see proof of Proposition 5.5).
We assume then |C ′ − H| effective and we seek for a contradiction. Since (C ′ − H) 2 = −4 this linear system must have a fixed integral curve E with negative self-intersection. It follows that E is rational and E 2 = −2, whence, in particular, |E| = |C ′ − H|. Then if p := deg(E) we have p ≤ v − 2. On the other hand, by (ii) above, there are uniquely determined integers x and y such that E ∼ xC ′ + yH, whence: (20) (v + 3)x + 4y = p and (21) 2(v − 1)x 2 + 2(v + 3)xy + 4y 2 = −2 Squaring (20) and subtracting (21) multiplied by 4 we get , whence x 2 < 1 by a simple computation, remembering that p ≤ v − 2. This is a contradiction and the proof of (a) is complete. Now we prove (b). Let ∆ be as in (18) with z ≥ 2 and let C be a curve such that ∂h N = ∆. Then we have ∂h N (d − 6) = 9 and ∂h N (d − 5) = 6. On the other hand, since d > 20, we have: whence h 0 (I C (4)) > 0 by Theorem 3.5.
The next example shows that there are smooth quartic surfaces, other than the Mori surfaces, containing curves with minimal nodal postulation.
Observe also that in this case α is not minimal (the least value being 11), in agreement with Theorem 4.10.
Note also that the quartic surface cannot be one of the surfaces considered in the proof of Proposition 5.7, because the classes of the two skew lines and of a hyperplane divisor are easily seen to be linearly independent.
Remark 5.9. If v > 3 the curves we produce in the proof of Proposition 5.7 are not aCM. Indeed the starting curve C ′ of degree v + 3 and genus v has non trivial H 1 (I C ′ (2)) for v > 3, and then we can use a bilinkage argument as in Remark 5.6.