ON THE GENUS OF PROJECTIVE CURVES

VERIFYING CERTAIN FLAG CONDITIONS

Luca Chiantini, Ciro Ciliberto and Vincenzo Di Gennaro

SUNTO. Assegnati degli interi r, s_{1}, ..., s_{l}, sia C(r; s_{1}, ..., s_{l}) l’insieme di tutte le curve C dello spazio proiettivo P^{r}, di grado s_{1}, ridotte ed irriducibili, tali che, per ogni i = 2, ..., l, C non sia contenuta in sottovariet di P^{r} di dimensione i e grado < s_{i}. Diremo che una curva C verifica una condizione di bandiera di tipo (r; s_{1}, ..., s_{l}) se C appartiene a C(r; s_{1}, ..., s_{l}). Nel presente lavoro, sotto l’ipotesi s_{1} >> ... >> s_{l}, si determina un limite superiore G^{h}(r; s_{1}, ..., s_{l}), dipendente solo da r, s_{1}, ..., s_{l}, per il numero G(r; s_{1}, ..., s_{l}) := max{p_{a}(C) : C ∈ C(r; s_{1}, ..., s_{l})}, dove p_{a}(C) denota il genere aritmetico di C. Si osservi che il problema del calcolo di G(r; s_{1}, ..., s_{l}) una generalizzazione di quello studiato da Halphen [H, 1883] e da Castelnuovo [Ca, 1889]. Poi si stabilisce una limitazione per l’errore G^{h}(r; s_{1}, ..., s_{l}) − G(r; s_{1}, ..., s_{l}) da cui si deduce che le curve C verificanti condizioni di bandiera di tipo (r; s_{1}, ..., s_{l}) e di genere massimo devono stare su una bandiera del tipo C = V_{s_{1}}^{1} ⊂ V_{s_{2}}^{2} ⊂ ... ⊂ V_{s_{l}}^{l} ⊂ P^{r}, dove V_{s}^{j} denota una sottovariet di P^{r} di grado s e dimensione j. Inoltre si prova che, almeno sotto certe condizioni, le curve sezioni lineari delle variet V_{s}^{j} della bandiera sono ancora curve di genere massimo rispetto alle condizioni di bandiera indotte. Ne risulta una struttura gerarchica della famiglia delle curve di genere massimo verificanti condizioni di bandiera. Gli autori rinviano alla nota

[CCD2] per risultati ottimali nel caso particolare r = 4, s_{1} >> s_{2} >> s_{3} ≥ 2. Infine si deduce una limitazione per la regolarit di Castelnuovo-Mumford delle curve verificanti condizioni di bandiera e si discute su alcune generalizzazioni delle condizioni di bandiera. In particolare si ottiene una limitazione per il genere di una curva in P^{r} non contenuta su una ipersuperficie di dato grado.

Contents

§0. Introduction.

§1. Numerical properties of the functions h(r; s_{1}, ..., s_{l}).

§2. The main theorem.

§3. Recurrence properties of some families of Hilbert functions.

References.

§0. Introduction.

Castelnuovo’s theorem. A classical result of Castelnuovo’s (see [Ca, 1889]) states that the arithmetic genus g := g(C) of an integral, non degenerate curve C of degree d in P^{r}, the projective space of dimension r over an algebraically closed field of characteristic 0, is bounded above by

Typeset by AMS-TEX

g ≤ G(r; d) := \binom{m}{2}(r − 1) + mϵ

where

d − 1 = m(r − 1) + ϵ, 0 ≤ ϵ ≤ r − 2.

Castelnuovo also proves that the bound is sharp and he classifies the curves with maximal genus, the so-called Castelnuovo curves.

We think it will be useful to briefly recall the main ideas of Castelnuovo’s proof. The first step is the basic inequality

g(C) ≤ ∑_{i=1}^{∞}(d − h_{Γ}(i))  (0.1)

where Γ is the general hyperplane section of C and by h_{V} we denote the Hilbert function of a scheme V in projective space (see [EH]). Castelnuovo also observes that the equality holds in (0.1) if and only if C is arithmetically Cohen-Macaulay (ACM).

Let C(r; d) be the set of all integral, non degenerate curves of degree d in P^{r} and let H(r; d) be the set of all Hilbert functions h_{Γ}, where Γ is the general hyperplane section of C, with C ∈ C(r; d). Notice that H(r; d) is partially ordered by the ordering

g ≤ g′ ⟺ g(n) ≤ g′(n), n ∈ N_{0}

defined for g, g′ any numerical functions. The inequality (0.1) shows that if k is a numerical function such that k(i) = d for i >> 0, and which is a lower bound for H(r; d) (such a function will be called by us a lower approximating function (LAF) for H(r; d)), then the number

G^{k}(r, d) := ∑_{i=1}^{∞}(d − k(i))

is an upper bound for the numerical set {g(C) : C ∈ C(r; d)}. We will say that a LAF k is a sharp function (SF) if there is a C ∈ C(r; d) such that g(C) = G^{k}(r, d). Notice that if k is a SF then k = h_{Γ} ∈ H(r; d), hence k is the minimum of H(r; d) and therefore it is uniquely determined. Furthermore the equality has to hold in (0.1), therefore C is ACM.

Castelnuovo’s strategy consists in finding a SF for H(r; d). He first proves, using the so-called uniform position principle (see [ACGH, pg.112]), the following basic inequality

h_{Γ}(i + j) ≥ min{d, h_{Γ}(i) + h_{Γ}(j) − 1}  (0.2)

whence he easily deduces that the function

h(r; d)(i) := min{d, i(r − 1) + 1}  (0.3)

is, with our terminology, a LAF. Then he proves that it is indeed a SF. This step has a different flavour according to whether d ≤ 2r or d > 2r. In the first case any linearly normal curve of degree d < 2r or a canonical curve if d = 2r in P^{r} is a curve of genus G(r; d) := G^{h}(r; d). In the latter case Castelnuovo observes that a curve C of genus G(r; d), if existing, must have h_{Γ}(2) = h(r; d)(2) = 2r − 1. At this point he proves another basic lemma (the so called Castelnuovo’s lemma) to the effect that if Γ is a set of points in general position in P^{r−1}, of degree d > 2r, such that h_{Γ}(2) = 2r − 1, then Γ lies on a rational normal curve which is the intersection of all quadrics through Γ. Since C is ACM, all these rational normal curves lift to a rational normal surface of degree r −1 in P^{r} on which C has to sit. By inspection of curves on such surfaces, Castelnuovo finally produces all curves of maximal genus G(r; d).

Halphen’s theorem and its generalization. It is now a natural question to determine the sharp upper bound G(r; d, s) for the arithmetic genus of an integral, non degenerate curve C of degree d in P^{r} not lying on any surface of degree smaller than s, where s ≥ r − 1 ≥ 2. Of course G(r; d, r − 1) = G(r; d). This problem goes back to Halphen [H, 1883] and Noether [N, 1883], who analysed the case r = 3 and proposed a solution for which it is essential to assume d sufficiently larger than s. The question has been reconsidered and solved in more recent times by Gruson and Peskine [GP, 1979] again for r = 3, any s and d > s^{2} − s, by Eisenbud and Harris [EH, 1982] for any r but s ≤ 2r − 2 and by the present authors [CCD, 1992] for any r and s and d >> s.

The basic ideas for the determination of G(r; d, s) are essentially the same as Castelnuovo’s as we recalled above. Specifically, one introduces the set C(r; d, s) of all integral, non degenerate curves C of degree d in P^{r} not lying on any surface of degree smaller than s, where s ≥ r − 1 ≥ 2, and H(r; d, s) the set of all Hilbert functions h_{Γ}, where Γ is the general hyperplane section of a curve C ∈ C(r; d, s). The problem consists in finding a SF for H(r; d, s). Again it is useful to spend a few words on the procedure we followed in [CCD].

First of all, in order to find a reasonable candidate for our SF, we consider a subproblem of the main one. Namely we define F(r; d, s) to be the set of all integral, non degenerate curves of degree d in P^{r} which lie on surfaces of degree s, and we accordingly define H_{F}(r; d, s) to be the corresponding set of Hilbert functions and we look for a SF for this set (notice that, by Bezout’s theorem, if d > s^{2} − s then F(r; d, s) is a subset of C(r; d, s)). Let C be a curve in F(r; d, s) lying on an integral surface S of degree s. Taking general hyperplane sections, we find Γ ⊂ D ⊂ P^{r−1}, where D is the general hyperplane section of S. Let d − 1 = ms + ϵ, where 0 ≤ ϵ ≤ s − 1. By the theorem of Bezout, one has

h_{Γ}(i) = h_{D}(i), i = 0, ..., m.

On the other hand, one knows that for any scheme V in projective space, one has

Δh_{V} (i) := h_{V} (i) − h_{V} (i − 1) ≥ h_{W} (i)  (0.4)

where W is the general hyperplane section of V (see [EH, pg.83]). Hence

h_{D}(i) ≥ ∑_{j=0}^{i} h_{Δ}(j)

where Δ is the general hyperplane section of D. From (0.3) one has h_{Δ}(j) ≥ h(r − 1; s)(j). Therefore one is led to expect that the sharp function h(r; d, s) for H_{F}(r; d, s) has the following recursive behaviour

h(r; d, s)(i) = ∑_{j=0}^{i} h(r − 1; s)(j), i = 0, ..., m.  (0.5)

The problem remains of determining the behaviour of any h_{Γ} ∈ H_{F}(r; d, s) for i ≥ m+1. We notice that this problem has been solved by Harris in [Hr, 1980] for the case of space curves. In [CCD, 1] it is solved for r ≥ 3 and d > s^{2} − s by a suitable refinement of Castelnuovo’s inequality (0.2). All the above analysis tells us that, when d > s^{2} − s, the function h(r; d, s) defined below is a LAF for H_{F}(r; d, s) [CCD, Proposition 4.2].

We define

h(r; d, s)(n) := ∑_{i=0}^{n} Δh(r; d, s)(i)

where

Δh(r; d, s)(i) := \{ \begin{matrix}0 & i < 0 \\ (r − 2)i + 1 & 0 ≤ i ≤ w \\ s & w < i ≤ m \\ s + k − (r − 2)(i − m) & m < i ≤ m + δ \\ s + k − (r − 2)(i − m) − 1 & m + δ < i ≤ w + m + e \\ 0 & i > w + m + e \end{matrix}

with m, w, k, δ, e defined as follows

d − 1 = ms + ϵ, 0 ≤ ϵ < s
s − 1 = (r − 2)w + v, 0 ≤ v < r − 2

if ϵ < w(r − 1 − v) ⇒ e = 0, ϵ = kw + δ, 0 ≤ δ < w
if ϵ ≥ w(r − 1 − v) ⇒ e = 1, ϵ + r − 2 − v = k(w + 1) + δ, 0 ≤ δ < w + 1.

Actually this function turns out to be a LAF also for H(r; d, s). This is technically the more complicated part of [CCD]. The main ingredients for the proof can be summarized to be:

(a) a lifting theorem to the effect that if C is in C(r; d, s) and d >> s, then Γ does not lie on any integral curve D ⊂ P^{r−1}, of degree σ < s (see also [CC]);

(b) a numerical property of the functions h(r; d, s), which says that for d >> s they are increasing in s.

Finally one proves that h(r; d, s) is a SF for d >> s by exhibiting suitable examples. It turns out that the extremal curves must lie on surfaces of degree s whose general hyperplane section is a Castelnuovo curve. Such surfaces in turn are contained in rational normal threefolds of degree r − 2 in P^{r}, if s > 2r − 2.

Flag conditions. The above discussion leads naturally to the consideration of a more general problem, namely the determination of a sharp upper bound for the genus of curves verifying certain flag conditions which we are going to define.

We will denote by V_{d}^{n} a complete, irreducible, non degenerate variety of degree d and dimension n in P^{r}. Let r, l, s_{1}, ..., s_{l} be integers such that r ≥ 3, 2 ≤ l ≤ r − 1, s_{i} ≥ r − i + 1. Let C(r; s_{1}, ..., s_{l}) be the set of all integral, non degenerate curves C of degree s_{1} in P^{r} not lying on any V_{s}^{i}, with s < s_{i}, for all i = 2, ..., l. A curve belonging to C(r; s_{1}, ..., s_{l}) is said to satisfy a flag condition of type (r; s_{1}, ..., s_{l}). We would like to compute

G(r; s_{1}, ..., s_{l}) := max{g(C), C ∈ C(r; s_{1}, ..., s_{l})}.

This of course amounts to compute a SF for the set H(r; s_{1}, ..., s_{l}) of all Hilbert functions h_{Γ}, where Γ is the general hyperplane section of a curve C ∈ C(r; s_{1}, ..., s_{l}). Unfortunately this problem seems to be in general quite untractable by using the same techniques we discussed above. Nonetheless, without finding a SF for H(r; s_{1}, ..., s_{l}), in this paper we will give an estimate for G(r; s_{1}, ..., s_{l}) which seems to us meaningiful. Moreover we refer the reader to the note [CCD2] for the computation of G(4; s_{1}, s_{2}, s_{3}), when s_{1} >> s_{2} >> s_{3} ≥ 2. Let us discuss now in more detail our main results.

As in the case l = 2 discussed above, we found it useful to start by a simpler problem. Indeed, we consider the set F(r; s_{1}, ..., s_{l}) of all integral, non degenerate curves C of degree s_{1} in P^{r} such that there is a flag

C := V_{s_{1}}^{1} ⊂ V_{s_{2}}^{2} ⊂ ... ⊂ V_{s_{l}}^{l}

of complete, integral, non degenerate varieties in P^{r} (as before, we note that if s_{i} > s_{i+1}^{2} − s_{i+1} for i = 1, ..., l − 1, then F(r; s_{1}, ..., s_{l}) is a subset of C(r; s_{1}, ..., s_{l})). Accordingly we first of all look for a LAF for the set H_{F}(r; s_{1}, ..., s_{l}) of corresponding Hilbert functions. It turns out that, exactly as in the case l = 2, (0.5), one is led to expect that a LAF h(r; s_{1}, ..., s_{l}) for H_{F}(r; s_{1}, ..., s_{l}) has the following recursive behaviour

h(r; s_{1}, ..., s_{l})(i) = ∑_{j=0}^{i} h(r − 1; s_{2}, ..., s_{l})(j), i = 0, ..., a_{1}  (0.6)

where we set

s_{i} − 1 = a_{i}s_{i+1} + b_{i}, 0 ≤ b_{i} < s_{i+1}, i = 1, ..., l; s_{l+1} := r − l.

Notice that in order that this definition makes sense, we have to assume a_{1} > 0, and by the recursive character of the function, we need a_{i} > 0 for all i = 1, ..., l, hence we need s_{1} > s_{2} > ... > s_{l} > s_{l+1}. Moreover, we have to assume s_{l−1} > s_{l}^{2} − s_{l}.

Again we have the problem of understanding the behaviour of a h_{Γ} ∈ H_{F}(r; s_{1}, ..., s_{l}) for i > a_{1}. This is really the crucial problem in this matter, which we are able to solve only in the particular case r = 4, l = 3 and s_{1} >> s_{2} >> s_{3} ≥ 2 (see the quoted paper [CCD2]). Unfortunately, when r ≥ 5 and l ≥ 3, we have been unable so far to find a complete solution for it, henceforth we are unable to write down a SF for H_{F}(r; s_{1}, ..., s_{l}). Notice that this difficulty already shows up for l = 3, and, in view of the recursive character indicated in (0.6), propagates for l > 3 also in the range 0 ≤ i ≤ a_{1}. In any event it is quite easy to find a LAF, though not a SF, for H_{F}(r; s_{1}, ..., s_{l}), by defining our function h(r; s_{1}, ..., s_{l}) in the range i > a_{1} by using the formula (0.2). In Proposition (1.1) we will see that, for s_{1} >> ... >> s_{l}, h(r; s_{1}, ..., s_{l})(i) = s_{1} for i >> 0; in other words we have that the function

h(r; s_{1}, ..., s_{l})(i) := \{ \begin{matrix}∑_{j=0}^{i} h(r − 1; s_{2}, ..., s_{l})(j) & 0 ≤ i ≤ a_{1} \\ min\{s_{1}, h(r; s_{1}, ..., s_{l})(a_{1}) + h(r; s_{1}, ..., s_{l})(i − a_{1}) − 1\} & i > a_{1} \end{matrix}

is a LAF for H_{F}(r; s_{1}, ..., s_{l}). We want to stress that the fact that our function h(r; s_{1}, ..., s_{l}) is only a LAF and not a SF is easily seen by observing that Δh(r; s_{1}, ..., s_{l}) is increasing for i > a_{1}, whereas we know by [CCD, Corollary 1.2], that Δh_{Γ} is decreasing in the same range for any function h_{Γ} ∈ H_{F}(r; s_{1}, ..., s_{l}).

However, the fact that we have got a LAF for H_{F}(r; s_{1}, ..., s_{l}), gives us the possibility of writing down the upper bound

G^{h}(r; s_{1}, ..., s_{l}) := ∑_{i=1}^{∞}(s_{1} − h(r; s_{1}, ..., s_{l})(i))

for the numerical set {g(C), C ∈ F(r; s_{1}, ..., s_{l})}. In Proposition (1.1) we find an explicit formula for G^{h}(r; s_{1}, ..., s_{l}), which shows that

G^{h}(r; s_{1}, ..., s_{l}) = \frac{s_{1}^{2}}{2s_{2}} + \frac{s_{1}}{2} (\frac{s_{2}}{s_{3}} + ... + \frac{s_{l}}{s_{l+1}} − l − \frac{2}{r − l} + η) + ρ + 1

where η and ρ are rational numbers depending only on r − l, s_{2}, ..., s_{l} and b_{1}, such that

|η| ≤ \frac{3}{4}, |ρ| ≤ (l^{2} + 2l + 9)\frac{s_{2}^{3}}{s_{3}^{2}} .

Turning again to the more general problem of bounding the numerical set {g(C), C ∈ C(r; s_{1}, ..., s_{l})}, we have been able to prove that if s_{1} >> s_{2} >> ... >> s_{l}, then G^{h}(r; s_{1}, ..., s_{l}) is indeed an upper bound, i.e. G(r; s_{1}, ..., s_{l}) ≤ G^{h}(r; s_{1}, ..., s_{l}). This is the content of Theorem (2.2), which is proved by using numerical properties of the function G^{h}(r; s_{1}, ..., s_{l}) shown in Propositions (1.1) and (1.4), and a suitable extension of our results in [CCD] given in Proposition (2.1).

Then we address the problem of how far is our bound G^{h}(r; s_{1}, ..., s_{l}) from G(r; s_{1}, ..., s_{l}). This question is analysed in Proposition (2.7) in which, by exhibiting suitable examples of curves in F(r; s_{1}, ..., s_{l}), we show that the gap between the two values is bounded by 6\frac{s_{2}^{3}}{s_{3}^{2}} if l = 3, and by \frac{3}{2}s_{1} + 2(l^{2} + 2l + 9)\frac{s_{2}^{3}}{s_{3}^{2}} for l > 3.

It is probably useful to point out that, with the above indicated procedure, we do find our upper bound G^{h}(r; s_{1}, ..., s_{l}) without finding a SF for H(r; s_{1}, ..., s_{l}). In fact this problem remains open. One reason, among many others pointed out above, why we were unable to approach this question has been the lack of a suitable lifting theorem. Notice however that in Theorem (2.2) we also prove that the curves in C(r; s_{1}, ..., s_{l}) with maximal genus must lie in F(r; s_{1}, ..., s_{l}). Therefore in order to compute G(r; s_{1}, ..., s_{l}) it would be sufficient to find a SF for H_{F}(r; s_{1}, ..., s_{l}). Actually it is another open question to prove the existence of such a SF in general. In any case, assuming its existence, we investigate some formal properties of such a function in 3. This leads us first to see that, in the range 0 ≤ i ≤ a_{1}, the definition of the function h(r; s_{1}, ..., s_{l}) is, in a certain sense, natural (see Proposition (3.1)); next it leads us to an understanding of the flags on which the curves of maximal genus belong: roughly speaking, the general curve sections of the varieties of the flag are again curves of maximal genus with respect to the appropriate flag conditions (see Theorem (3.4)). From this result stems a hyerarchical structure of the family of curves with maximal genus subject to flag conditions.

All the above discussion shows that the first relevant case to be investigated for the search of a SF for H_{F}(r; s_{1}, ..., s_{l}) is the case l = 3, and particularly interesting is the case r = 4, s_{1} >> s_{2} >> s_{3}. Recall that the main problem here is the understanding of the behaviour of a function in H_{F}(4; s_{1}, s_{2}, s_{3}) for i > a_{1}. As we said before, in the paper [CCD2], we have been able to determine the SF for H_{F}(4; s_{1}, s_{2}, s_{3}) and so to compute G(4; s_{1}, s_{2}, s_{3}). We guess that the ideas which we used for solving the case r = 4 (see for instance [CCD2, Proposition 4], which enables us to handle the Hilbert function of points in P^{3}) are not sufficient, at least without substantial improvement, to solve the general problem.

Moreover, in this paper we will prove that, when l = 3 but r ≥ 4, the curves C of maximal genus G(r; s_{1}, s_{2}, s_{3}), satisfying a flag condition of type (r; s_{1}, s_{2}, s_{3}), with s_{1} >> s_{2} >> s_{3}, are ACM and lie on a flag of type:

C := V_{s_{1}}^{1} ⊂ V_{s_{2}}^{2} ⊂ V_{s_{3}}^{3} ⊂ P^{r}

where V_{s_{2}}^{2} is a surface whose general hyperplane sections are curves of maximal genus in C(r−1; s_{2}, s_{3}) and V_{s_{3}}^{3} is a threefold whose general curves sections are curves of maximal genus in C(r − 2; s_{3}) (see Corollary (3.4)).

Finally we refer the reader to Remark (2.9) for a discussion on further generalizations of flag conditions and related results and problems. In particular we give a bound for the genus of a curve in P^{r} which does not lie on any hypersurface of given degree.

§1. Numerical properties of the functions h(r; s_{1}, ..., s_{l}).

The aim of this section is to prove a few numerical properties of the functions h(r; s_{1}, ..., s_{l}) defined above, which will turn out to be essential in the rest of the paper. We keep all the notation of the Introduction and, occasionally, when there is no possibility of confusion, we denote by h the function h(r; s_{1}, ..., s_{l}).

Proposition 1.1. Fix integers r, s_{1}, ..., s_{l} such that 2 ≤ l ≤ r − 1 and s_{l} ≥ r − l + 1. Assume that:

s_{i} ≥ 8(l − 1)[(l − i + 1)^{2} + 2(l − i + 1) + 9]\begin{matrix}s_{i+1} & 3 \\ s_{i+2} & 2 \end{matrix}

for all i = 1, ..., l − 1. Then, for the function h := h(r; s_{1}, ..., s_{l}), we have:

(i) h(n) ≥ 0 for all n ≥ 0 and h is not decreasing; (ii) h(n) ≤ s_{1} for all n ≥ 0; (iii) h(a_{1} + ... + a_{l} + 2) = s_{1}.

Moreover, if we put

G^{h}(r; s_{1}, ..., s_{l}) := ∑_{i=1}^{∞}(s_{1} − h(r; s_{1}, ..., s_{l})(i))

then we have: (iv) h(n) = ns_{2} − G^{h}(r − 1; s_{2}, ..., s_{l}) + 1, for a_{2} + ... + a_{l} + 2 ≤ n ≤ a_{1};

(v) there are rational numbers R_{i}, i = 2, ..., l, depending only on s_{l−i+2}, ..., s_{l}, r − l, b_{l−i+1}, such that

|R_{i}| ≤ 2(i^{2} + 2i + 9)\frac{s_{l−i+2}^{3}}{s_{l−i+3}^{2}} , i = 2, ..., l

and that

G^{h}(r; s_{1}, ..., s_{l}) = \frac{s_{1}^{2}}{2s_{2}} + \frac{s_{1}}{2} (\frac{s_{2}}{s_{3}} + ... + \frac{s_{l}}{s_{l+1}} − l − \frac{2}{r − l} + η) + ρ + 1

where

η := ∑_{i=2}^{l−1} \frac{R_{l+1−i}}{s_{i}} + \frac{(1 − b_{l})(b_{l} + 1 − r + l)}{s_{l}(r − l)}, ρ := \frac{R_{l}}{2}

and

|η| ≤ \frac{3}{4}, |ρ| ≤ (l^{2} + 2l + 9)\frac{s_{2}^{3}}{s_{3}^{2}} .

Proof. First of all we remark that the bounds on |η| and |ρ| can be easily deduced, with elementary calculations which we omit, from the hypotheses on the s_{i}’s, once the rest of the assertion has been proved. This we will make by induction on l.

Let l = 2. In this case the properties (i),(ii),(iii) and (iv) are easy to verify, and therefore we leave them to the reader. In order to prove (v) for l = 2, it is useful to set in general

ρ_{i} := \{ \begin{matrix}− ∑_{j=1}^{a_{2}+...+a_{l}+1} h(r; s_{1}, ..., s_{l})(j) & for i = 1 \\ (1 − G^{h}(r − 1; s_{2}, ..., s_{l}))(a_{2} + ... + a_{l} + 1) + s_{2}\binom{a_{2}+...+a_{l}+2}{2} & for i = 2 \\ \frac{1+b_{1}}{s_{2}} (1 − G^{h}(r − 1; s_{2}, ..., s_{l})) + \frac{1+b_{1}}{2s_{2}} (s_{2} − 1 − b_{1}) & for i = 3 \\ ∑_{j=a_{1}+1}^{a_{1}+...+a_{l}+1}(s_{1} − h(r; s_{1}, ..., s_{l})(j)) & for i = 4. \end{matrix}

Now, in the case l = 2, we have (note that the hypotheses on the s_{i}’s imply a_{1} > a_{2} +2):

G^{h}(r; s_{1}, s_{2}) = ∑_{i=1}^{a_{1}+a_{2}+1}(s_{1} − h(r; s_{1}, s_{2})(i)) =

= ∑_{i=1}^{a_{2}+1}(s_{1} − h(r; s_{1}, s_{2})(i)) + ∑_{i=a_{2}+2}^{a_{1}}(s_{1} − h(r; s_{1}, s_{2})(i)) + ∑_{i=a_{1}+1}^{a_{1}+a_{2}+1}(s_{1} − h(r; s_{1}, s_{2})(i)).

By (iv) we deduce that:

G^{h}(r; s_{1}, s_{2}) = \frac{s_{1}^{2}}{2s_{2}} + \frac{s_{1}}{2s_{2}} [ − 2 + 2G^{h}(r − 1; s_{2}) − s_{2}] + ρ_{1} + ρ_{2} + ρ_{3} + ρ_{4}.

Since G^{h}(r − 1; s_{2}) = \binom{a_{2}}{2}(r − 2) + b_{2}, we obtain:

G^{h}(r; s_{1}, s_{2}) = \frac{s_{1}^{2}}{2s_{2}} + \frac{s_{1}}{2} [\frac{s_{2}}{r − 2} − 2 − \frac{2}{r − 2} + \frac{(1 − b_{2})(b_{2} + 3 − r)}{s_{2}(r − 2)}] + ρ_{1} + ρ_{2} + ρ_{3} + ρ_{4}.

Now we can conclude the proof for l = 2 by showing that in this case

|ρ_{i}| ≤ \{ \begin{matrix}6\frac{s_{2}^{3}}{(r−2)^{2}} & for i = 1 \\ 5\frac{s_{2}^{3}}{(r−2)^{2}} & for i = 2 \\ \frac{s_{2}^{2}}{r−2} + \frac{s_{2}}{2} & for i = 3 \\ 2\frac{s_{2}^{3}}{(r−2)^{2}} + 2\frac{s_{2}^{2}}{r−2} & for i = 4. \end{matrix}

Indeed the above inequalities yield

|ρ_{1} + ρ_{2} + ρ_{3} + ρ_{4}| ≤ 17\frac{s_{2}^{3}}{(r − 2)^{2}} − 1

which proves the assertion. In order to prove the above estimates for the |ρ_{i}|’s, we argue as follows. Since Δh(r; s_{1}, s_{2})(i) ≤ s_{2} for all i, we have

|ρ_{1}| ≤ ∑_{j=1}^{a_{2}+1}(∑_{j=0}^{i} s_{2}) ≤ s_{2}\binom{a_{2} + 3}{2}

whence the assertion, by taking into account that a_{2} ≤ \frac{s_{2}}{s_{3}} . The estimates for |ρ_{2}| and |ρ_{3}| follow immediately from the same argument as above and from the fact that G^{h}(r − 1; s_{2}) ≤ \frac{s_{2}^{2}}{r−2}, whose easy proof we omit. Finally, we have

|ρ_{4}| ≤ ∑_{j=a_{1}+1}^{a_{1}+a_{2}+1}(s_{1} − h(r; s_{1}, s_{2})(a_{1})) = ∑_{j=a_{1}+1}^{a_{1}+a_{2}+1}(s_{1} − a_{1}s_{2} + 1 + G^{h}(r − 1; s_{2})) =

= ∑_{j=a_{1}+1}^{a_{1}+a_{2}+1}(b_{1} + G^{h}(r − 1; s_{2})) = (a_{2} + 1)(b_{1} + G^{h}(r − 1; s_{2}))

which easily yields our estimate.

Now we turn to the case l ≥ 3 and assume, by induction, that the proposition is true for 2, ..., l − 1. First of all we prove property (iv). As in the case l = 2, one can easily see that the hypotheses on the s_{i}’s imply a_{1} > a_{2} + ... + a_{l} + 2, hence we have

h(r; s_{1}, ..., s_{l})(a_{2} + ... + a_{l} + 2) = 1 + ∑_{i=1}^{a_{2}+...+a_{l}+2} h(r − 1; s_{2}, ..., s_{l})(i) =

= 1 + s_{2}(a_{2} + ... + a_{l} + 2) − G^{h}(r − 1; s_{2}, ..., s_{l}).

Property (iv) is now a consequence of the definition of h(r; s_{1}, ..., s_{l}), since, by induction, we have h(r − 1; s_{2}, ..., s_{l})(i) = s_{2} for i ≥ a_{2} + ... + a_{l} + 2. Property (iv) implies in turn that

h(r; s_{1}, ..., s_{l})(a_{1}) = a_{1}s_{2} + 1 − G^{h}(r − 1; s_{2}, ..., s_{l}) ≤ s_{1} − G^{h}(r − 1; s_{2}, ..., s_{l}) ≤ s_{1}

since, by induction, (ii) holds and therefore G^{h}(r − 1; s_{2}, ..., s_{l}) ≥ 0. Then, again by induction and by the definition of h(r; s_{1}, ..., s_{l}), property (i) and (ii) clearly hold. In order to prove (iii), it suffices to prove that

h(a_{1}) + h(a_{2} + ... + a_{l} + 2) − 1 ≥ s_{1}.

By (iv) we have

h(a_{1}) + h(a_{2} + ... + a_{l} + 2) − 1 = 1 − 2G^{h}(r − 1; s_{2}, ..., s_{l}) + s_{2}(a_{1} + ... + a_{l} + 2).

By induction

G^{h}(r−1; s_{2}, ..., s_{l}) ≤ \frac{s_{2}^{2}}{2s_{3}} +\frac{s_{2}}{2} (\frac{s_{3}}{s_{4}} +...+\frac{s_{l}}{r − l} −l+1−\frac{2}{r − l} +\frac{3}{4})+[(l−1)^{2}+2(l−1)+9]\begin{matrix}s_{3} & 3 \\ s_{4} & 2 \end{matrix} +1

hence, by using the assumpions on the s_{i}’s, we find

h(a_{1})+h(a_{2}+...+a_{l}+2)−1 ≥ \frac{5s_{2}}{4} −1+a_{1}s_{2}−2[(l−1)^{2}+2(l−1)+9]\begin{matrix}s_{3} & 3 \\ s_{4} & 2 \end{matrix} ≥ s_{2}+a_{1}s_{2} ≥ s_{1}

proving (iii). We conclude by proving (v). As in the case l = 2, we have

G^{h}(r; s_{1}, ..., s_{l}) = ∑_{i=1}^{a_{1}+...+a_{l}+1}(s_{1} − h(r; s_{1}, ..., s_{l})(i)) =

= ∑_{i=1}^{a_{2}+...+a_{l}+1}(s_{1} − h(r; s_{1}, ..., s_{l})(i)) + ∑_{i=a_{2}+...+a_{l}+2}^{a_{1}}(s_{1} − h(r; s_{1}, ..., s_{l})(i))+

+ ∑_{i=a_{1}+1}^{a_{1}+...+a_{l}+1}(s_{1} − h(r; s_{1}, ..., s_{l})(i)).

Hence:

G^{h}(r; s_{1}, ..., s_{l}) = [s_{1}(a_{2}+...+a_{l}+1)+ρ_{1}]+[s_{1}(a_{1}−a_{2}−...−a_{l}−1)− ∑_{i=a_{2}+...+a_{l}+2}^{a_{1}} h(i)]+ρ_{4}.

Then using (iv) we find

G^{h}(r; s_{1}, ..., s_{l}) = [s_{1}(a_{2} + ... + a_{l} + 1) + ρ_{1}] + [s_{1}(a_{1} − a_{2} − ... − a_{l} − 1)−

− ∑_{i=a_{2}+...+a_{l}+2}^{a_{1}}(is_{2} + 1 − G^{h}(r − 1; s_{2}, ..., s_{l}))] + ρ_{4}

and therefore, with an elementary computation which can be omitted, we find

G^{h}(r; s_{1}, ..., s_{l}) = \frac{s_{1}^{2}}{2s_{2}} + \frac{s_{1}}{2s_{2}} [ − 2 + 2G^{h}(r − 1; s_{2}, ..., s_{l}) − s_{2}] + ρ_{1} + ρ_{2} + ρ_{3} + ρ_{4}.

We can now plug into this expression the value of G^{h}(r − 1; s_{2}, ..., s_{l}) which is obtained by induction using (v). As a result we get the desired expression of G^{h}(r; s_{1}, ..., s_{l}), where the numbers R_{i}, i = 2, ..., l − 1, are defined by induction, and

R_{l} := 2(ρ_{1} + ρ_{2} + ρ_{3} + ρ_{4} − 1).

Again by induction we know that the required bounds for |R_{i}|, i = 2, ..., l − 1, hold. Hence we only need to prove the bound for |R_{l}|. This proof runs in the general case quite similarly to the case l = 2, by suitably estimating the numbers |ρ_{i}|, i = 1, ..., 4. Indeed, we have

|ρ_{i}| ≤ \{ \begin{matrix}\frac{l^{2}s_{2}^{3}}{2s_{3}^{2}} & for i = 1 \\ \frac{ls_{2}^{3}}{s_{3}^{2}} (1 + \frac{l}{2}) & for i = 2 \\ \frac{s_{2}^{2}}{s_{3}} + \frac{s_{2}}{2} & for i = 3 \\ \frac{ls_{2}^{2}}{s_{3}} (1 + \frac{s_{2}}{s_{3}} ) & for i = 4. \end{matrix}

The above inequalities, by our assumptions on the s_{i}’s, imply

|ρ_{1} + ρ_{2} + ρ_{3} + ρ_{4} − 1| ≤ (l^{2} + 2l + 1)\frac{s_{2}^{3}}{s_{3}^{2}}

which yields the assertion. As for the above estimates for the |ρ_{i}|’s, again the proof is quite similar to the case l = 2 and therefore we omit it. We only point out that the crucial, but easy, inequality G^{h}(r − 1, s_{2}) ≤ \frac{s_{2}^{2}}{r−2}, which we used in the case l = 2, has to be replaced here by the inequality

G^{h}(r − 1; s_{2}, ..., s_{l}) ≤ \frac{s_{2}^{2}}{s_{3}}  (1.2)

which can be proved with elementary computations, by using the inductive formula (v) for G^{h}(r − 1; s_{2}, ..., s_{l}) and the assumptions on the s_{i}’s.♢

Remark 1.3 The hypotheses on the s_{i}’s in Proposition (1.1) are certainly not the sharpest in order to prove the result. They are only of the simplest form we were able to conceive. Since Proposition (1.1) is a result on the asymptotic behaviour of the function G^{h}(r; s_{1}, ..., s_{l}), we cared for qualitative rather than quantitative hypotheses. Also we want to point out that, by suitably changing the hypotheses on the s_{i}’s, one can obtain sharper bounds on the |R_{i}|’s, and accordingly on |η| and |ρ|. For instance, |η| can be made smaller than any positive number, by suitable assumptions of the type s_{i} >> s_{i+1}, for all i = 1, ..., l − 1.

The next proposition is another technical result on the functions we introduced above. Precisely, we need to show some properties of monotony of the function G^{h}(r; s_{1}, ..., s_{l}).

Proposition 1.4. Assume that

s_{i} ≥ 8(l − 1)[(l − i + 1)^{2} + 2(l − i + 1) + 9]\frac{(s_{i+1} + 1)^{3}}{r − i − 1}

for i = 1, ..., l − 1 and let us set

δ_{i} := \frac{s_{1}}{2} [\frac{s_{i−1}}{s_{i}(s_{i} + 1)} − 3\frac{s_{i}}{r − i}]

where i = 2, ..., l. Then we have:

(i) 0 < δ_{l} < δ_{l−1} < ... < δ_{3} < δ_{2}; (ii) δ_{i} ≤ G^{h}(r; s_{1}, ..., s_{l}) − G^{h}(r; s_{1}, ..., s_{i−1}, s_{i} + 1), for all i = 2, ..., l.

Proof. Property (i) is an immediate consequence of the hypotheses on the s_{i}’s. As for property (ii), one has to estimate G^{h}(r; s_{1}, ..., s_{l}) − G^{h}(r; s_{1}, ..., s_{i−1}, s_{i} + 1). We use (v) of Proposition (1.1) which can be applied, inasmuch as one sees that the hypotheses we have here enable us to do so. For example, in case i = 2, we have

G^{h}(r; s_{1}, ..., s_{l}) − G^{h}(r; s_{1}, s_{2} + 1) =

= \frac{s_{1}}{2} [\frac{s_{1}}{s_{2}(s_{2} + 1)} + ... + \frac{s_{l} − 2}{r − l} − l − \frac{s_{2} − 2}{r − 2} + 2 + (η − η′) + 2\frac{ρ − ρ′}{s_{1}} ]

where η′, ρ′ have, with respect to G^{h}(r; s_{1}, s_{2} + 1), the same meaning as ρ and η have with respect to G^{h}(r; s_{1}, ..., s_{l}) (see Proposition (1.1)). Our hypotheses on the s_{i}’s imply

|2\frac{ρ − ρ′}{s_{1}} | ≤ \frac{1}{2}.

Furthermore Proposition (1.1), (v), yields

|η − η′| ≤ \frac{3}{2}.

The assertion then easily follows. The proof of the cases i = 3, ..., l − 1 is quite similar, and therefore we omit it. ♢

§2. The main theorem.

In this section we prove our main result on the bound of the genus of a curve verifying flag conditions (see Theorem (2.2) below). First of all we need a preliminary proposition, which will play an essential role in what follows. It can also be seen as an extension of the main theorem of [CCD]. As before, we keep all the notation we introduced in §0.

Proposition 2.1. Let l ≥ 2 be an integer and let us consider a flag

C := V_{s_{1}}^{1} ⊂ V_{s_{2}}^{2} ⊂ ... ⊂ V_{s_{l−1}}^{l−1}

of complete, irreducible, non degenerate varieties in P^{r}. Let us assume that V_{s_{l−1}}^{l−1} is not contained in any V_{s}^{l} with s < s_{l}. Moreover put s_{l+1} = r − l and assume that:

s_{i} ≥ 8(l − 1)[(l − i + 1)^{2} + 2(l − i + 1) + 9]\begin{matrix}s_{i+1} & 3 \\ s_{i+2} & 2 \end{matrix}

for all i = 1, ..., l − 1 and that:

s_{l−1} > max\{\frac{s_{l}^{2}}{r − l} + (2r − 2)s_{l}, \frac{2s_{l}}{r − l} ∏_{j=1}^{r−l} \sqrt[r−l+1−j]{(r − l + 1)!s_{l}}\}.

Then

h_{Γ} ≥ h(r; s_{1}, ..., s_{l}).

In particular:

g(C) ≤ G^{h}(r; s_{1}, ..., s_{l}).

Proof. The case l = 2 is the main theorem of [CCD]. Now we assume l ≥ 3 and argue by induction. Let H = P^{r−1} be a general hyperplane in P^{r}. We let V_{i,H} := V_{s_{i}}^{i} ∩ H, which is a V_{s_{i}}^{i−1} for i = 2, ..., l − 1 and is Γ for i = 1. Thus we have a new flag in H

V_{2,H} ⊂ ... ⊂ V_{l−1,H}

and we can apply the induction to it. Indeed, the numerical conditions on the s_{i}’s are clearly verified. The condition that V_{l−1,H} is not contained in any variety of dimension l − 1 and degree s < s_{l} holds in view of the numerical hypotheses on s_{l−1} and s_{l} and of the lifting theorem of [CC].

Now we notice that, since Γ ⊂ V_{2,H}, by Bezout’s theorem we have

h_{Γ}(i) = h_{V_{2,H}} (i), i = 0, ..., a_{1}.

Let D be the general hyperplane section of V_{2,H}. By (0.4) we have

h_{V_{2,H}} (i) ≥ ∑_{j=0}^{i} h_{D}(j)

for all i. But now, by induction and by the definition of h(r; s_{1}, ..., s_{l}), we conclude that

h_{Γ}(i) = h_{V_{2,H}} (i) ≥ ∑_{j=0}^{i} h_{D}(j) ≥ ∑_{j=0}^{i} h(r − 1; s_{2}, ..., s_{l})(j) = h(r; s_{1}, ..., s_{l})(i)

for all i = 0, ..., a_{1}. The assertion for i > a_{1} follows by the definition of h(r; s_{1}, ..., s_{l}) in this range (see §0) and by applying Castelnuovo’s inequality (0.2), which yields that

h_{Γ}(i) ≥ min{s_{1}, h_{Γ}(a_{1}) + h_{Γ}(i − a_{1}) − 1},

and arguing by induction on i.

Finally the inequality g(C) ≤ G^{h}(r; s_{1}, ..., s_{l}) follows by (0.1). ♢

Now we are ready for the main theorem, but before that we want to remind to the reader that the explicit expression of G^{h}(r; s_{1}, ..., s_{l}), which appears in its statement, has been given in Proposition (1.1), (v) and the number δ_{l} was defined in the statement of Proposition (1.4).

Theorem 2.2. Let C be an irreducible non degenerate curve of degree s_{1} and arithmetic genus g := g(C) in P^{r}. Let l = 2, ..., r − 1 be an integer and let s_{2}, ..., s_{l} be positive integers (with s_{l} ≥ r − l + 1) such that:

s_{i} ≥ 8(l − 1)[(l − i + 1)^{2} + 2(l − i + 1) + 9]\frac{(s_{i+1} + 1)^{3}}{r − i − 1}

and that

s_{i} > max\{\frac{(s_{i+1} + 1)^{2}}{r − i − 1} +(2r−2)(s_{i+1}+1), 2\frac{(s_{i+1} + 1)}{r − i − 1} ∏_{j=1}^{r−i−1} \sqrt[r−i−j]{(r − i)!(s_{i+1} + 1)}\}

for all i = 1, ..., l −1. Let us also assume that for all i = 2, ..., l, there is no V_{s}^{i} containing C with s < s_{i}. Then we have:

(i) g ≤ G^{h}(r; s_{1}, ..., s_{l}); (ii) if G^{h}(r; s_{1}, ..., s_{l}) − g < δ_{l}, there is a unique flag

C := V_{s_{1}}^{1} ⊂ V_{s_{2}}^{2} ⊂ ... ⊂ V_{s_{l}}^{l} ⊂ P^{r}

and h_{Γ} ≥ h.

Proof. (i) If C is not contained in any surface of degree s < s_{2} +1, then by Propositions (1.4) and (2.1) we have

g ≤ G^{h}(r; s_{1}, s_{2} + 1) < G^{h}(r; s_{1}, ..., s_{l}).

Hence we may assume that C lies on a V_{s_{2}}^{2}, which, by the hypotheses on C, does not lie on any V_{s}^{3}, with s < s_{3}. If V_{s_{2}}^{2} does not lie on any V_{s}^{3} with s < s_{3} + 1, then, as before, by Propositions (1.4) and (2.1) we have:

g ≤ G^{h}(r; s_{1}, s_{2}, s_{3} + 1) < G^{h}(r; s_{1}, ..., s_{l}).

So we may assume that C ⊂ V_{s_{2}}^{2} ⊂ V_{s_{3}}^{3}. Continuing in this fashion, one proves that either g < G(r; s_{1}, ..., s_{l}) or that there is a flag

(2.3) C := V_{s_{1}}^{1} ⊂ V_{s_{2}}^{2} ⊂ ... ⊂ V_{s_{l−1}}^{l−1} ⊂ P^{r}

where, by the assumptions on C, we may assume that V_{s_{l−1}}^{l−1} is not contained in any V_{s}^{l} with s < s_{l}. Then by applying again Proposition (2.1) one sees that g ≤ G^{h}(r; s_{1}, ..., s_{l}).

(ii) Assume there is no surface V_{s_{2}}^{2} containing C. Then again by Proposition (1.4) and by Proposition (2.1), we have

g ≤ G^{h}(r; s_{1}, s_{2} + 1) ⟹ G^{h}(r; s_{1}, ..., s_{l}) − g ≥ δ_{2} ≥ δ_{l}

a contradiction. In this way we find a V_{s_{2}}^{2} containing C. Iterating the procedure, one proves the existence of the flag. The uniqueness follows by Bezout’s theorem and by the hypotheses on the s_{i}’s. ♢

In the same circle of ideas of the proof of the previous theorem, we can prove the following regularity result:

Proposition 2.4. With the same assumptions as in Theorem (2.2), one has:

(i) H^{1}(P^{r}, I_{Γ}(n)) = 0 for all n ≥ a_{1} + ... + a_{l} + 2; (ii) H^{1}(C, O_{C}(n)) = 0 for all n ≥ a_{1} + ... + a_{l} + 1; (iii) H^{1}(P^{r}, I_{C}(n)) = 0 for all n ≥ a_{1} + ... + a_{l} + 2 + G^{h}(r; s_{1}, ..., s_{l}) − g; (iv) moreover, if the SF for H_{F}(r; s_{1}, ..., s_{l}) exists, then H^{1}(P^{r}, I_{C}(n)) = 0 for all n ≥ a_{1} + ... + a_{l} + 2 + G(r; s_{1}, ..., s_{l}) − g.

Proof. (i) Suppose that C is not contained in a V_{s}^{2}, with s < s_{2} + 1. Then, by Propositions (1.1) and (2.1), h_{Γ}(n) = d for n ≥ \overline{a}_{1} + \overline{a}_{2} + 2, where

s_{1} − 1 = \overline{a}_{1}(s_{2} + 1) + \overline{b}_{1}, 0 ≤ \overline{b}_{1} ≤ s_{2}
s_{2} = \overline{a}_{2}(r − 2) + \overline{b}_{2}, 0 ≤ \overline{b}_{2} ≤ r − 3.

In view of our hypotheses on the s_{i}’s, we have \overline{a}_{1} + \overline{a}_{2} + 2 ≤ a_{1} + ... + a_{l} + 2, hence the assertion. Iterating the argument we may assume that there is a flag of type (2.3). Then, again by Propositions (1.1) and (2.1), we have h_{Γ}(n) ≥ h(r; s_{1}, ..., s_{l})(n) = d for n ≥ a_{1} + ... + a_{l} + 2, whence the assertion follows. Property (ii) is a direct consequence of (i).

(iii) By (i) and by Castelnuovo regularity (see [M]), we have that the homogeneous ideal of Γ is generated in degree t ≤ a_{1} + ... + a_{l} + 3. Now we make the following:

Claim. If n ≥ a_{1} + ... + a_{l} + 2 then h^{1}(P^{r}, I_{C}(n + 1)) ≤ max{0, h^{1}(P^{r}, I_{C}(n)) − 1}.

Indeed we consider the exact sequence

... → H^{0}(P^{r}, I_{C}(n+1)) → H^{0}(P^{r}, I_{Γ}(n+1)) → H^{1}(P^{r}, I_{C}(n)) → H^{1}(P^{r}, I_{C}(n+1)) → 0

and therefore we may assume h^{1}(P^{r}, I_{C}(n)) > 0. The claim will be proved if we prove that the map H^{0}(P^{r}, I_{C}(n + 1)) → H^{0}(P^{r}, I_{Γ}(n + 1)) is not surjective. Assume the contrary. Then all maps H^{0}(P^{r}, I_{C}(m)) → H^{0}(P^{r}, I_{Γ}(m)), m ≥ n + 1, would be surjective, and therefore h^{1}(P^{r}, I_{C}(m)) = h^{1}(P^{r}, I_{C}(n)) > 0 for all m ≥ n + 1, a contradiction.

The claim yields that

h^{1}(P^{r}, I_{C}(n)) = 0, n ≥ a_{1} + ... + a_{l} + 2 + h^{1}(P^{r}, I_{C}(a_{1} + ... + a_{l} + 2)).

So we only need to prove that

h^{1}(P^{r}, I_{C}(a_{1} + ... + a_{l} + 2)) ≤ G^{h}(r; s_{1}, ..., s_{l}) − g.

First of all we assume that C is not contained in a V_{s}^{2} with s < s_{2} + 1. Then, as in (i), we consider \overline{a}_{1} and \overline{a}_{2}. Since \overline{a}_{1} + \overline{a}_{2} ≤ a_{1} + ... + a_{l}, by the claim applied to the case l = 2 and to the sequence (s_{1}, s_{2} + 1), we have that

h^{1}(P^{r}, I_{C}(a_{1} + ... + a_{l} + 2)) ≤ h^{1}(P^{r}, I_{C}(\overline{a}_{1} + \overline{a}_{2} + 2)).

Now by (i) and by Propositions (1.1) and (2.1), we have:

h^{1}(P^{r}, I_{C}(\overline{a}_{1} + \overline{a}_{2} + 2)) = h^{0}(C, O_{C}(\overline{a}_{1} + \overline{a}_{2} + 2)) − h_{C}(\overline{a}_{1} + \overline{a}_{2} + 2) =

= 1 − g + (\overline{a}_{1} + \overline{a}_{2} + 2)s_{1} − h_{C}(\overline{a}_{1} + \overline{a}_{2} + 2) ≤

≤ 1 − g + (\overline{a}_{1} + \overline{a}_{2} + 2)s_{1} − ∑_{i=1}^{\overline{a}_{1}+\overline{a}_{2}+2} h_{Γ}(i) ≤

≤ 1 − g + (\overline{a}_{1} + \overline{a}_{2} + 2)s_{1} − ∑_{i=1}^{\overline{a}_{1}+\overline{a}_{2}+2} h(r; s_{1}, s_{2} + 1)(i) ≤

≤ G^{h}(r; s_{1}, s_{2} + 1) − g ≤ G^{h}(r; s_{1}, ..., s_{l}) − g.

By iterating this procedure, we may assume the existence of a flag like (2.3). But then, by arguing as above, one proves the bound for h^{1}(P^{r}, I_{C}(a_{1} + ... + a_{l} + 2)).

(iv) We will apply Proposition (2.7) below, which is independently proved, in order to use the fact that G^{h}(r; s_{1}, ..., s_{l}) − G(r; s_{1}, ..., s_{l}) < δ_{l}. As before, we only prove that

h^{1}(P^{r}, I_{C}(a_{1} + ... + a_{l} + 2)) ≤ G(r; s_{1}, ..., s_{l}) − g.

First suppose that C is not contained in a V_{s}^{2} with s < s_{2} + 1. Then, as in (iii) and using Proposition (1.4), we have:

h^{1}(P^{r}, I_{C}(a_{1} + ... + a_{l} + 2)) ≤ h^{1}(P^{r}, I_{C}(\overline{a}_{1} + \overline{a}_{2} + 2)) ≤ G^{h}(r; s_{1}, s_{2} + 1) − g ≤

≤ G^{h}(r; s_{1}, ..., s_{l}) − δ_{2} − g < G(r; s_{1}, ..., s_{l}) + δ_{l} − δ_{2} − g ≤ G(r; s_{1}, ..., s_{l}) − g.

Iterating this argument, we may assume the existence of a flag like: C := V_{s_{1}}^{1} ⊂ V_{s_{2}}^{2} ⊂ ... ⊂ V_{s_{l}}^{l} ⊂ P^{r}. Denote by k(r; s_{1}, ..., s_{l}) the SF for H_{F}(r; s_{1}, ..., s_{l}). Then, arguing as in (iii), one proves:

h^{1}(P^{r}, I_{C}(a_{1} + ...a_{l} + 2))) ≤ 1 − g + (a_{1} + ...a_{l} + 2)s_{1} − ∑_{i=1}^{a_{1}+...a_{l}+2} h_{Γ}(i) ≤

≤ 1 − g + (a_{1} + ...a_{l} + 2)s_{1} − ∑_{i=1}^{a_{1}+...a_{l}+2} k(r; s_{1}, ..., s_{l})(i) ≤ G(r; s_{1}, ..., s_{l}) − g. ♢

We want to finish this section by addressing the question of sharpness of the bound G^{h}(r; s_{1}, ..., s_{l}) which we found with Theorem (2.2). First we make the following:

Remark 2.5. There is no hope for the bound G^{h}(r; s_{1}, ..., s_{l}) to be sharp in general if l ≥ 3. In order to convince the reader of this, we start by considering the case l = 3. We will see later on (see Corollary (3.4)) that a curve C ∈ C(r; s_{1}, s_{2}, s_{3}) of maximal genus G(r; s_{1}, s_{2}, s_{3}) is ACM and, moreover, one has h_{Γ} = h(r; s_{1}, s_{2}, s_{3}) in the interval [0, a_{1}]. At this point the error for h(r; s_{1}, s_{2}, s_{3}), and therefore for G^{h}(r; s_{1}, s_{2}, s_{3}), resides in the interval I := [a_{1} +1, a_{1} +a_{2} +a_{3} +1]. We notice that our definition of h(r; s_{1}, s_{2}, s_{3}) in I is certainly consistent with Castelnuovo’s inequality (0.2) and therefore somehow justified by this. But the behaviour of h(r; s_{1}, s_{2}, s_{3}) in I cannot be the real behaviour of h_{Γ} in this range. In fact Δh_{Γ} in I is strictly decreasing, as shown by Corollary (1.2) of [CCD], whereas Δh(r; s_{1}, s_{2}, s_{3}) is actually increasing. We want to point out however that the error which our approximation via Castelnuovo theory of h_{Γ} causes, affects only the constant term of G^{h}(r; s_{1}, s_{2}, s_{3}), regarded as a polynomial in s_{1}. This is easy to verify by estimating ∑_{i∈I}(s_{1} − h(r; s_{1}, s_{2}, s_{3})(i)) as we saw in Proposition (1.1). Of course if l > 3 the error propagates to the terms η and ρ appearing in the formula for G^{h}(r; s_{1}, ..., s_{l}). However, as we shall see in a moment, it resides only there (see Proposition (2.7)).

The reader should also take into account that if C is a complete intersection of type (t_{1}, ..., t_{r−1}) in P^{r} with t_{1} ≥ ... ≥ t_{r−1}, and if we set s_{i} := t_{i}...t_{r−1}, i = 1, ..., r − 1, then the genus g of C is

g = \frac{s_{1}^{2}}{2s_{2}} + \frac{s_{1}}{2} (\frac{s_{2}}{s_{3}} + ... + s_{r−1} − r − 1) + 1.

Comparing with the general formula for G^{h}(r; s_{1}, ..., s_{r−1}), we see that it differs from it only for the terms η and ρ. On the other hand it is reasonable to expect that G(r; s_{1}, ..., s_{r−1}) coincides with the arithmetic genus of a complete intersection as above, as suggested by the index theorem [Hr, pg.196].

To prove the result we announced in the previous remark, we need a:

Lemma 2.6. Let X be an irreducible, reduced, non degenerate, arithmetically Cohen-Macaulay (ACM) curve in P^{n}. Let Y be the cone over X in P^{n+1}. Let D be a reduced divisor contained in the smooth locus of X such that the linear series |D| is base point free and contains |O_{X}(1)|. Then there is an irreducible, reduced, ACM curve Z on Y such that its general hyperplane section Γ has the same Hilbert function as D.

Proof. Let Z′ be the cone over D in Y; by Bertini’s theorem and by our assumptions on D we know that Z′ is linearly equivalent (as a Weil divisor) to a reduced, irreducible curve Z ⊂ Y. Since Z′ is ACM, then also Z is so; in particular these two curves have the same Hilbert function. We conclude by observing that the Hilbert function of any hyperplane sections of Z is the first difference of the Hilbert function of Z and the same happens for Z′.♢

Proposition 2.7. Let l = 2, ..., r −1 be an integer and let s_{1}, s_{2}, ..., s_{l} be integers (with s_{l} ≥ r − l + 1) such that:

s_{i} ≥ 8(l − 1)[(l − i + 1)^{2} + 2(l − i + 1) + 9]\begin{matrix}s_{i+1} & 3 \\ s_{i+2} & 2 \end{matrix}

for all i = 1, ..., l − 1 and that:

s_{l−1} > \frac{2s_{l}}{r − l} ∏_{j=1}^{r−l} \sqrt[r−l+1−j]{(r − l + 1)!s_{l}}.

Then there is an irreducible non degenerate curve C in P^{r} of degree s_{1} and genus g such that

g = \frac{s_{1}^{2}}{2s_{2}} + \frac{s_{1}}{2} [\frac{s_{2}}{s_{3}} + ... + \frac{s_{l}}{s_{l+1}} − l − \frac{2}{r − l} + \overline{η}] + \overline{ρ} + 1

with |\overline{η}| ≤ \frac{3}{4} and |\overline{ρ}| ≤ (l^{2} + 2l + 9)\frac{s_{2}^{3}}{s_{3}^{2}} and moreover, if l = 3, with \overline{η} = η and 0 ≤

ρ − \overline{ρ} ≤ 6\frac{s_{2}^{3}}{s_{3}^{2}} , η and ρ being the numbers appearing in the formula of G^{h}(r; s_{1}, s_{2}, s_{3}) in Proposition (1.1). In particular there are curves verifying the hypotheses of (ii) of Theorem (2.2).

Proof. Let us consider a flag V_{s_{l−1}}^{1} ⊂ V_{s_{l}}^{2} ⊂ P^{r−l+2}, where V_{s_{l−1}}^{1} has genus G^{h}(r − l + 2; s_{l−1}, s_{l}) and V_{s_{l}}^{2} has general hyperplane sections of maximal genus. The existence of such flags is proved in [CCD], examples (6.4), (6.5), under conditions which are easily seen to be verified in our situation. Let g_{l−1} be the arithmetic genus of V_{s_{l−1}}^{1} and Γ_{l−1} be its general hyperplane section. Over V_{s_{l−1}}^{1} we fix a divisor D_{l−1} of degree s_{l−2}. Since

by our assumptions and by (1.2) we have s_{l−2} ≥ s_{l−1} + 2\frac{s_{l−1}^{2}}{s_{l}} ≥ s_{l−1} + 2g_{l−1}, then D_{l−1} can be taken in such a way to satisfy the hypothesis of the previous Lemma. So we can find an irreducible, reduced, ACM curve V_{s_{l−2}}^{1} on the cone V_{s_{l−1}}^{2} ⊂ P^{r−l+3} over V_{s_{l−1}}^{1}. Call g_{l−2} its genus, Γ_{l−2} its general hyperplane section. Since V_{s_{l−2}}^{1} is ACM, by the construction of the function h(r − l + 3; s_{l−2}, s_{l−1}, s_{l}) (see §0) and by Proposition (1.1), one gets

g_{l−2} = ∑_{i=1}^{a_{l}+a_{l−1}+a_{l−2}+2}(s_{l−2} − h_{Γ_{l−2}}(i)).

Using Bezout’s theorem and the fact that V_{s_{l−1}}^{1} is ACM, we have also, for i = 1, ..., a_{l−2},

h_{Γ_{l−2}}(i) = ∑_{j=0}^{i} h_{Γ_{l−1}}(j).

Now h_{Γ_{l−1}} is known by [CCD] (see also §0), and by the definition of the function h(r − l + 3; s_{l−2}, s_{l−1}, s_{l}) one has

∑_{j=0}^{i} h_{Γ_{l−1}}(j) = h(r − l + 3; s_{l−2}, s_{l−1}, s_{l})(i)

for i = 1, ..., a_{l−2}. This discussion shows that the difference G^{h}(r −l +3; s_{l−2}, s_{l−1}, s_{l})− g_{l−2} is

∑_{i=a_{l−2}+1}^{a_{l}+a_{l−1}+a_{l−2}+2}(h_{Γ_{l−2}}(i) − h(r − l + 3; s_{l−2}, s_{l−1}, s_{l})(i))

which can be bounded exactly in the same way as we did for ρ_{4} in the proof of Proposition (1.1). If l = 3 this ends the proof. Otherwise, in order to accomplish the proof, one has to iterate the argument and proceed, in order to compute the genus of V_{s_{i}}^{1}, in the same way as, in the proof of Proposition (1.1), we computed the number G(r − i + 1; s_{i}, ..., s_{l}), i = 1, ..., l − 2. We omit the details.

As for the last assertion, one sees by elementary calculations, which can be omitted, that if the hypotheses on the s_{i}’s of Theorem (2.2) hold, then the curves we constructed in the present proof verify (ii) of Theorem (2.2). ♢

As a relevant direct consequence of the Theorem (2.2) and of the previous Proposition (2.7), we want to point out the following:

Corollary 2.8. Let l = 2, ..., r − 1 be an integer and let s_{1}, ..., s_{l} be positive integers (with s_{l} ≥ r − l + 1) such that:

s_{i} ≥ 8(l − 1)[(l − i + 1)^{2} + 2(l − i + 1) + 9]\frac{(s_{i+1} + 1)^{3}}{r − i − 1}

and that

s_{i} > max\{\frac{(s_{i+1} + 1)^{2}}{r − i − 1} +(2r−2)(s_{i+1}+1), 2\frac{(s_{i+1} + 1)}{r − i − 1} ∏_{j=1}^{r−i−1} \sqrt[r−i−j]{(r − i)!(s_{i+1} + 1)}\}

for all i = 1, ..., l − 1. Then the curves C verifying a flag condition of type (r; s_{1}, ..., s_{l}) and having maximal genus G(r; s_{1}, ..., s_{l}) must lie on a flag of type C := V_{s_{1}}^{1} ⊂ V_{s_{2}}^{2} ⊂ ... ⊂ V_{s_{l}}^{l} ⊂ P^{r}. In particular:

G(r; s_{1}, ..., s_{l}) = max{g(C) : C ∈ F(r; s_{1}, ..., s_{l})}.

Remark 2.9 In conclusion, let us remark that the flag conditions that we considered in the present paragraph could be generalized. Precisely one could say that an irreducible, non degenerate curve C ⊂ P^{r} verifies a generalized flag condition of type (r; s_{1}, s_{2}^{(i_{2})}, ..., s_{l}^{(i_{l})}) if C is of degree s_{1} and does not lie on any variety V_{s}^{i_{j}}, with s < s_{j}, j = 2, ..., l. Then one could try to determine or estimate the maximum genus of curves verifying such a generalized flag condition. Unfortunately we are not jet ready to deal with this problem in its generality. However we want to briefly discuss a particular instance of it, on which we are able to provide some information and give some flavour of what should be the general answer.

Precisely we consider a flag condition of type (r; d, t^{(r−1)}), i.e. we would like to estimate the arithmetic genus g of an irreducible, non degenerate curve C ⊂ P^{r} of degree d not lying on any hypersurface of degree s < t. Such an estimate is not difficult to find. In fact, let s(r; t) be the maximum integer such that any integral surface of degree s < s(r; t) in P^{r} lies on a hypersurface of degree t − 1. Then, if d >> 0, by the main result of [CCD], one has

g ≤ G^{h}(r; d, s(r; t)).

This bound is asymptotically sharp. Indeed, let S be an integral surface of degree s(r; t) not lying on any hypersurface of degree t − 1; a general hypersurface section of S of very high degree has genus which is close to the above bound.

Of course the problem remains of determining s(r; t). This is a question of independent interest which we would like to point out to the reader. It is clear that this problem is related to the maximal rank problem for surfaces in P^{r}. However, it is not so difficult to give an estimate of s(r; t):

s(r; t) > max\{s ∈ N : \frac{st(t − 1)}{2} + t < h^{0}(P^{r}, O_{P^{r}}(t − 1))\}.

Indeed one verifies that for every integral surface S of degree s in P^{r} one has

h_{S}(n) ≤ \frac{sn(n + 1)}{2} + n + 1.

In conclusion one finds with the above estimate

g ≤ \frac{r!d^{2}}{4t^{r−2}} + O(d)  (2.10)

and we remark that, using a surface S in P^{r} complete intesection of r − 2 hypersurfaces of degree t, and taking as above hypersurface sections of S of very high degree, we find curves verifying the desired flag condition, whose genus is

g = \frac{d^{2}}{2t^{r−2}} + O(d).

§3. Recurrence properties of some families of Hilbert functions.

A consequence of Corollary (2.8) is that in order to compute G(r; s_{1}, ..., s_{l}) it would be sufficient to find a SF for H_{F}(r; s_{1}, ..., s_{l}). We will denote by k(r; s_{1}, ..., s_{l}) such a SF: as we said in §0, it is an open question to prove the existence of k(r; s_{1}, ..., s_{l}) in general. In this section we point our attention to the set H_{F}(r; s_{1}, ..., s_{l}) and we investigate some formal properties of its sharp function, if existing. Then we draw some consequence which leads us to a justification of the definition (0.6) of the function h(r; s_{1}, ..., s_{l}) (see Proposition (3.1)) and to an understanding of the flags on which the curves of maximal genus belong (see Theorem (3.3)). In Corollary (3.4) we specialize these results in the case l = 3.

Convention. In this section, where no explicit assumptions on the s_{i}’s are made, we assume the same numerical hypotheses as in Theorem (2.2).

Taking into account Proposition (2.1), first we slightly refine the notion of a lower approximating function given in §0. A numerical function f is defined to be a lower approximating function (LAF) for H_{F}(r; s_{1}, ..., s_{l}) if for all g ∈ H_{F}(r; s_{1}, ..., s_{l}) one has

g ≥ f ≥ h(r; s_{1}, ..., s_{l}).

Of course h(r; s_{1}, ..., s_{l}) and k(r; s_{1}, ..., s_{l}) are LAF. We note that for any such a function f one can define the function

G^{f}(r; s_{1}, ..., s_{l}) := ∑_{i=1}^{∞}(s_{1} − f(i)).

It is worth remarking that if there is a SF k := k(r; s_{1}, ..., s_{l}), then

G(r; s_{1}, ..., s_{l}) = G^{k}(r; s_{1}, ..., s_{l}).

One has

G(r; s_{1}, ..., s_{l}) ≤ G^{f}(r; s_{1}, ..., s_{l}) ≤ G^{h}(r; s_{1}, ..., s_{l})

and hence G^{f}(r; s_{1}, ..., s_{l}) can replace G^{h}(r; s_{1}, ..., s_{l}) in the statement of Theorem (2.2).

Among all lower approximating functions, we want to pick a family of functions which is of particular interest, as it will be clear in a moment.

Given the integers r, s_{1}, ..., s_{l} as usual, and a LAF f_{i} for H_{F}(r−i+1; s_{i}, ..., s_{l}), we say that the family ϕ := (f_{i})_{i=1,...,l−1} is a lower approximating recursive sequence (LARS) for H_{F}(r; s_{1}, ..., s_{l}) if

f_{i}(n) = ∑_{j=0}^{n} f_{i+1}(j)

for all i = 1, ..., l − 2 and for all n = 0, ..., a_{i}. For instance the family (h(r − i + 1; s_{i}, ..., s_{l}))_{i=1,...,l−1} is a LARS.

There are two reasons why the LARS’s are, in our opinion, relevant in the study of our subject. The first is explained by the:

Proposition 3.1. Let r, s_{1}, ..., s_{l} be integers such that for every i = 1, ..., l − 1 there is the sharp function k(r − i + 1; s_{i}, ..., s_{l}) in H_{F}(r − i + 1; s_{i}, ..., s_{l}). Then the family (k(r − i + 1; s_{i}, ..., s_{l}))_{i=1,...,l−1} is a LARS for H_{F}(r; s_{1}, ..., s_{l}).

Proof. For l = 2 the sharp functions exist and are described in §0 of the present paper. Of course, they verify the recursivity property.

Now we assume l ≥ 3 and argue by induction. Then we may assume the existence of a flag

D = V_{s_{2}}^{1} ⊂ V_{s_{3}}^{2} ⊂ ... ⊂ V_{s_{l}}^{l−1} ⊂ P^{r−1}

with D an ACM curve whose general hyperplane section has Hilbert function k(r − 1; s_{2}, ..., s_{l}). We can consider the cone from a point in P^{r} over each variety of the previous flag, and we get a new flag

V_{s_{2}}^{2} ⊂ V_{s_{3}}^{3} ⊂ ... ⊂ V_{s_{l}}^{l} ⊂ P^{r}.

By Lemma (2.6) we can find a C = V_{s_{1}}^{1} of degree s_{1} on V_{s_{2}}^{2}. Since D is ACM, by Bezout’s theorem and by the definition of D, we have

h_{Γ}(n) = h_{D}(n) = ∑_{j=0}^{n} k(r − 1; s_{2}, ..., s_{l})(j) ≥ k(r; s_{1}, ..., s_{l})(n)

where, as usual, Γ is the general hyperplane section of C and n = 0, ..., a_{1}. To prove the converse inequality, fix a flag

C = V_{s_{1}}^{1} ⊂ V_{s_{2}}^{2} ⊂ ... ⊂ V_{s_{l}}^{l} ⊂ P^{r}

such that h_{Γ} = k(r; s_{1}, ..., s_{l}). We call D the general hyperplane section of V_{s_{2}}^{2} and D′ its general hyperplane section. Then, using again Bezout’s theorem, we have

k(r; s_{1}, ..., s_{l})(n) = h_{Γ}(n) = h_{D}(n) ≥ ∑_{j=0}^{n} h_{D′}(j) ≥ ∑_{j=0}^{n} k(r − 1; s_{2}, ..., s_{l})(j)

for n = 0, ..., a_{1}.♢

Before proceeding further, we want to stress here that the hypotheses of the previous proposition are certainly verified for l = 2 (see [CCD]) and for r = 4 and l = 3 (see [CCD2]).

The second reason for the importance of lower approximating recursive sequences is given by the:

Proposition 3.2. Let ϕ := (f_{i})_{i=1,...,l−1} be a LARS for H_{F}(r; s_{1}, ..., s_{l}). Consider a flag of type:

C = V_{s_{1}}^{1} ⊂ V_{s_{2}}^{2} ⊂ ... ⊂ V_{s_{l}}^{l} ⊂ P^{r}.

Let us fix i = 2, ..., l and let C_{i} be a general curve section of V_{s_{i}}^{i} in P^{r+1−i}. Then for any integer α ≥ 1 one has

G^{f_{1}}(r; s_{1}, ..., s_{l}) − g(C) < α\binom{\frac{s_{i−1}}{s_{i}} − s_{i} + i − 2}{i − 1} ⟹ G^{f_{i}}(r + 1 − i; s_{i}, ..., s_{l}) − g(C_{i}) < α.

Proof. We will work out only the case i = 2, since the other cases i > 2 can be proved in exactly the same fashion.

Assume that G^{f_{2}}(r − 1; s_{2}, ..., s_{l}) − g(C_{2}) ≥ α. By the definition of lower approximating recursive sequences, we have

f_{1}(n) = 1 + ns_{2} − G^{f_{2}}(r − 1; s_{2}, ..., s_{l}), if s_{2} ≤ n ≤ a_{1}

since the hypotheses on the s_{i}’s yield s_{2} > a_{2} + ... + a_{l} + 2. By the main result of [GLP], we have

h_{C_{2}}(n) = 1 − g(C_{2}) + ns_{2}, if s_{2} ≤ n ≤ a_{1}.

Using, as usual, Bezout’s theorem, it follows that

h_{Γ}(n) = h_{C_{2}}(n) = 1 − g(C_{2}) + ns_{2} ≥ α − G^{f_{2}}(r − 1; s_{2}, ..., s_{l}) + 1 + ns_{2} = α + f_{1}(n), if s_{2} ≤ n ≤ a_{1}. Whence

G^{f_{1}}(r; s_{1}, ..., s_{l}) − g(C) ≥ ∑_{n=s_{2}}^{a_{1}}(h_{Γ}(n) − f_{1}(n)) ≥ α(a_{1} − s_{2} + 1) ≥ α(\frac{s_{1}}{s_{2}} − s_{2})

which finishes the proof for i = 2. ♢

Putting together the information contained in Proposition (3.1) and Propostion (3.2), we can prove the following theorem, which expresses one the main feature of curves with maximal genus, i.e. they lie on flags of varieties whose general curve sections still have maximal genus.

Theorem 3.3. Suppose that for every i = 1, ..., l − 1 there is the sharp function k(r − i + 1; s_{i}, ..., s_{l}) in H_{F}(r − i + 1; s_{i}, ..., s_{l}). Let C be a curve in C(r; s_{1}, ...s_{l}) such that g(C) = G(r; s_{1}, ..., s_{l}). Then there is a unique flag of type

C = V_{s_{1}}^{1} ⊂ V_{s_{2}}^{2} ⊂ ... ⊂ V_{s_{l}}^{l} ⊂ P^{r}

such that for all i = 2, ..., l, the general curve section C_{i} ⊂ P^{r+1−i} of V_{s_{i}}^{i} has the property

g(C_{i}) = G(r − i + 1; s_{i}, ..., s_{l}).

Proof. The flag exists by Corollary (2.8). Then we can apply Proposition (3.2) with α = 1 and we get the result. ♢

We want to point out another consequence of Propostion (3.2) concerning the case l = 3.

Corollary 3.4. Let r, s_{1}, s_{2}, s_{3} be integers such that r ≥ 4, s_{3} ≥ r − 2,

s_{2} > max\{272\frac{(s_{3} + 1)^{3}}{r − 3} , 2\frac{s_{3} + 1}{r − 3} ∏_{j=1}^{r−3} \sqrt[r−2−j]{(r − 2)!(s_{3} + 1)}\},

s_{1} > max\{384\frac{(s_{2} + 1)^{3}}{r − 2} , 6\frac{s_{2}^{4}}{s_{3}^{2}} + s_{2}, 2\frac{s_{2} + 1}{r − 2} ∏_{j=1}^{r−2} \sqrt[r−1−j]{(r − 1)!(s_{2} + 1)}\}.

Let C be a curve in C(r; s_{1}, s_{2}, s_{3}) of arithmetic genus g = G(r; s_{1}, s_{2}, s_{3}) and consider the flag on which C lies :

C = V_{s_{1}}^{1} ⊂ V_{s_{2}}^{2} ⊂ V_{s_{3}}^{3} ⊂ P^{r}.  (3.5)

Then we have:

(i) C is ACM; (ii) V_{s_{2}}^{2} is a surface whose general hyperplane sections are curves of maximal genus in C(r − 1; s_{2}, s_{3}); (iii) V_{s_{3}}^{3} is a threefold whose general curves sections are curves of maximal genus in C(r − 2; s_{3}); (iv) for all i = 0, ..., a_{1} one has h_{Γ}(i) = h(r; s_{1}, s_{2}, s_{3})(i).

In particular, if we denote by F^{0}(r; s_{1}, s_{2}, s_{3}) the set of all ACM curves C ∈ F(r; s_{1}, s_{2}, s_{3}) such that there is a flag C = V_{s_{1}}^{1} ⊂ V_{s_{2}}^{2} ⊂ V_{s_{3}}^{3} ⊂ P^{r} with V_{s_{2}}^{2} and V_{s_{3}}^{3} ACM varieties, then

G(r; s_{1}, s_{2}, s_{3}) = max{g(C), C ∈ F^{0}(r; s_{1}, s_{2}, s_{3})}.

Proof. First we show (ii). By Proposition (2.7) we know that G^{h}(r; s_{1}, s_{2}, s_{3}) − g ≤ 6\frac{s_{2}^{3}}{s_{3}^{2}} which is, in view of our hypotheses on s_{1}, strictly less than \frac{s_{1}}{s_{2}} − s_{2}. Hence we can apply Proposition (3.2) for the LARS (h(r − i + 1; s_{i}, ..., s_{3}))_{i=1,2}, for α = 1 and for i = 2, obtaining (ii).

In order to prove (iii), we note that the general hyperplane section of V_{s_{2}}^{2} lies on a Castelnuovo surface of degree s_{3} in P^{r−1} by [CCD, Corollary 6.3]. These surfaces can be lifted in P^{r} to a threefold T of degree s_{3} by the main result of [CC] and one has T = V_{s_{3}}^{3} by Bezout’s theorem.

Now we prove (i). First we notice that, since V_{s_{2}}^{2} is ACM, we can argue as in the proof of the Proposition (2.7) and so we can find an irreducible, reduced, ACM curve D in C(r; s_{1}, s_{2}, s_{3}) such that its general hyperplane section Δ has the same Hilbert function as Γ. Hence:

G(r; s_{1}, s_{2}, s_{3}) = g(C) ≤ ∑_{i=1}^{∞}(d − h_{Γ}(i)) = ∑_{i=1}^{∞}(d − h_{Δ}(i)) = g(D) ≤ G(r; s_{1}, s_{2}, s_{3}).

It follows that g(C) = ∑_{i=1}^{∞}(d − h_{Γ}(i)) and C is ACM.

Finally (iv) follows by arguing as in the proof of the Proposition (2.1), taking into account that C and V_{s_{2}}^{2} are ACM and using (ii) and the definition of h(r; s_{1}, s_{2}, s_{3})(i).♢

Remark 3.5 Let H_{F^{0}}(r; s_{1}, s_{2}, s_{3}) be the set of Hilbert functions corresponding to F^{0}(r; s_{1}, s_{2}, s_{3}). By using similar arguments as in the proof of Corollary (3.4), one can prove that the Hilbert function of the general hyperplane section of a curve C ∈ C(r; s_{1}, s_{2}, s_{3}) of maximal genus G(r; s_{1}, s_{2}, s_{3}), is a minimal element in the ordered set H_{F^{0}}(r; s_{1}, s_{2}, s_{3}).

References

[ACGH] E.Arbarello, M.Cornalba, P.A.Griffiths and J.Harris, Geometry of algebraic curves, Volume I, Springer-Verlag, A series of Comprehensive Studies in Mathematics, 1985, 267.

[Ca] G.Castelnuovo, Ricerche di geometria sulle curve algebriche, Zanichelli, Bologna, 1937.

[CC] L.Chiantini and C.Ciliberto, A few remarks on the lifting problem, Astérisque 218** (1993).

[CCD] L.Chiantini, C.Ciliberto and V.Di Gennaro, The genus of projective curves, Duke Math. J., Vol.70, No.2, May 1993, 229-245.

[CCD2] L.Chiantini, C.Ciliberto and V.Di Gennaro, The genus of curves in P^{4} verifying certain flag conditions , preprint, 1995.

[EH] D.Eisenbud and J.Harris,Curves in projective space, Sém. Math. Sup. 85, Les Presses de l’Université de Montréal, Montréal, 1982.

[GLP] L.Gruson, R.Lazarsfeld and C.Peskine, On a theorem of Castelnuovo and the equations defining space curves, Invent. Math. 72 (1983), 491-506.

[GP] L.Gruson and C.Peskine, “Genre des courbes dans l’espace projectif” in Algebraic Geometry: Proceedings, Norway, 1977, Lecture Notes in Math. 687, Springer Verlag, New York, 1978, 31-59.

[H] G.Halphen, Mémoire sur la classification des courbes gauches algébriques, Oeuvres complètes t.III.

[Hr] J.Harris, The genus of space curves, Math. Ann. 249, 191-204.

[M] D.Mumford, Lectures on Curves on an Algebraic Surface, Ann. of Math. Stud. 59, Princeton Univ. Press, Princeton, N.J., 1966.

[N] M.Noether, Zur Grundlegung der Theorie der Algebraischen Raumcurven, Königlichen Akad. der Wissenschaften, 1883.

L. Chiantini: Dipartimento Metodi e Modelli Matematici, Via Scarpa 800161 Roma, Italy

C.Ciliberto: Dipartimento di Matematica, Università di Roma “Tor Vergata” 00133 Roma, Italy

V.Di Gennaro: Dipartimento di Matematica, Università di Roma “Tor Vergata” 00133 Roma, Italy