HIGHER RAMIFICATION AND VARIETIES OF SECANT DIVISORS ON THE GENERIC CURVE

GAVRIL FARKAS

ABSTRACT

Abstract. For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. The aim of this paper is to clarify this problem in the case of the generic curve C of given genus. We determine precisely under which conditions the cycle of e-secant k-planes in non-empty and we compute its dimension. We also precisely determine the dimension of the variety of linear series on C carrying e-secant k-planes.

For a smooth projective curve C of genus g, we denote by C_{e} the e-th symmetric product of C and by G_{d}^{r}(C) the variety of linear series of type 𝔤_{d}^{r} on C, that is,

G_{d}^{r}(C) := {(L, V) : L ∈ Pic^{d}(C), V ∈ G(r + 1, H^{0}(L))}.

The main result of Brill-Noether theory states that if [C] ∈ M_{g} is a general curve then G_{d}^{r}(C) is a smooth variety of dimension equal to ρ(g, r, d) := g − (r + 1)(g − d + r). For a linear series l = (L, V) ∈ G_{d}^{r}(C) and an effective divisor D ∈ C_{e}, using the natural inclusion H^{0}(C, L ⊗ O_{C}(−D)) ⊂ H^{0}(C, L), we can define a new linear series l(−D) := (L⊗O_{C}(−D), V ∩H^{0}(L⊗O_{C}(−D))). We fix integers 0 ≤ f < e and introduce the determinantal cycle

V_{e}^{e−f}(l) := {D ∈ C_{e} : dim l(−D) ≥ r − e + f}

of effective divisors of degree e which impose at most e − f independent conditions on

l. If l is very ample and we view C ↪^{l} P^{r} as an embedded curve, then V_{e}^{e−f}(l) parameterizes e-secant (e − f − 1)-planes to C. Each irreducible component of V_{e}^{e−f}(l) has dimension at least e − f(r + 1 − e + f). The cycles V_{e}^{e−f}(l) have been extensively studied in classical enumerative geometry. The virtual class [V_{e}^{e−f}(l)]^{virt} ∈ A^{f(r+1−e+f)}(C_{e}) has been computed by MacDonald and its expression is tremendously complicated and thus of limited practical use (see [ACGH], Chapter VIII). One case when we have a manageable formula is for e = 2r − 2 and f = r − 1, when [V_{2r−2}^{r−1}(l)]^{virt} computes the (virtual) number of (r − 2)-planes in P^{r} which are (2r − 2)-secant to C (cf. [Ca]).

Research partially supported by an Alfred P. Sloan Fellowship, the NSF Grants DMS-0450670 and DMS-0500747 and a 2006 Texas Summer Research Assignment.

2000 Mathematics Subject Classification 14H10, 14C20.

Surprisingly little is known about the validity of these classical enumerative formulas (see [H] and [LB1] for partial results in the case of curves in P^{3}). The aim of this paper is to clarify this problem for a general curve [C] ∈ M_{g}. For every linear series l ∈ G_{d}^{r}(C) we determine precisely under which conditions the cycle V_{e}^{e−f}(l) is nonempty and has the expected dimension. Then having fixed [C] ∈ M_{g}, we determine the dimension of the family of linear series l ∈ G_{d}^{r}(C) with an e-secant (e − f − 1)-plane. For our first result, we use degeneration techniques together with a few facts about the ample cone of the moduli space \overline{M}_{0,g} to prove the following:

Theorem 0.1. Let [C] ∈ M_{g} be a general curve and we fix non-negative integers 0 ≤ f < e, r and d, such that r − e + f ≥ 0. Then we have that

dim{l ∈ G_{d}^{r}(C) : V_{e}^{e−f}(l) ≠ ∅} ≤ ρ(g, r, d) − f(r + 1 − e + f) + e.

In particular, if ρ(g, r, d) − f(r + 1 − e + f) + e < 0, then V_{e}^{e−f}(l) = ∅, for every l ∈ G_{d}^{r}(C).

More precisely, in Section 2 we prove the following dimensionality estimate

dim{(D, l) ∈ C_{e} × G_{d}^{r}(C) : D ∈ V_{e}^{e−f}(l)} ≤ ρ(g, r, d) − f(r + 1 − e + f) + e,

which obviously implies Theorem 0.1. This result generalizes the Brill-Noether theorem. Indeed, when l = K_{C}, then V_{e}^{e−f}(K_{C}) = C_{e}^{f} := {D ∈ C_{e} : h^{0}(O_{C}(D)) ≥ f + 1}. Since the fibres of the Abel-Jacobi map C_{e}^{f} → W_{e}^{f}(C) are at least f-dimensional, clearly G_{e}^{f}(C) ≠ ∅ implies that dim C_{e}^{f} ≥ f. Our result reads G_{e}^{f}(C) = ∅ when ρ(g, f, e) < 0, which is the non-existence part of the classical Brill-Noether theorem (cf. [EH1]). More generally, we have the following result in the case ρ(g, r, d) = 0:

Corollary 0.2. Suppose ρ(g, r, d) = 0 and e < f(r + 1 − e + f). Then for a general curve [C] ∈ M_{g} we have that V_{e}^{e−f}(l) = ∅ for every l ∈ G_{d}^{r}(C), that is, no linear series of type 𝔤_{d}^{r} on C has any e-secant (e − f − 1)-planes.

An immediate consequence of Theorem 0.1 is a proof of the following conjecture of Coppens and Martens (cf. [CM2] Theorem 3.3.1, for a proof in the case f = 1):

Corollary 0.3. Let [C] ∈ M_{g} be a general curve and we fix integers 0 ≤ f < e, d and r such that r − e + f ≥ 0. Let l be a general linear series of type 𝔤_{d}^{r} belonging to an irreducible component of G_{d}^{r}(C). Assuming that V_{e}^{e−f}(l) is not empty, then e − f(r + 1 − e + f) ≥ 0. Moreover V_{e}^{e−f}(l) is equidimensional and dim V_{e}^{e−f}(l) = e − f(r + 1 − e + f).

We note that when f = 1, Theorem 0.1 concerns the higher order very ampleness of linear series on a general curve. We recall that a linear series l ∈ G_{d}^{r}(C) is said to be (e − 1)-very ample if dim l(−p_{1} − ⋯ − p_{e}) = r − e, for any choice of (not necessarily distinct) e points p_{1}, …, p_{e} ∈ C. Thus 0-very ampleness is equivalent to generation by global sections and 1-very ampleness reduces to the classical notion of very ampleness.

Corollary 0.4. Let [C] ∈ M_{g} be a general curve and e, r, d be non-negative integers such that ρ(g, r, d) + 2e − 2 − r < 0. Then every linear series l ∈ G_{d}^{r}(C) is (e − 1)-very ample.

Theorem 0.1 does not address the issue of existence of linear series with e-secant (e−f −1)-planes. We prove the following existence result for secant planes corresponding to linear series 𝔤_{d}^{r} on an arbitrary smooth curve of genus g.

Theorem 0.5. Let [C] ∈ M_{g} be a general smooth curve and we fix integers 0 ≤ f < e ≤ g, d and r, such that f(r + 1 − e + f) ≥ e, d ≥ 2e − f − 1, g − d + r ≥ 0,

ρ(g, r, d) − f(r + 1 − e + f) + e ≥ 0 andρ(g, r − e + f, d − e) ≥ 0.

Assume moreover that we are in one of the following situations:

(i) 2f ≤ e − 1, (ii) e = 2r − 2 andf = r − 1, (iii) e < 2(r + 1 − e + f), or
(iv) ρ(g, r, d) ≥ f(r + 1 − e + f) − (g − d + r).

Then there exists a linear series l ∈ G_{d}^{r}(C) such that V_{e}^{e−f}(l) ≠ ∅. Moreover, one has that the following dimensionality statement:

dim{(D, l) ∈ C_{e} × G_{d}^{r}(C) : D ∈ V_{e}^{e−f}(l)} = ρ(g, r, d) − f(r + 1 − e + f) + e.
The inequalities ρ(g, r − e + f, d − e) ≥ 0 and ρ(g, r, d) + e − f(r + 1 − e + f) ≥ 0

are obvious necessary conditions for the existence of l ∈ G_{d}^{r}(C) with V_{e}^{e−f}(l) ≠ ∅ on a general curve [C] ∈ M_{g}. To give an example, an elliptic quartic curve C ⊂ P^{3} has no 3-secant lines even though ρ(g, r, d) + e − f(r + 1 − e + f) > 0 (note that e = 3 and f = 1 in this case). Theorem 0.5 is stated in the range f(r + 1 − e + f) ≥ e, corresponding to the case when linear series l ∈ G_{d}^{r}(C) with V_{e}^{e−f}(l) ≠ ∅ are expected to be special in the Brill-Noether cycle G_{d}^{r}(C). It is clear though that the methods of this paper can be applied to the case e ≥ f(r + 1 − e + f) as well. In that range however, when one expects existence of e-secant (e − f − 1)-planes for every l ∈ G_{d}^{r}(C), there are nearly optimal existence results obtained by using positivity for Chern classes of certain vector bundles in the style of [FL]: For every curve [C] ∈ M_{g} and l ∈ G_{d}^{r}(C), assuming that d ≥ 2e − 1 and e − f(r + 1 − e + f) ≥ r − e + f, one knows that V_{e}^{e−f}(l) ≠ ∅ (cf. [CM1], Theorem 1.2). For l ∈ G_{d}^{r}(C) such that g − d + r ≤ 1 (e.g. when l is non-special), if we keep the assumption e − f(r + 1 − e + f) ≥ 0, it is known that V_{e}^{e−f}(l) ≠ ∅ if and only if ρ(g, r − e + f, d − e) ≥ 0 (cf. [ACGH], pg. 356). This appears to be the only case when MacDonald’s formula displays some positivity features that can be used to derive existence results on V_{e}^{e−f}(l). In the case l = K_{C}, one recovers of course the existence theorem from classical Brill-Noether theory. We finally mention that Theorem 0.5 holds independent of the assumptions (i) − (iii), whenever a certain transversality condition (18) concerning a general curve [Y, p] ∈ M_{e,1} is satisfied (see Section 3 for details). Theorem 0.5 is then proved by verifying this condition (18) in each of the cases

(i) − (iii).

We now specialize to the case when e = f(r + 1 − e + f) which is covered by Theorem 0.5. One can write r = (u − 1)(f + 1) and e = uf for some u ≥ 1, and we obtain the following result concerning the classical problem of existence of uf-secant secant (uf − f − 1)-planes to curves in P^{r}:

Corollary 0.6. Let C be a smooth curve of genus g. We fix integers d, u, f ≥ 2 and assume that the inequalities g ≥ uf, d ≥ 2uf −f −1, ρ(g, uf +u−f −1, d) ≥ 0 and ρ(g, u−1, d−uf) ≥ 0 hold. Then there exists an embedding C ⊂ P^{(u−1)(f+1)} with deg(C) = d, such that C has a uf-secant (uf − f − 1)-plane. If moreover, [C] ∈ M_{g} is general in moduli, then the embedded

curve C ↪^{l} P^{(u−1)(f+1)} corresponding to a general linear series l ∈ G_{d}^{(u−1)(f+1)}(C) has only a finite number of uf-secant (uf − f − 1)-planes.

If [C] ∈ M_{g} is suitably general we can prove that the Cayley-Castelnuovo numbers predicting the number of (2r − 2)-secant (r − 2)-planes of a curve in C ⊂ P^{r} have a precise enumerative meaning:

Theorem 0.7. Let [C] ∈ M_{g} be a general curve. We fix integers d, r ≥ 3 such that d ≥ 3r − 2,

ρ(g, r, d) ≥ ∅ and ρ(g, 1, d − 2r + 2) ≥ 0. Then if C ↪^{l} P^{r} is an embedding corresponding to a general linear series l ∈ G_{d}^{r}(C), then C has only finitely many (2r − 2)-secant (r − 2)-planes. Their number (counted with multiplicities) is

C(d, g, r) = ∑_{i=0}^{r−1} \frac{(−1)^{i}}{r − i} \binom{d − r − i + 1}{r − 1 − i}\binom{d − r − i}{r − 1 − i}\binom{g}{i}.

A modern proof of the formula for C(d, g, r) is due to MacDonald and appears in [ACGH] Chapter VIII. The original formula is due to Castelnuovo (cf. [Ca]). When r = 3, we recover Cayley’s formula for the number of 4-secant lines of a smooth space curve C ⊂ P^{3} of degree d (cf. [C]):

C(d, g, 3) = \frac{1}{12}(d − 2)(d − 3)^{2}(d − 4) − \frac{g}{2}(d^{2} − 7d + 13 − g).

To make a historical remark, there have been various attempts to rigorously justify the so-called functional method that Cayley (1863), Castelnuovo (1889) and Severi (1900) used to derive their enumerative formulas and to determine their range of applicability (see [LB1], [V]). For instance, Cayley’s formula is shown to hold for an arbitrary smooth curve in P^{3}, provided that C(d, g, 3) is defined as the degree of a certain 0-cycle Sec_{4}(C) in G(1, 3) (cf. [LB2]). The drawback of this approach is that it becomes very difficult to determine when this newly defined invariant is really enumerative. For instance Le Barz only shows that this happens for very special curves in P^{3} (rational curves and generic complete intersections) and one of the aims of this paper is to establish the validity of such formulas for curves that are general with respect to moduli.

The second topic we study concerns ramification points of powers of linear series on curves. This question appeared first in a particular case in [F1]. We recall that for a pointed curve [C, p] ∈ M_{g,1} and a linear series l = (L, V) ∈ G_{d}^{r}(C), the vanishing sequence of l at p

a^{l}(p) : a_{0}^{l}(p) < … < a_{r}^{l}(p) ≤ d

is obtained by ordering the set {ord_{p}(σ)}_{σ∈V}. The weight of p with respect to l is defined as w^{l}(p) := ∑_{i=0}^{r}(a_{i}^{l}(p) − i). One says that p is a ramification point of l if w^{l}(p) ≥ 1 and we denote by R(l) the finite set of ramification points of l. If [C, p] ∈ M_{g,1} and \overline{α} : 0 ≤ α_{0} ≤ … ≤ α_{r} ≤ d − r is a Schubert index of type (r, d), the cycle

G_{d}^{r}(C, p, \overline{α}) := \{l ∈ G_{d}^{r}(C) : a_{i}^{l}(p) ≥ α_{i} + i for i = 0 … r\}

can be realized as a generalized determinantal variety inside G_{d}^{r}(C) having virtual dimension ρ(g, r, d, \overline{α}) := ρ(g, r, d) − ∑_{j=0}^{r} α_{j}. For a general pointed curve [C, p] ∈ M_{g,1}, it is known that the virtual dimension equals the actual dimension, that is,

dim G_{d}^{r}(C, p, \overline{α}) = ρ(g, r, d, \overline{α}) (cf. [EH2] Theorem 1.1).

We address the following question: suppose l = (L, V) ∈ G_{d}^{r}(C) is a linear series with a prescribed ramification sequence \overline{α} at a fixed point p ∈ C. Is then p a ramification point of any of the powers L^{⊗n} for n ≥ 2? If so, can we describe the sequence a^{L^{⊗n}}(p)? One certainly expects that under suitable genericity assumptions on C and L, the points in ⋃_{n≥1} R(L^{⊗n}) should be uniformly distributed on C. For example, it is known that for every C and L ∈ Pic^{d}(C), the set ⋃_{n≥1} R(L^{⊗n}) is dense in C with respect to the classical topology (cf. [N]). Silverman and Voloch showed that for any L ∈ Pic^{d}(C) there exist finitely many points p ∈ C such that the set {n ≥ 1 : p ∈ R(L^{⊗n})} is infinite (cf. [SV]).

We prove that on a generic pointed curve [C, p], a linear series (L, V) and its multiples L^{⊗n} share no ramification points, that is R(l) and R(L^{⊗n}) are as transverse as they can be expected to be and moreover, the vanishing sequence a^{L^{⊗n}}(p) is close to being minimal:

Theorem 0.8. We fix a general pointed curve [C, p] ∈ M_{g,1}, integers r, d ≥ 1, n ≥ 3 and a Schubert index \overline{α} : 0 ≤ α_{0} ≤ … ≤ α_{r} ≤ d − r. We also set m := [(n + 1)/2]. Then for every linear series l = (L, V) ∈ G_{d}^{r}(C, p, \overline{α}) and every positive integer

a < nd − ρ(g, r, d, \overline{α}) − g − [\frac{g}{m}],

we have that h^{0}(C, L^{⊗n}(−ap)) = h^{0}(C, L^{⊗n})−a = nd+1−g−a. In other words, a_{i}^{L^{⊗n}}(p) = i for 0 ≤ i ≤ a − 1.

In the case n = 2, when we compare R(l) and R(L^{⊗2}) our results are sharper:

Theorem 0.9. We fix a general pointed curve [C, p] ∈ M_{g,1}, integers r, d ≥ 1 and a Schubert index \overline{α} : α_{0} ≤ … ≤ α_{r} ≤ d − r. Then for every (L, V) ∈ G_{d}^{r}(C, p, \overline{α}) and every positive integer

a < max\{2d + 2 − 2g − ρ(g, r, d, \overline{α}) + [\frac{g − 1}{2}], 2d + 2 − 2g − 2ρ(g, r, d, \overline{α}) + 2[\frac{g}{3}]\},

we have that h^{0}(C, L^{⊗2}(−ap)) = h^{0}(C, L^{⊗2}) − a = 2d + 1 − g − a.

Comparing the bounds on a given in Theorems 0.8 and 0.9 with the obvious necessary condition a ≤ nd − g + 1 which comes from the Riemann-Roch theorem, we see that our results are essentially optimal for relatively small values of ρ(g, r, d, \overline{α}) when the linear series (L, V) ∈ G_{d}^{r}(C, p, \overline{α}) have a strong geometric characterization. On the other hand, if for instance ρ(g, r, d, \overline{α}) = g, then L ∈ Pic^{d}(C) and p ∈ C are arbitrary and one cannot expect to prove a uniform result about the vanishing of H^{1}(C, L^{⊗n}(−ap)).

Theorems 0.8 and 0.9 concern line bundles L with prescribed ramification at a given point p ∈ C. Such bundles are of course very special in Pic^{d}(C). If instead, we try to describe ⋃_{n≥1} R(L^{⊗n}) for a general line bundle L ∈ Pic^{d}(C), the answer turns out to be particularly simple. We give a short proof of the following result:

Theorem 0.10. Let C be a smooth curve of genus g and L ∈ Pic^{d}(C) a very general line bundle.

(1) All the ramification points of the powers L^{⊗n} are ordinary, that is, w^{L^{⊗n}}(p) ≤ 1 for all p ∈ C and n ≥ 1.

(2) R(L^{⊗a}) ∩ R(L^{⊗b}) = ∅ for a ≠ b, that is, a point p ∈ C can be a ramification point for at most a single power of L.

After this paper has been written I have learnt that Theorem 0.10 has also been proved independently by M. Coppens in [Co]. I would like to thank the referee for a very careful reading of this paper and for pointing out that the initial proof of Theorem 0.1 was not complete.

1. Ramification points of multiples of linear series

In this section we use the technique of limit linear series to prove Theorems 0.8 and 0.9. We start by fixing a Schubert index \overline{α} : 0 ≤ α_{0} ≤ … ≤ α_{r} ≤ d − r and two integers a ≥ 0, n ≥ 2. We also set m := [(n + 1)/2].

We assume that for every [C, p] ∈ M_{g,1} there exists a linear series l = (L, V) ∈ G_{d}^{r}(C, p, \overline{α}) such that H^{0}(K_{C} ⊗ L^{⊗(−n)} ⊗ O_{C}(ap)) ≠ 0. By a degeneration argument we are going to show that this implies the inequalities

(1)  a ≥ nd − g − ρ(g, r, d, \overline{α}) − [\frac{g}{m}], when n ≥ 3,

(2)  a ≥ 2d + 2 − 2g − ρ(g, r, d, \overline{α}) + [\frac{g − 1}{2}],

and

(3)  a ≥ 2d + 2 − 2g − 2ρ(g, r, d, \overline{α}) + 2[\frac{g}{3}], when n = 2.

This will prove both Theorems 0.8 and 0.9.

We degenerate [C, p] to a stable curve [X_{0} := E_{0}∪_{p_{1}} E_{1}∪_{p_{2}} …∪_{p_{g−1}} E_{g−1}, p_{0}], where E_{i} is a general elliptic curve, p_{i}, p_{i+1} ∈ E_{i} are points such that p_{i+1} − p_{i} ∈ Pic^{0}(E_{i}) is not a torsion class and moreover E_{i} ∩ E_{i+1} = {p_{i+1}} for 0 ≤ i ≤ g − 2. Thus X_{0} is a string of g elliptic curves and the marked point p_{0} specializes to a general point lying on the first component E_{0}. We also consider a 1-dimensional family π : X → B together with a section σ : B → X, such that B = Spec(R) with R being a discrete valuation ring having uniformizing parameter t. We assume that X is a smooth surface and that there exists an isomorphism between X_{0} and π^{−1}(0). Under this isomorphism we also assume that σ(0) = p_{0} ∈ X_{0}. Here 0 ∈ B is the point corresponding to the maximal ideal of R and we denote by η and \overline{η} the generic and geometric generic point of B respectively. By assumption, there exists a linear series l\overline{η} = (L\overline{η}, V\overline{η}) ∈ G_{d}^{r}(X\overline{η}, σ(\overline{η}), \overline{α}), such that H^{0}(X\overline{η}, ω_{X\overline{η}} ⊗ L_{X\overline{η}}^{⊗(−n)} ⊗ O_{X\overline{η}}(aσ(\overline{η}))) ≠ 0. By possibly blowing up X at the nodes of X_{0} and thus replacing the central fibre by a curve X obtained from X_{0} by inserting chains of smooth rational curves at the points p_{1}, …, p_{g−1}, we may assume that l\overline{η} comes from a linear series l_{η} = (L_{η}, V_{η}) ∈ G_{d}^{r}(X_{η}, σ(η), \overline{α}) on the generic fibre X_{η}.

We denote by l_{E_{i}} = (L_{E_{i}}, V_{E_{i}}) ∈ G_{d}^{r}(E_{i}) the E_{i}-aspect of the limit linear series on X induced by l_{η}: Precisely, if L is a line bundle on X extending L_{η}, then L_{E_{i}} ∈ Pic^{d}(E_{i}) is the restriction to E_{i} of the unique twist L_{E_{i}} of L along components of π^{−1}(0) such that deg_{Z}(L_{i|Z}) = 0 for any irreducible component Z ≠ E_{i} of π^{−1}(0) (see also [EH1], p. 348). Since we gave ourselves the freedom of blowing-up X at the nodes of π^{−1}(0), we can also assume that {l_{E_{i}}}_{i=0}^{g−1} constitutes a limit 𝔤_{d}^{r} on X_{0} which is obtained from a refined limit 𝔤_{d}^{r} on X by retaining only the aspects of the elliptic components of X. The compatibility relations between the vanishing orders of the l_{E_{i}}’s imply the following inequality between Brill-Noether numbers:

(4) ρ(g, r, d, \overline{α}) ≥ ρ(l_{E_{0}}, p_{0}, p_{1})+ρ(l_{E_{1}}, p_{1}, p_{2})+⋯+ρ(l_{E_{g−2}}, p_{g−2}, p_{g−1})+ρ(l_{E_{g−1}}, p_{g−1}),

where ρ(l_{E_{i}}, p_{i}, p_{i+1}) := ρ(1, r, d) − w^{l_{E_{i}}} (p_{i}) − w^{l_{E_{i}}} (p_{i+1}). By assumption, there exists a non-zero section ρ_{η} ∈ H^{0}(X_{η}, ω_{X_{η}} ⊗ L_{η}^{⊗(−n)} ⊗ O_{X_{η}}(aσ(η))). This implies that if we denote by L̃_{i} the unique line bundle on the surface X such that (1) L̃_{i|X_{η}} = L_{η}, and (2)

deg_{Z}(ω_{X} ⊗L̃_{i}^{⊗(−n)} ⊗O_{X}(ap_{0})) = 0, for every component Z of X such that Z ≠ E_{i}, then H^{0}(E_{i}, ω_{X} ⊗ L̃_{i}^{⊗(−n)} ⊗ O_{X}(ap_{0}) ⊗ O_{E_{i}}) ≠ 0. We set

M_{i} := ω_{π} ⊗ L̃_{i}^{⊗(−n)} ⊗ O_{X} (a σ(B)) ∈ Pic(X).

Then M_{i|E_{i}} = O_{E_{i}}((a + 2i) · p_{i} + (2g − 2 − 2i) · p_{i+1} ⊗ L_{E_{i}}^{⊗(−n)}) for all 0 ≤ i ≤ g − 1. For each such i we denote by n_{i} the smallest integer such that ρ̃_{i} := t^{n_{i}}ρ_{η} ∈ π_{∗}(M_{i}) and we set

ρ_{i} := ρ̃_{i|E_{i}} ∈ H^{0}(E_{i}, M_{i|E_{i}}).

Thus 0 ≠ ρ_{i} ∈ H^{0}(E_{i}, O_{E_{i}}((a + 2i) · p_{i} + (2g − 2 − 2i) · p_{i+1} ⊗ L_{E_{i}}^{⊗(−n)})) and in a way similar to [EH1] Proposition 2.2, we can prove that

(5)  ord_{p_{i}}(ρ_{i}) + ordp_{i}(ρ_{i−1}) ≥ 2g − 2 − nd + a = deg(M_{i|E_{i}}).

One also has the inequalities ord_{p_{i}}(ρ_{i}) + ord_{p_{i+1}}(ρ_{i}) ≤ 2g − 2 − nd + a (and similar inequalities when passing through the rational components of X), from which it follows that one can write down a non-decreasing sequence of vanishing orders

(6)  0 ≤ ord_{p_{0}}(ρ_{0}) ≤ ord_{p_{1}}(ρ_{1}) ≤ … ≤ ord_{p_{i}}(ρ_{i}) ≤ … ≤ ord_{p_{g−1}}(ρ_{g−1}).

Since ρ_{g−1} is a non-zero section of a line bundle of degree 2g − 2 − nd + a on E_{g−1}, we must have that ord_{p_{g−1}}(ρ_{g−1}) ≤ 2g − 2 − nd + a. This inequality will eventually lead to the bound on the constant a.

Let us suppose now that we have fixed one of the elliptic components of X, say E_{i}, such that ρ(l_{E_{i}}, p_{i}, p_{i+1}) = 0. By counting dimensions, we see that for every 0 ≤ j ≤ r there exists a section u_{j} ∈ V_{E_{i}} such that div(u_{j}) ≥ a_{j}^{l_{E_{i}}} (p_{i}) · p_{i} + a_{r−j}^{l_{E_{i}}}(p_{i+1}) · p_{i+1}. In

particular, we have that a_{j}^{l_{E_{i}}} (p_{i}) + a_{r−j}^{l_{E_{i}}}(p_{i+1}) ≤ d. Since p_{i+1} − p_{i} ∈ Pic^{0}(E_{i}) is not a

torsion class, it follows that the equality a_{j}^{l_{E_{i}}} (p_{i}) + a_{r−j}^{l_{E_{i}}}(p_{i+1}) = d can hold for at most one value 0 ≤ j ≤ r. Because ρ(l_{E_{i}}, p_{i}, p_{i+1}) = 0, this implies that

a_{j}^{l_{E_{i}}} (p_{i}) + a_{r−j}^{l_{E_{i}}}(p_{i+1}) ≥ d − 1 for all 0 ≤ j ≤ r,

and there exists precisely one such index j such that a_{j}^{l_{E_{i}}} (p_{i}) + a_{r−j}^{l_{E_{i}}}(p_{i+1}) = d. In this

case we get that div(u_{j}) = a_{j}^{l_{E_{i}}} (p_{i}) · p_{i} + a_{r−j}^{l_{E_{i}}}(p_{i+1}) · p_{i+1}, and for degree reasons we

must have that L_{E_{i}} = O_{E_{i}}(a_{j}^{l_{E_{i}}} (p_{i}) · p_{i} + a_{r−j}^{l_{E_{i}}}(p_{i+1}) · p_{i+1}) ∈ Pic^{d}(E_{i}).

To summarize, if ρ(l_{E_{i}}, p_{i}, p_{i+1}) = 0, then the vanishing sequence a^{l_{E_{i+1}}} (p_{i+1}) of the E_{i+1}-aspect of the limit 𝔤_{d}^{r} on X, is obtained from the vanishing sequence a^{l_{E_{i}}} (p_{i}) by raising all entries by 1, except one single entry which remains unchanged. Thus,

a_{j}^{l_{E_{i}}} (p_{i}) = a_{j}^{l_{E_{i+1}}} (p_{i+1}) for one index 0 ≤ j ≤ r and a_{k}^{l_{E_{i+1}}} (p_{i+1}) = a_{k}^{l_{E_{i}}} (p_{i}) + 1 for k ≠ j.

We now study what happens to the non-decreasing sequence (6) as we pass through a component E_{i} with ρ(l_{E_{i}}, p_{i}, p_{i+1}) = 0. Assume that ord_{p_{i}}(ρ_{i}) = ord_{p_{i+1}}(ρ_{i+1}) := b. This implies that ord_{p_{i+1}}(ρ_{i}) = 2g − 2 − nd + a − b and

L_{E_{i}}^{⊗n} = O_{E_{i}}((a + 2i − b) · p_{i} + (nd − a + b − 2i) · p_{i+1}) ∈ Pic^{nd}(E_{i}).

Because ρ(l_{E_{i}}, p_{i}, p_{i+1}) = 0, as we have seen, L_{E_{i}} can be represented by an effective divisor which is supported only at p_{i} and p_{i+1}. Precisely, we can write that L_{E_{i}} = O_{E_{i}}(a_{j}^{l_{E_{i}}} (p_{i}) · p_{i} + a_{r−j}^{l_{E_{i}}}(p_{i+1}) · p_{i+1}) for a unique 0 ≤ j ≤ r. Since L_{E_{i}} cannot admit two different representations by effective divisors supported only at p_{i} and p_{i+1}, we must have that

(7)  L_{E_{i}} = O_{E_{i}}(\frac{a + 2i − b}{n} · p_{i} + \frac{nd − a + b − 2i}{n} · p_{i+1}).

In particular, we have that (a + 2i − b)/n ∈ ℤ and a_{j}^{l_{E_{i}}} (p_{i}) = (a + 2i − b)/n.

We consider a connected subcurve Y ⊂ X containing m + 1 elliptic components E_{i} and we measure the increase in (6) as we pass through the components of Y.

Lemma 1.1. We fix m := [(n + 1)/2] and integers i and b such that bm ≤ i ≤ g − 1. We denote by R(i) := #{0 ≤ l ≤ i − 1 : ρ(l_{E_{l}}, p_{l}, p_{l+1}) ≥ 1}. Then the following inequality holds:

ord_{p_{i}}(ρ_{i}) + R(i) ≥ b(m − 1).

Proof. We proceed by induction on b. For b = 0 there is nothing to prove. We set b ≥ 1, i := (b − 1)m and we assume that ord_{p_{i}}(ρ_{i}) + R(i) ≥ (b − 1)(m − 1). We are going to prove that the following inequality holds:

(8)  ord_{p_{i+m}}(ρ_{i+m}) − ord_{p_{i}}(ρ_{i}) + R(i + m) − R(i) ≥ m − 1.

Assume this is not the case. Then there exist integers 0 ≤ l < j ≤ m − 1 such that the following relations hold: (i) ρ(l_{E_{i+l}}, p_{i+l}, p_{i+l+1}) = ρ(l_{E_{i+j}}, p_{i+j}, p_{i+j+1}) = 0 and

(ii) ord_{p_{i+l}}(ρ_{i+l}) = ord_{p_{i+l+1}}(ρ_{i+l+1}) := b, ord_{p_{i+j}}(ρ_{i+j}) = ord_{p_{i+j+1}}(ρ_{i+j+1}) := c.

Using (7) this implies that

L_{E_{i+l}} = O_{E_{i+l}}(\frac{a + 2i + 2l − b}{n} · p_{i+l} + \frac{nd − a + b − 2i − 2l}{n} · p_{i+l+1}), and

L_{E_{i+j}} = O_{E_{i+j}}(\frac{a + 2i + 2j − c}{n} · p_{i+j} + \frac{nd − a + c − 2i − 2j}{n} · p_{i+j+1}).

In particular, (2j − 2l − c + b)/n ∈ ℤ, hence we can write c = b − kn + 2(j − l) for some k ∈ ℤ. If k ≥ 1, since c ≥ b, we obtain that m − 1 ≥ j − l ≥ n/2, which is a contradiction. Therefore we must have that k ≤ 0, and this holds for every pair (j, l) satisfying (i) and (ii). We choose now the pair 0 ≤ l < j ≤ m − 1 satisfying (i) and (ii) and for which moreover, the difference j − l is maximal.

For each integer 0 ≤ e ≤ l − 1 we have that either ρ(l_{E_{i+e}}, p_{i+e}, p_{i+e+1}) ≥ 1 or ord_{p_{i+e+1}}(ρ_{i+e+1}) > ord_{p_{i+e}}(ρ_{i+e}). This fact leads to the inequality

(9)  ord_{p_{i+l}}(ρ_{i+l}) − ord_{p_{i}}(ρ_{i}) + R(i + l) − R(i) ≥ l.

Similarly, by studying the subcurve of Y containing E_{i+j+1}, …, E_{i+m−1}, we find that

(10) ord_{p_{i+m}}(ρ_{i+m}) − ord_{p_{i+j+1}}(ρ_{i+j+1}) + R(i + m) − R(i + j + 1) ≥ m − j − 1.

Finally, we look at the subcurve of X containing E_{i+l}, …, E_{i+j} and we can write

(11) ord_{p_{i+j}}(ρ_{i+j}) − ord_{p_{i+l}}(ρ_{i+l}) + R(i + j + 1) − R(i + l) ≥ c − b ≥ 2(j − l) ≥ j − l + 1.

By adding (9), (10) and (11) together we obtain (8) which proves the Lemma. □

When n = 2 we have a slightly better estimate than in the general case:

Lemma 1.2. (n = 2) (1) Let i be an integer such that 2b ≤ i ≤ g−1. Then ord_{p_{i}}(ρ_{i})+R(i) ≥ b.

(2) We fix 0 ≤ i ≤ g − 4 and let Y be a connected subcurve of X containing precisely three elliptic curves E_{i}, E_{i+1} and E_{i+2}. If R(i + 3) = R(i), that is,

ρ(l_{E_{i}}, p_{i}, p_{i+1}) = ρ(l_{E_{i+1}}, p_{i+1}, p_{i+2}) = ρ(l_{E_{i+2}}, p_{i+2}, p_{i+3}) = 0,

then we have the inequality ord_{p_{i+3}}(ρ_{i+3}) ≥ ord_{p_{i}}(ρ_{i}) + 2.

Proof. We only prove (2), the remaining statement being analogous to Lemma 1.1. We may assume that ord_{p_{i}}(ρ_{i}) = ord_{p_{i+1}}(ρ_{i+1}) := b. Hence (a + 2i − b)/2 ∈ ℤ and there exists an index 0 ≤ j ≤ r such that

a_{j}^{l_{E_{i}}} (p_{i}) = a_{j}^{l_{E_{i+1}}} (p_{i+1}) = \frac{a + 2i − b}{2}, while a_{k}^{l_{E_{i+1}}} (p_{i+1}) = a_{k}^{l_{E_{i}}} (p_{i}) + 1 for k ≠ j.

If ord_{p_{i+2}}(ρ_{i+2}) = ord_{p_{i+1}}(ρ_{i+1}) = b, then (7) implies that (a + 2i + 2 − b)/2 is an entry in the vanishing sequence a^{l_{E_{i+1}}} (p_{i+1}). But this is impossible, because (a + 2i − b)/2 was an entry in the sequence a^{l_{E_{i}}} (p_{i}), hence we must have that ord_{p_{i+2}}(ρ_{i+2}) ≥ b + 1. Next, if ord_{p_{i+3}}(ρ_{i+3}) = b + 1, this implies that ord_{p_{i+3}}(ρ_{i+3}) = ord_{p_{i+2}}(ρ_{i+2}) = b + 1, hence again (a + 2(i + 2) − (b + 1))/2 ∈ ℤ, which is not possible for parity reasons. Thus we must have that ord_{p_{i+3}}(ρ_{i+3}) ≥ b + 2. □

Proof of Theorem 0.8. We complete the proof of our result in the case n ≥ 3. We write g = bm + c with 0 ≤ c ≤ m − 1 and we set i := bm. From Lemma 1.1 we obtain that ord_{p_{i}}(ρ_{i}) + R(i) ≥ b(m − 1). Using the reasoning of Lemma 1.1 for the connected subcurve of X which contains E_{i}, E_{i+1}, …, E_{i+c−1} = E_{g−1}, we get that

(12)  ord_{p_{g−1}}(ρ_{g−1}) − ord_{p_{i}}(ρ_{i}) + R(g − 1) − R(i) ≥ c − 2.

Using (12), together with the inequality R(g − 1) ≤ ρ(g, r, d, \overline{α}), we can write that

deg(K_{C} ⊗L^{⊗(−n)}⊗O_{C}(ap)) = 2g−2−nd+a ≥ ord_{p_{g−1}}(ρ_{g−1}) ≥ g−[\frac{g}{m}]−ρ(g, r, d, \overline{α})−2,

which finishes the proof of Theorem 0.8. □

Proof of Theorem 0.9. From Lemma 1.2 part (1), we obtain that

ord_{p_{g−1}}(ρ_{g−1}) + R(g − 1) ≥ [(g − 1)/2].

Since R(g − 1) ≤ ρ(g, r, d, \overline{α}), this leads to the inequality a ≥ 2d + 2 − 2g + [(g − 1)/2] − ρ(g, r, d, \overline{α}). To prove (3) we divide X into e := [g/3] + 1 connected subcurves Y_{1}, …, Y_{e} such that Y_{1}, …, Y_{e−1} each contain three elliptic components, #(Y_{i} ∩ Y_{i+1}) = 1 for all 1 ≤ i ≤ e − 2 and Y_{e} := \overline{(∪_{i=1}^{e−1}Y_{i})^{c}}. The curves Y_{i} fall into two categories: those for which there exists an elliptic component E_{l} ⊂ Y_{i} such that ρ(l_{E_{l}}, p_{l}, p_{l+1}) ≥ 1 (and there are at most ρ(g, r, d, \overline{α}) such Y_{i}’s), and those for which ρ(l_{E_{l}}, p_{l}, p_{l+1}) = 0 for each elliptic component E_{l} ⊂ Y_{i}. Lemma 1.2 part (2) gives that ord_{p_{g−1}}(ρ_{g−1}) ≥ 2([g/3]−ρ(g, r, d, \overline{α})). This proves (2) and finishes the proof of Theorem 0.9. □

Remark 1.3. It is natural to ask how close to being optimal are the bounds we obtained above. For ρ(g, r, d, \overline{α}) relatively small, when any L ∈ G_{d}^{r}(C, p, \overline{α}) has a strong geometric characterization, the inequalities (1), (2) and (3) are in fact optimal. To see an example, we set g = 3, r = 3, d = 6 and ρ(g, r, d, \overline{α}) = 0. Thus we look at 𝔤_{6}^{3}’s on a general [C, p] ∈ M_{3,1} having ramification at p equal to (0 ≤ α_{0} ≤ α_{1} ≤ α_{2} ≤ α_{3} ≤ 3), where ∑_{i=0}^{3} α_{i} = 3. Theorem 0.9 gives us that H^{0}(K_{C} ⊗L^{⊗(−2)}⊗O_{C}(a·p)) = 0 for every integer a ≤ 9. We show that this is optimal by noting that when a = 10 and \overline{α} = (0, 0, 1, 2), we have that

H^{0}(K_{C} ⊗ L^{⊗(−2)} ⊗ O_{C}(10p)) ≠ 0, for every L ∈ G_{6}^{3}(C, p, \overline{α}).

Indeed, any such linear series is of the form L = K_{C} ⊗ A^{∨} ⊗ O_{C}(5p) ∈ W_{6}^{3}(C), where A ∈ W_{3}^{1}(C) is such that h^{0}(A(−2p)) ≥ 1. A non-hyperelliptic curve of genus 3 has two

such 𝔤_{3}^{1}’s. Precisely, if z, t ∈ C are the two points the tangent line at p to C ↪^{|K_{C}|} P^{2} meets C again, then A = O_{C}(2p + z) or A = O_{C}(2p + t). Say, we choose A = O_{C}(2p + z). By direct calculation we obtain that L^{⊗2} ⊗ O_{C}(−10p) = K_{C}^{⊗2} ⊗ A^{⊗(−2)} = O_{C}(2t), hence h^{0}(K_{C} ⊗ L^{⊗(−2)} ⊗ O_{C}(10p)) = 1.

2. Varieties of secant planes to the general curve

We fix a smooth curve [C] ∈ M_{g} and two integers 0 ≤ f < e. In this section we study the varieties V_{e}^{e−f}(l) of e-secant (e−f −1)-planes corresponding to a linear series l ∈ G_{d}^{r}(C). We first define the correspondence

Σ_{C} := {(D, l) ∈ C_{e} × G_{d}^{r}(C) : dim l(−D) ≥ r − e + f},

and denote by π_{1} : Σ_{C} → C_{e} and π_{2} : Σ_{C} → G_{d}^{r}(C) the two projections. We assume that Σ_{C} ≠ ∅ for the general curve [C] ∈ M_{g}. Under this assumption, we show that

(13)  dim(Σ_{C}) ≤ ρ(g, r, d) − f(r + 1 − e + f) + e.

(We recall that the dimension of a scheme is the maximum of the dimensions of its irreducible components). Since Σ_{C} is a determinantal subvariety of C_{e} × G_{d}^{r}(C), it follows that for a general [C] ∈ M_{g}, if non-empty, the scheme Σ_{C} is equidimensional and dim(Σ_{C}) = ρ(g, r, d) − f(r + 1 − e + f) + e. Note that this result does not establish the non-emptiness of Σ_{C} which is an issue that we will deal with in Section 3. In any event,

(13) implies the dimensional estimate

dim{l ∈ G_{d}^{r}(C) : V_{e}^{e−f}(l) ≠ ∅} ≤ ρ(g, r, d) − f(r + 1 − e + f) + e.

This will prove Theorem 0.1 as well as Corollaries 0.3 and 0.4.

We start by setting some notation. We denote by j : \overline{M}_{0,g} → \overline{M}_{g} the “flag” map obtaining by attaching to each stable curve [R, x_{1}, …, x_{g}] ∈ \overline{M}_{0,g} fixed elliptic tails E_{1}, …, E_{g} at the points x_{1}, …, x_{g} respectively. Thus j([R, x_{1}, …, x_{g}]) := [R̃] = [R ∪_{x_{1}} E_{1} ∪ … ∪_{x_{g}} E_{g}] and for such a curve, we denote by p_{R} : R̃ → R the projection onto R, that is, p_{R}(E_{i}) = \{x_{i}\} for 1 ≤ i ≤ g. We denote by \overline{C}_{g,n} = \overline{M}_{g,n+1} the universal curve and by π : \overline{C}_{g,n} → \overline{M}_{g,n} the morphism forgetting the (n + 1)-st marked point. We write π_{e} : \overline{C}_{g,n}^{e} → \overline{M}_{g,n} for the e-fold fibre product of \overline{C}_{g,n} over \overline{M}_{g,n} and we introduce a map χ : \overline{M}_{0,g} ×_{\overline{M}_{g}} \overline{C}_{g}^{e} → \overline{C}_{0,g}^{e} which collapses the elliptic tails. Thus χ is defined by

χ([R, x_{1}, …, x_{g}], (y_{1}, …, y_{e})) := ([R, x_{1}, …, x_{g}], p_{R}(y_{1}), …, p_{R}(y_{e})),

for points y_{1}, …, y_{e} ∈ R̃. Let W ⊂ \overline{C}_{g}^{e} be the closure of the locus

{[C, y_{1}, …, y_{e}] ∈ C_{g}^{e} : ∃l ∈ G_{d}^{r}(C) with dim l(−y_{1} − ⋯ − y_{e}) ≥ r − e + f}.

By assumption π_{e}(W) = \overline{M}_{g} and we define the locus U := χ(W ∩ (\overline{M}_{0,g} ×_{\overline{M}_{g}} \overline{C}_{g}^{e})). Then π_{e}(U) = \overline{M}_{0,g} and we denote by e − m the minimal fibre dimension of the map π_{e|U} : U → \overline{M}_{0,g}. Thus 0 ≤ m ≤ e and dim(U ∩ π_{e}^{−1}[R, x_{1}, …, x_{g}]) ≥ e − m, for every [R, x_{1}, …, x_{g}], with equality for a general point [R, x_{1}, …, x_{g}] ∈ \overline{M}_{0,g}.

We recall that for every choice of 4 marked points {i, j, k, l} ⊂ {1, …, g}, one has a fibration π_{ijkl} : \overline{M}_{0,g} → \overline{M}_{0,4} obtained by forgetting the marked points with labels in the set {i, j, k, l}^{c} and stabilizing the resulting rational curve. If we single out the first 3 marked points x_{1}, x_{2}, x_{3} as being 0, 1 and ∞, in this way we obtain a birational map π_{123} = (π_{1234}, …, π_{123i}, …, π_{123g}) : \overline{M}_{0,g} → \overline{M}_{0,4}^{g−3} = (P^{1})^{g−3} defined by

π_{123}([R, x_{1}, …, x_{g}]) := ([R, x_{1}, x_{2}, x_{3}, x_{4}], [R, x_{1}, x_{2}, x_{3}, x_{5}], …, [R, x_{1}, x_{2}, x_{3}, x_{g}]).

The map π_{123} expresses \overline{M}_{0,g} as a blow-up of (P^{1})^{g−3} such that all exceptional divisors of π_{123} are boundary divisors of \overline{M}_{0,g} (cf. [K]). In a similar manner, one has a birational map f : \overline{C}_{0,g}^{e} → \overline{M}_{0,4}^{g−3+e} = (P^{1})^{g−3+e} defined by f([R, x_{1}, …, x_{g}], y_{1}, …, y_{e}) :=

:= ([R, x_{1}, x_{2}, x_{3}, x_{4}], …, [R, x_{1}, x_{2}, x_{3}, x_{g}], [R, x_{1}, x_{2}, x_{3}, y_{1}], …, [R, x_{1}, x_{2}, x_{3}, y_{e}]).

For simplicity, sometimes we write f([R, x_{1}, …, x_{g}], y_{1}, …, y_{e}) = (x_{4}, …, x_{g}, y_{1}, …, y_{e}). The maps f and π_{123} fit in a commutative diagram, where p_{1} : (P^{1})^{g−3+e} → (P^{1})^{g−3} is the projection on the first g − 3 factors:

\overline{C}_{0,g}^{e} ⟶^{f} (P^{1})^{g−3+e} = \overline{M}_{0,4}^{g−3+e}
↓^{π_{e}}  ↓^{p_{1}}
\overline{M}_{0,g} ⟶^{π_{123}} (P^{1})^{g−3} = \overline{M}_{0,4}^{g−3}

Finally, for 2 ≤ k ≤ e we define the diagonal loci Δ_{k} ⊂ (P^{1})^{g−3+e} as consisting of those points (x_{4}, …, x_{g}, y_{1}, …, y_{e}) for which at least k of the points y_{1}, …, y_{e} coincide. We need the following result concerning existence of sublinear limit linear series of a fixed limit 𝔤_{d}^{r}, having prescribed vanishing sequence at a given point:

Lemma 2.1. Let X be a curve of compact type, Y ⊂ X an irreducible component and let p ∈ Y be a smooth point of X. Assume that l is a (refined) limit 𝔤_{d}^{r} on X and let (a_{0} < a_{1} < … < a_{r}) be the vanishing sequence a^{l}(p). We fix a subsequence (a_{j_{0}} < a_{j_{1}} < … < a_{j_{b}}) of a^{l}(p), where 0 ≤ b ≤ r. Then there exists a limit 𝔤_{d}^{b} on X, say l′ ⊂ l, such that a^{l′}(p) = (a_{j_{0}}, …, a_{j_{b}}).

Proof. Let us denote by l := {l_{Z} = (L_{Z}, V_{Z})}_{Z⊂X} the original limit 𝔤_{d}^{r} on X. For each integer 0 ≤ k ≤ b there exists a section σ_{j_{k}} ∈ V_{Y} such that ord_{p}(σ_{j_{k}}) = a_{j_{k}}. We consider the subspace W_{Y} :=< σ_{j_{0}}, …, σ_{j_{b}} >⊂ V_{Y}. Since #{ord_{p}(σ)}_{σ∈W_{Y}} = b + 1, we obtain that dim(W_{Y}) = b + 1 and we set l′_{Y} := (L_{Y} , W_{Y}) ∈ G_{d}^{b}(Y). Suppose now that Z is a component of X meeting Y in a point q. We denote by (c_{j_{0}} < c_{j_{1}} < … < c_{j_{b}}) the vanishing sequence a^{l′_{Y}} (q). Let (e_{j_{0}} < e_{j_{1}} < … < e_{j_{b}}) be the complementary sequence, that is, e_{j_{k}} = d−c_{j_{b−k}} for each 0 ≤ k ≤ b. Then we can choose a section τ_{k} ∈ V_{Z} such that ord_{q}(τ_{k}) = e_{j_{k}}. We define W_{Z} :=< τ_{0}, …, τ_{b} >⊂ V_{Z}. Because all the entries (e_{j_{k}})_{k=0}^{b} are distinct, we get that dim(W_{Z}) = b + 1 and then set l′_{Z} := (L_{Z}, W_{Z}) ∈ G_{d}^{b}(Z). We continue inductively, and for each irreducible component Z′ ⊂ X we obtain an aspect l′_{Z′} = (L_{Z′}, W_{Z′}) ∈ G_{d}^{b}(Z′). The collection {l′_{Z}}_{Z⊂X} is the desired limit 𝔤_{d}^{b} on X. □

Next we explain how the assumption that for every [C] ∈ M_{g} there exists a linear series l ∈ G_{d}^{r}(C) with V_{e}^{e−f}(l) ≠ ∅, can be used to construct a flag curve R̃ ∈ j(\overline{M}_{0,g}) such that all the e points coming from the limit of an effective divisor D ∈ V_{e}^{e−f}(l) specialize to a connected subcurve of R̃ having arithmetic genus at most min{g, e}.

Proposition 2.2. Let U ⊂ \overline{C}_{0,g}^{e} be an irreducible component of the closure of the locus of limits of e-secant divisors with respect to linear series 𝔤_{d}^{r} on flag curves from \overline{M}_{g}. Assuming that dim(U) = g − 3 + e − m with 0 ≤ m ≤ e, there exists a point ([R, x_{1}, …, x_{g}], ỹ_{1}, …, ỹ_{e}) ∈ W ∩ (\overline{M}_{0,g} ×_{\overline{M}_{g}} \overline{C}_{g}^{e}) corresponding to a genus g flag curve

R̃ = R ∪_{x_{1}} E_{1} ∪ … ∪_{x_{g}} E_{g} and pointsỹ_{1}, …, ỹ_{e} ∈ R̃,

such that either (i) ỹ_{1} = ⋯ = ỹ_{e} ∈ R − {x_{1}, …, x_{g}}, or else, (ii) all the points ỹ_{1}, …, ỹ_{e} lie on a connected subcurve Y ⊂ R̃ satisfying p_{a}(Y) ≤ min\{m, g\} and #(Y ∩ (\overline{R̃ − Y})) ≤ 1.

Proof. We start by noting that if m = 0 then U = \overline{C}_{0,g}^{e} and possibility (i) is satisfied. Thus we may assume that m ≥ 1. First, we claim that dim f(U) = dim U = g − 3 + e − m. Indeed, since π_{e}(U) = \overline{M}_{0,g} it follows that p_{1}(f(U)) = (P^{1})^{g−3} and we choose a general point t = (x_{4}, …, x_{g}) ∈ (P^{1} − {0, 1, ∞})^{g−3}, such that x_{i} ≠ x_{j} for i ≠ j. Then π_{e}^{−1}(t) = (P^{1})^{e} and f_{|π_{e}^{−1}(t)} is an isomorphism onto its image, hence f_{|U} is birational onto its image as well. Obviously, when m ≥ g we can take Y = R̃. From now on we shall assume that 1 ≤ m ≤ g − 1.

Let us assume first that f(U) ∩ Δ_{e} ≠ ∅. Then dim(f(U) ∩ Δ_{e}) ≥ g − m − 2. For

dimension reasons, there must exist a point z = (x_{4}, …, x_{g}, y_{1}, …, y_{1}) ∈ f(U) ∩ Δ_{e} such that either (i) at least g − m − 3 of the points x_{j} with 4 ≤ j ≤ g are mutually distinct and belong to the set P^{1} − {0, 1, ∞, y_{1}} and y_{1} ∈ P^{1} − {0, 1, ∞}, or (ii) at least g − m − 2 of the x_{j}’s (4 ≤ j ≤ g) are mutually distinct and belong to the set P^{1} − {0, 1, ∞, y_{1}} and then y_{1} ∈ P^{1} may, or may not be equal to one of the points 0, 1 or ∞. Suppose we are in situation (i), the remaining case being similar. We fix a point ([R, x_{1}, …, x_{g}], y_{1}, …, y_{e}) ∈ f^{−1}(z), hence y_{1}, …, y_{e} ∈ R. If Z ⊂ R denotes the minimal connected subcurve of R containing all the points y_{1}, …, y_{e}, then x_{1}, x_{2}, x_{3} ∈ R − Z, unless y_{1} = ⋯ = y_{e}. (In the latter case either y_{1} ∈ R − {x_{1}, …, x_{g}} which corresponds to the situation when all the points ỹ_{i} = y_{i} specialize to the same smooth point of R̃ lying on the rational spine, or else, if y_{1} = x_{j} for some 4 ≤ j ≤ g, then we can find a connected subcurve of R̃ of genus 1 containing ỹ_{1}, …, ỹ_{e}, where p_{R}(ỹ_{i}) = y_{i} for 1 ≤ i ≤ e). Since at least g −m = 3+(g −m−3) of the points x_{1}, …, x_{g} lie on Z^{c}, it follows that ỹ_{1}, …, ỹ_{e} lie on a connected subcurve of R̃ of genus ≤ m, which completes the proof in this case.

We are left with the possibility f(U) ∩ Δ_{e} = ∅ and we denote by k ≤ e − 1 the largest integer for which f(U) ∩ Δ_{k} ≠ ∅ and by L an irreducible component of f(U) ∩ Δ_{k}. Since by definition f(U) ∩ Δ_{k+1} = ∅, it follows that there exists a point t_{0} = (p_{1}, …, p_{e}) ∈ (P^{1})^{e} such that L ⊂ (P^{1})^{g−3} × {t_{0}}. In particular, the projection map p_{1|L} : L → p_{1}(L) is 1 : 1 and then dim p_{1}(L) = dim(L) ≥ g − m + (e − k − 2) ≥ g − m, unless k = e − 1, when dim p_{1}(L) ≥ g − m − 1. In the first case it follows that there exists a point (x_{4}, …, x_{g}, p_{1}, …, p_{e}) ∈ f(U) ∩ Δ_{k} such that at least g − m of the points x_{4}, …, x_{g} are equal to a fixed point r ∈ P^{1} − {p_{1}, …, p_{e}}. In the second case, that is, when k = e − 1, since #{p_{i}}_{i=1}^{e} = 2, one of the points 0, 1 or ∞, say 0, does not appear among the p_{i}’s. Then we can find a point (x_{4}, …, x_{g}, p_{1}, …, p_{e}) ∈ f(U) ∩ Δ_{e−1} with at least g − m of the x_{j}’s equal to 0.

The conclusion in both cases is that there exists a point ([R, x_{1}, …, x_{g}], y_{1}, …, y_{e}) ∈ W ∩ (\overline{M}_{0,g} ×_{\overline{M}_{g}} \overline{C}_{g}^{e}) corresponding to the flag curve R̃ = R ∪_{x_{1}} E_{1} ∪ … ∪_{x_{g}} E_{g}, such that

the points y_{1}, …, y_{e} lie on a connected subcurve Y ⊂ R̃ where #(Y ∩ (\overline{R̃ − Y})) ≤ 1 and p_{a}(Y) ≤ m ≤ e. □

Proof of Theorem 0.1. We choose R̃ = R ∪_{x_{1}} E_{1} ∪ … ∪_{x_{g}} E_{g} as above and denote by Y ⊂ R̃ a connected subcurve onto which the points y_{1}, …, y_{e} specialize. We know that either (a) p_{a}(Y) = m ≤ min{e, g}, or (b) y_{1} = ⋯ = y_{e} ∈ R − {x_{1}, …, x_{g}}.

We first deal with case (a) and dispose of (b) at the end using [EH2]. If m < g we set Z := \overline{R̃ − Y} and \{p\} := Y ∩ Z and we denote by Y′ and Z′ the components of Y and Z respectively, containing the point p. When m = g, then necessarily e ≥ g and Y := R̃, Z = ∅ and p ∈ R̃ is a general (smooth) point. By assumption, [R̃, y_{1}, …, y_{e}] ∈ W, hence there exists a proper flat morphism ϕ : X → B satisfying the following properties:

• X is a smooth surface, B is a smooth affine curve, 0 ∈ B is a point such that ϕ^{−1}(0) is a curve stably equivalent to R̃ and X_{t} = ϕ^{−1}(t) is a smooth projective curve of genus g for t ≠ 0. Moreover, there are e sections σ_{i} : B → X of ϕ satisfying the condition σ_{i}(0) = y_{i} ∈ ϕ^{−1}(0)_{reg} for all 1 ≤ i ≤ e.

• If X_{η} := X − ϕ^{−1}(0), then there exists a line bundle L_{η} ∈ Pic(X_{η}) of relative degree d and a subvector bundle V_{η} ⊂ ϕ_{∗}(L_{η}) having rank r + 1, such that for t ≠ 0 we have that

dim V_{t} ∩ H^{0}(X_{t}, L_{t}(− ∑_{j=1}^{e} σ_{j}(t))) = r + 1 − e + f.

After possibly making a finite base change and resolving the resulting singularities, the pair (L_{η}, V_{η}) induces a (refined) limit 𝔤_{d}^{r} on R̃, which we denote by 𝔩. The vector bundle V_{η} ∩ ϕ_{∗}(L_{η} ⊗ O_{X_{η}}(− ∑_{j=1}^{e} σ_{j}(B − {0}))) induces a limit linear series 𝔤_{d−e}^{r−e+f} on ϕ^{−1}(0) which we denote by 𝔪. For a component A of ϕ^{−1}(0), if (L_{A}, V_{A}) ∈ G_{d}^{r}(A) denotes the A-aspect of 𝔩, then there exists a unique effective divisor D_{A} ∈ A_{e} supported only at the points from (A ∩ ⋃_{j=1}^{e} σ_{j}(B)) ⋃(A ∩ \overline{ϕ^{−1}(0) − A}) such that the A-aspect of 𝔪 is of the form

𝔪_{A} = (M_{A} := L_{A} ⊗ O_{A}(−D_{A}), W_{A} ⊂ V_{A} ∩ H^{0}(M_{A})) ∈ G_{d−e}^{r−e+f}(A).

The collection 𝔪_{Y} := {𝔪_{A}}_{A⊂Y} forms a limit 𝔤_{d−e}^{r−e+f} on Y. We denote by (a_{0} < … < a_{r}) the vanishing sequence of 𝔩_{Y′} at p, thus {a_{i}}_{i=0}^{r} = {ord_{p}(σ)}_{σ∈V_{Y′}} and we denote by (b_{0} < … < b_{r}) the vanishing sequence a^{𝔩_{Z′}} (p). By ordering the set {ord_{p}(σ)}_{σ∈W_{Y′}} we obtain a subsequence (a_{i_{0}} < … < a_{i_{r−e+f}}) of a^{𝔩_{Y′}} (p). When we order the entries in {a_{i}}_{i=0}^{r} − {a_{i_{k}}}_{k=0}^{r−e+f} we obtain a new sequence (a_{j_{0}} < a_{j_{1}} < … < a_{j_{e−f−1}}). Using Lemma 2.1, we find that there exists a limit linear series 𝔩′_{Y} of type 𝔤_{d}^{e−f−1} on Y with the property that a^{𝔩′_{Y}} (p) = (a_{j_{0}}, a_{j_{1}}, …, a_{j_{e−f−1}}).

Let us assume first that we are in the situation m < g, hence Z ≠ ∅. The point p ∈ Y lies on a rational component which implies the following inequality corresponding to Y (see also [EH2], Theorem 1.1):

(14)  V_{1} := ρ(m, e − f − 1, d) − ∑_{k=0}^{e−f−1} a_{j_{k}} + \binom{e − f}{2} ≥ 0.

Applying the same principle for the limit linear series 𝔪_{Y} on Y, we find that the adjusted Brill-Noether number with respect to the point p is non-negative:

(15)  V_{2} := ρ(m, r − e + f, d − e) − ∑_{k=0}^{r−e+f} a_{i_{k}} + \binom{r + 1 − e + f}{2} ≥ 0.

Next we turn our attention to Z and use the fact that the point p ∈ Z does not lie on an elliptic component, hence [Z, p] satisfies the ”strong” pointed Brill-Noether theorem:

(16)  V_{3} := ρ(g − m, r, d) − ∑_{k=0}^{r} b_{k} + \binom{r + 1}{2} ≥ 0.

If we add (14), (15) and (16) together and use that ∑_{k=0}^{r} b_{k} +∑_{k=0}^{r−e+f} a_{i_{k}} +∑_{k=0}^{e−f−1} a_{j_{k}} = (r + 1)d, we obtain the inequality

ρ(g, r, d) − f(r + 1 − e + f) + e ≥ e − m ≥ 0.

The case m = g, when Y = R̃, is similar but simpler. We add together (14) and (15) (now there is no (16)) and we write the following inequalities:

ρ(g, r, d) + e − f(r + 1 − e + f) = (ρ(g, r − e + f, d − e) − ∑_{k=0}^{r−e+f} a_{i_{k}} + \binom{r + 1 − e + f}{2})+

+(ρ(g, e−f−1, d)− ∑_{k=0}^{e−f−1} a_{j_{k}}+\binom{e − f}{2})+ ∑_{k=0}^{r−e+f} a_{i_{k}}+ ∑_{k=0}^{e−f−1} a_{j_{k}}−\binom{r + 1}{2}+e−g ≥ e−g ≥ 0,

since ∑_{k=0}^{r−e+f} a_{i_{k}} + ∑_{k=0}^{e−f−1} a_{j_{k}} ≥ \binom{r+1}{2}. Thus we obtain the same numerical conclusion as in the case m < g.

Assume now that we are in the case (b) when y_{1} = ⋯ = y_{e} ∈ R − {x_{1}, …, x_{g}}. Then reasoning as above, we find a limit 𝔤_{d}^{r} on R̃ having vanishing sequence at y_{1} at least (0, 1, …, e − f − 1, e, e + 1, …, r + f − 1, r + f). Using once more [EH2], Theorem 1.1, we obtain the inequality

ρ(g, r, d) + e − f(r + 1 − e + f) ≥ ρ(g, r, d) − f(r + 1 − e + f) ≥ 0.

Using the semicontinuity of the dimension of the fibres, it follows that for a general curve [C] ∈ M_{g}, if π_{1} : Σ_{C} → C_{e} is the first projection, then the minimal fibre dimension of π_{1} cannot exceed the dimension of the space of pairs of limit linear series 𝔩 ⊃ 𝔪 consisting of a 𝔤_{d}^{r} ⊃ 𝔤_{d−e}^{r−e+f} on the flag curve ϕ^{−1}(0) such that 𝔪 = 𝔩(−D_{e}), where D_{e} is a degree e effective divisor on ϕ^{−1}(0) with the property that supp(D_{e}) ⊂ Y ∩ ϕ^{−1}(0)_{reg}. Since the map (𝔩 ⊃ 𝔪, 𝔪_{Y} , 𝔩′_{Y}) ↦ (𝔪_{Y} , 𝔩′_{Y} , 𝔩_{Z}) ∈ G̃_{d−e}^{r−e+f}(Y) × G̃_{d}^{e−f−1}(Y) × G̃_{d}^{r}(Z) is injective, it follows that for a general divisor D_{gen} ∈ π_{1}(Σ_{C}) we have the estimate

dim π_{1}^{−1}(D_{gen}) ≤ V_{1} + V_{2} + V_{3} = ρ(g, r, d) − f(r + 1 − e + f) + m,

hence dim(Σ_{C}) = dim π_{1}^{−1}(D_{gen}) + e − m ≤ ρ(g, r, d) − f(r + 1 − e + f) + e. This finishes the proof of Theorem 0.1. □

3. Existence of linear series with secant planes

We turn our attention to showing existence of linear series which possess e-secant (e−f −1)-planes. The strategy we pursue is to construct limit linear series 𝔤_{d}^{r} on a curve of compact type [Y ∪_{p} Z] ∈ \overline{M}_{g}, where (Y, p) and (Z, p) are suitably general smooth pointed curves of genus e and g − e respectively. These limit 𝔤_{d}^{r}’s will carry a sublinear series 𝔤_{d−e}^{r−e+f} = 𝔤_{d}^{r}(−D_{e}), where D_{e} is a degree e effective divisor on Y. Like in the proof of Theorem 0.1, such 𝔤_{d}^{r}’s are determined by their Z-aspect and by a pair of linear series (𝔤_{d−e}^{r−e+f}, 𝔤_{d}^{e−f−1}) on Y. We determine the dimension of the space of such pairs, which will enable us to show that the original pair (𝔤_{d−e}^{r−e+f}, 𝔤_{d}^{e−f−1}) on Y ∪_{p} Z can be smoothed to every smooth curve of genus g. This will finish the proof of Theorem 0.5.

We start by choosing two general pointed curves [Y, p] ∈ M_{e,1} and [Z, p] ∈ M_{g−e,1} such that both (Y, p) and (Z, p) satisfy the Brill-Noether theorem with prescribed ramification (cf. [EH2], Theorem 1.1 and Proposition 1.2): If \overline{α} : 0 ≤ α_{0} ≤ … ≤ α_{r} ≤ d − r is a Schubert index of type (r, d), then (Y, p) possesses a 𝔤_{d}^{r} with ramification sequence ≥ \overline{α} at the point p, if and only if

(17)  ∑_{i=0}^{r} max{α_{i} + g(Y) − d + r, 0} ≤ g(Y).

In case this inequality is satisfied, then dim G_{d}^{r}(Y, p, \overline{α}) = ρ(g, r, d, \overline{α}) (One obviously has a similar statement for [Z, p]).

We denote by π : X → (T, 0) the versal deformation space of the stable curve π^{−1}(0) = X_{0} := Y ∪_{p} Z. Let Δ ⊂ T be the boundary divisor corresponding to singular curves, and we write π^{−1}(Δ) = Δ_{e} + Δ_{g−e}, where Δ_{e} (resp. Δ_{g−e}) is the divisor corresponding to the marked point lying on the component of genus e (resp. g − e). We consider the e-fold fibre product U := (X − Δ_{g−e}) ×_{T} ⋯ ×_{T} (X − Δ_{g−e}), the projection ϕ : U → T and the induced curve p_{2} : X ×_{T} U → U. Then we introduce the stack of limit linear series of type 𝔤_{d}^{r} over U

σ : 𝔊̃_{d}^{r}(X ×_{T} U/U) → U, where 𝔊̃_{d}^{r}(p_{2}) = 𝔊̃_{d}^{r}(X ×_{T} U/U) = 𝔊̃_{d}^{r}(π) ×_{T} U,

and we write τ := ϕ ∘ σ : 𝔊̃_{d}^{r}(p_{2}) → T (see [EH1] Theorem 3.4, for details on the construction of 𝔊̃_{d}^{r}(π)). The fibre τ^{−1}(t) corresponding to a point t ∈ Δ (in which case one can write π^{−1}(t) = Y_{t} ∪ Z_{t}, with g(Y_{t}) = e, g(Z_{t}) = g − e), parameterizes limit 𝔤_{d}^{r}’s on Y_{t} ∪ Z_{t} together with e-tuples (x_{1}, …, x_{e}) ∈ (Y_{t} − Y_{t} ∩ Z_{t})^{e}. Let us denote by L_{Y} a degree d Poincaré bundle on π_{2} : X ×_{T} 𝔊̃_{d}^{r}(p_{2}) → 𝔊̃_{d}^{r}(p_{2}) characterized by the property that its restriction to curves of type Y_{t} ∪ Z_{t} are line bundles of bidegree (d, 0). We also write V_{Y} ⊂ (π_{2})_{∗}(L_{Y}) for the rank r + 1 tautological bundle whose fibres correspond to the global sections of the genus e-aspect of each limit 𝔤_{d}^{r}. Finally, for 1 ≤ j ≤ e, we denote by D_{j} ⊂ X ×_{T} 𝔊̃_{d}^{r}(p_{2}) the diagonal divisor corresponding to pulling back the diagonal under the map X ×_{T} 𝔊̃_{d}^{r}(p_{2}) → X ×_{T} X which projects onto the j-th factor, that is, (x, l, x_{1}, …, x_{e}) ↦ (x, x_{j}) where x, x_{1}, …, x_{e} ∈ π^{−1}(t). There exists an evaluation vector bundle morphism over 𝔊̃_{d}^{r}(p_{2})

χ : V_{Y} → (π_{2})_{∗}(L_{Y} ⊗ O_{∑_{j=1}^{e} D_{j}})

and we denote by H the rank e − f degeneracy locus of the map χ. Set-theoretically, H consists of those points (t, l, x_{1}, …, x_{e}) with ϕ(x_{1}, …, x_{e}) = t ∈ T and l ∈ G̃_{d}^{r}(π^{−1}(t)), satisfying the condition that dim l(−x_{1} − ⋯ − x_{e}) ≥ r + 1 − e + f. The dimension of every irreducible component of H is at least ρ(g, r, d) + dim T + e − f(r + 1 − e + f).

In order to show that τ : H → T is dominant, it suffices to prove that τ^{−1}(0) has at least one irreducible component of dimension ρ(g, r, d)+e−f(r +1−e+f). This in fact will prove the stronger statement that Σ_{C} ≠ ∅ for every [C] ∈ M_{g}. Indeed, even though τ : 𝔊̃_{d}^{r}(p_{2}) → T is not a proper morphism, the restriction τ_{τ^{−1}(T −Δ)} : τ^{−1}(T −Δ) → T −Δ is proper, hence there exists an irreducible component of H which maps onto T − Δ. Since π : X → (T, 0) can be chosen in such a way that there exists a point t ∈ T with π^{−1}(t) ≅ C, this proves our contention. We set the integer

α_{0} := [\frac{ρ(e, r − e + f, d − e)}{r + 1 − e + f}] = [\frac{e}{r + 1 − e + f} ] + d − r − f − e,

thus we can write ρ(e, r − e + f, d − e) = α_{0} · (r + 1 − e + f) + c, where 0 ≤ c ≤ r − e + f. Then there exists a unique Schubert index of type (r − e + f, d − e),

\overline{α} : 0 ≤ α_{0} ≤ α_{1} ≤ … ≤ α_{r−e+f} ≤ d − r − f,

with α_{r−e+f} − α_{0} ≤ 1, such that ∑_{j=0}^{r−e+f} α_{j} = ρ(e, r − e + f, d − e). We have that α_{j} = α_{0} for 0 ≤ j ≤ r−e+f −c and α_{j} = α_{0}+1 for r−e+f −c+1 ≤ j ≤ r−e+f. Note that since α_{0} + g(Y) − (d − e) + r − e + f = [e/(r + 1 − e + f)] ≥ 0, condition (17) is verified and the variety G_{d−e}^{r−e+f}(Y, p, \overline{α}) is non-empty of dimension ρ(e, r −e+f, d−e)−∑_{j=0}^{r−e+f} α_{j} = 0.

Next we set β_{0} := [e/(e−f)] and write e = β_{0} ·(e−f)+c̃, where 0 ≤ c̃ ≤ e−f −1. Then there exists a unique Schubert index of type (e − f − 1, 2e − f − 1)

\overline{β} : 0 ≤ β_{0} ≤ β_{1} ≤ … ≤ β_{e−f−1} ≤ e,

such that β_{e−f+1} −β_{0} ≤ 1 and ∑_{j=0}^{e−f−1} β_{j} = e. Precisely, β_{j} = β_{0} for 0 ≤ j ≤ e−f −c̃−1 and β_{j} = β_{0} + 1 for e − f − c̃ ≤ j ≤ e − f − 1. By (17), the variety G_{2e−f−1}^{e−f−1}(Y, p, \overline{β}) is non-empty and of dimension e − ∑_{j=0}^{e−f−1} β_{j} = 0.

First we are going to prove Theorem 0.5 under the assumption that there exist two linear series (A, W_{A}) ∈ G_{d−e}^{r−e+f}(Y, p, \overline{α}) and (L, W_{L}) ∈ G_{2e−f−1}^{e−f−1}(Y, p, \overline{β}) satisfying the condition

(18)  H^{0}(Y, L ⊗ A^{∨} ⊗ O_{Y} ((d + f − 2e) · p)) = 0.

Note that deg(L ⊗ A^{∨} ⊗ O_{Y} ((d + f − 2e) · p) = g(Y) − 1, and (18) states that a suitable translate of at least one of the finitely many line bundles of type L ⊗ A^{∨} lies outside the theta divisor of Y.

Remark 3.1. Condition (18) is a subtle statement concerning [Y, p]. It is not true that (18) holds for every choice of (A, W_{A}) ∈ G_{d−e}^{r−e+f}(Y, p, \overline{α}) and (L, W_{L}) ∈ G_{2e−f−1}^{e−f−1}(Y, p, \overline{β}). For instance, in the case e = 2r − 2 and f = r − 1, corresponding to (2r − 2)-secant (r − 2)- planes which every curve Y ⊂ P^{r} is expected to possess in finite number, we obtain that A = B ⊗ O_{Y} ((d − 3r + 2) · p), where B ∈ W_{r}^{1}(Y) and L ⊗ O_{Y} (−2p) ∈ W_{3r−6}^{r−2}(Y). By Riemann-Roch, we can write that L = K_{Y} ⊗ O_{Y} (2 · p) ⊗ B̃^{∨}, where B̃ ∈ W_{r}^{1}(Y) and then (18) translates into the vanishing statement H^{0}(Y, B ⊗ B̃ ⊗ O_{Y} (−3 · p)) = 0. The curve Y has \frac{(2r−2)!}{r!(r−1)!} pencils 𝔤_{r}^{1}. If we choose B ≠ B̃ ∈ W_{r}^{1}(Y), then h^{0}(Y, B ⊗B̃) ≥ 4 and (18) has no chance of being satisfied. If B = B̃, then the Gieseker-Petri theorem implies that the map H^{0}(Y, B) ⊗ H^{0}(Y, K_{Y} ⊗ B^{∨}) → H^{0}(Y, K_{Y}) is an isomorphism, whence h^{0}(Y, B^{⊗2}) = 3. Choosing p ∈ Y outside the set of ramification points of the finitely many line bundles B^{⊗2} where B ∈ W_{r}^{1}(Y), we obtain that H^{0}(B^{⊗2} ⊗ O_{Y} (−3 · p)) = 0. Therefore in this case, condition (18) is equivalent to the Gieseker-Petri theorem.

We shall study when (18) is actually satisfied. We note that by the Riemann-Roch theorem, (18) also implies that h^{0}(Y, L ⊗ A^{∨} ⊗ O_{Y} ((d + f − 2e + 1) · p)) = 1. Assuming that (A, W_{A}) ∈ G_{d−e}^{r−e+f}(Y, p, \overline{α}) and (L, W_{L}) ∈ G_{2e−f−1}^{e−f−1}(Y, p, \overline{β}) satisfy (18), it follows from Riemann-Roch that there exists a unique effective divisor of degree e

D ∈ |L ⊗ O_{Y} ((d − 2e + f + 1) · p) ⊗ A^{∨}|,

and moreover p ∉ supp(D). We introduce the space of sections

V_{Y} := W_{A} + W_{L} ⊂ H^{0}(Y, L ⊗ O_{Y} ((d − 2e + f + 1) · p)), where we view

W_{A} ⊂ H^{0}(L⊗O_{Y} ((d−2e+f+1)·p−D)) and W_{L} ⊂ H^{0}(L) ⊂ H^{0}(L⊗O_{Y} ((d−2e+f+1)·p)).

We claim that dim(V_{Y}) = r+1, hence 𝔩_{Y} = (L⊗O_{Y} ((d−2e+f+1)·p), V_{Y}) ∈ G_{d}^{r}(Y). Moreover, 𝔩_{Y} has the following vanishing sequence at p: (19) a^{𝔩_{Y}} (p) = (α_{0}, …, α_{r−e+f} +r−e+f, β_{0}+d−2e+f +1, β_{1}+d−2e+f +2, …, β_{e−f−1}+d−e).

Indeed, our original assumption f(r + 1 − e + f) ≥ e is equivalent with the inequality α_{r−e+f} + r − e + f < d − 2e + f + 1, which shows that the sequence (19) contains r + 1 distinct entries. Since p ∉ supp(D), we obtain that the vanishing orders of the sections from W_{A} ⊂ H^{0}(L ⊗ O_{Y} ((d − 2e + f + 1) · p)) are precisely

α_{0}, α_{1} + 1, …, α_{r−e+f} + r − e + f,

while those of the sections from W_{L} ⊂ H^{0}(L ⊗ O_{Y} ((d − 2e + f + 1) · p)) are precisely

β_{0}+d−2e+f +1, β_{1}+d−2e+f +2, …, β_{e−f−1}+e−f −1+d−2e+f +1 = β_{e−f−1}+d−e.

We have found r + 1 sections from V_{Y} having distinct vanishing orders at the point p, hence dim(V_{Y}) = r + 1. Moreover, a^{𝔩_{Y}} (p) is equal to the sequence (19).

Next we choose a linear series 𝔩_{Z} ∈ G_{d}^{r}(Z, p) such that {𝔩_{Y} , 𝔩_{Z}} is a refined limit 𝔤_{d}^{r}. Then the ramification sequence of 𝔩_{Z} at the point p must be equal to

α^{𝔩_{Z}}(p) = \overline{γ} := (e−β_{e−f−1}, e−β_{e−f−2}, …, e−β_{0}, d−r−α_{r−e+f}, …, d−r−α_{1}, d−r−α_{0}).

We claim that condition (17) is satisfied for Z and that the variety G_{d}^{r}(Z, p, \overline{γ}) is nonempty and of dimension ρ(g − e, r, d, \overline{γ}) = ρ(g, r, d) + e − f(r + 1 − e + f). For this to happen, one has to check that the following inequality holds:

(20)  ∑_{j=0}^{r} max{α_{j}^{𝔩_{Z}}(p) + g − e − d + r, 0} ≤ g − e.

There are two things to notice: First, that by direct computation we have that

α_{e−f}^{𝔩_{Z}}(p) + g − e − d + r = g − e − α_{r−e+f} = (g − d + r) + [f − \frac{e}{r + 1 − e + f} ] ≥ 0,

hence α_{j}^{𝔩_{Z}}(p)+g−e−d+r ≥ 0 for all e−f ≤ j ≤ r. Second, that since 0 ≤ β_{e−f−1}−β_{0} ≤ 1, in order to estimate the sum of the first e − f terms in the sum (20), there are two cases to consider. Either α_{0}^{𝔩_{Z}}(p) + g − e − d + r ≥ 0, in which case we find that

∑_{j=0}^{r} max{α_{j}^{𝔩_{Z}}(p) + g − e − d + r, 0} = ∑_{j=0}^{r}(α_{j}^{𝔩_{Z}}(p) + g − e − d + r) =

= g − e − ρ(g − e, r, d, \overline{γ}) = g − e − (ρ(g, r, d) + e − f(r + 1 − e + f)) ≤ g − e.

Else, if α_{0}^{𝔩_{Z}}(p)+g −e−d+r ≤ −1, then also α_{j}^{𝔩_{Z}}(p)+g −e−d+r ≤ 0 for 0 ≤ j ≤ e−f −1 and the left hand side of (20) equals

∑_{j=e−f}^{r}(α_{j}^{𝔩_{Z}}(p)+g−e−d+r) = (r+1−e+f)(g−e)− ∑_{i=0}^{r−e+f} α_{i} = g−e−ρ(g, r−e+f, d−e) ≤ g−e.

In both cases the inequality (17) is satisfied which proves our claim.

Since the chosen (A, W_{A}) ∈ G_{d−e}^{r−e+f}(Y, p, \overline{α}) and (L, W_{L}) ∈ G_{2e−f−1}^{e−f−1}(Y, p, \overline{β}) are isolated points in their corresponding varieties of linear series on Y, it follows that limit 𝔤_{d}^{r}’s on X_{0} constructed in the way we just described, fill-up a component of τ^{−1}(0) ⊂ H.

Indeed, suppose (𝔫_{Y} , 𝔫_{Z}, D̃) ∈ H is a point lying in the same irreducible component of τ^{−1}(0) as (𝔩_{Y} , 𝔩_{Z}, D). Here, 𝔫_{Y} ∈ G_{d}^{r}(Y), 𝔫_{Z} ∈ G_{d}^{r}(Z, p, \overline{γ}) and D̃ ∈ Y_{e} is a divisor such that p ∉ supp(D̃). Then a^{𝔫_{Y}} (p) = a^{𝔩_{Y}} (p) which is given by (19), therefore 𝔫_{Y} (−(d − 2e + f + 1) · p) ∈ G_{e−f−1}^{2e−f−1}(Y, p, \overline{β}) which is a reduced 0-dimensional variety. This implies that 𝔫_{Y} (−(d − 2e + f + 1) · p) = (L, W_{L}). Next, we consider the linear series 𝔫_{Y} (−D̃) ∈ G_{d−e}^{r−e+f}(Y). Since p ∉ supp(D̃), the vanishing sequence of this linear series is a subsequence of length r + 1 − e + f of a^{𝔩_{Y}} (p). Necessarily, α^{𝔫_{Y} (−D̃)}(p) ≥ \overline{α} and because ρ(e, r − e + f, d − e, \overline{α}) = 0, we must have that 𝔫_{Y} (−D̃) ∈ G_{d−e}^{r−e+f}(Y, p, \overline{α}) which is a discrete set, hence 𝔫_{Y} (−D̃) = (A, W_{A}) and D̃ = D ∈ Y_{e}. This shows that 𝔫_{Y} = 𝔩_{Y} and every point of this component of τ^{−1}(0) is determined by the 𝔫_{Z}. The dimension of this component is thus equal to

ρ(e, r−e+f, d−e, \overline{α})+ρ(g−e, r, d, \overline{γ})+ρ(e, e−f−1, 2e−f−1, \overline{β}) = ρ(g, r, d)−f(r+1−e+f)+e, which finishes the proof of Theorem 0.5, subject to proving assumption (18).

Remark 3.2. A slight variation of the argument described above, enables us to prove Theorem 0.5 even in some cases when we cannot establish (18). We start with a linear series (A, W_{A}) ∈ G_{d−e}^{r−e+f}(Y, p, \overline{α}) and assume that the following condition holds:

(21)  H^{0}(Y, O_{Y} ((d − 1) · p) ⊗ A^{∨}) = 0.

There exists a unique divisor D ∈ |O_{Y} (d·p)⊗A^{∨})| and (21) guarantees that p ∉ supp(D). We define the space of sections

V_{Y} := H^{0}(O_{Y} (2e − f − 1) · p) + W_{A} ⊂ H^{0}(O_{Y} (d · p)), where W_{A} ⊂ H^{0}(O_{Y} (d · p − D)).

Reasoning along the same lines as in the previous case, since p ∉ supp(D) we find that dim(V_{Y}) = r + 1, hence 𝔩_{Y} = (O_{Y} (d · p), V_{Y}) ∈ G_{d}^{r}(Y). Moreover, we can check that

a^{𝔩_{Y}} (p) = (α_{0}, α_{1} + 1, …, α_{r−e+f} + r − e + f, d − 2e + f + 1, d − 2e + f + 2, …, d − e − 1, d).

Like in the previous situation, we choose a linear series 𝔩_{Z} ∈ G_{d}^{r}(Z, p) such that {𝔩_{Y} , 𝔩_{Z}} is a refined limit 𝔤_{d}^{r}. Thus we must have the following ramification sequence at p:

α^{𝔩_{Z}}(p) = \overline{γ} := (0, e, …, e, d − r − α_{r−e+f}, …, d − r − α_{1}, d − r − α_{0}).

Condition (17) which guarantees the existence of 𝔩_{Z} is satisfied if and only if

ρ(g, r, d) ≥ f(r + 1 − e + f) − (g − d + r), in the case g − d + r < e

and

ρ(g, r, d) ≥ f(r + 1 − e + f) − e, in the case g − d + r ≥ e.

Since we are always working under the hypothesis ρ(g, r, d) − f(r + 1 − e + f) + e ≥ 0, we see that the previous condition holds whenever g − d + r ≥ e, and that, in general, 𝔩_{Z} ∈ G_{d}^{r}(Z, p, \overline{γ}) exists if and only if

(22)  ρ(g, r, d) ≥ f(r + 1 − e + f) − (g − d + r).

Assuming (22), the variety G_{d}^{r}(Z, p, \overline{γ}) is non-empty of dimension ρ(g − e, d, r, \overline{γ}) = ρ(g, r, d) − f(r + 1 − e + f) + e. The same argument as before shows that limit 𝔤_{d}^{r}’s on X_{0} constructed in such a way, fill-up a component of τ^{−1}(0) ⊂ H of expected dimension ρ(g, r, d) − f(r + 1 − e + f) + e, which finishes the proof.

Now we complete the proof of Theorem 0.5 by discussing under which assumptions we can establish (18):

Proof of Theorem 0.5. We retain the notation introduced above and show that there exist two linear series (A, W_{A}) ∈ G_{d−e}^{r−e+f}(Y, p, \overline{α}) and (L, W_{L}) ∈ G_{2e−f−1}^{e−f−1}(Y, p, \overline{β}) satisfying (18) whenever one of the following conditions is satisfied:

(i) 2f ≤ e − 1, (ii) e = 2r − 2 and f = r − 1, (iii) e < 2(r + 1 − e + f).

As we already explained, (18) in case (ii) is a consequence of the Gieseker-Petri theorem.

We now treat case (i) when β_{0} = 1 and c̃ = f ≤ e − f − 1. By Riemann-Roch we find that L = K_{Y} ⊗ O_{Y} ((e − 2f + 2) · p) ⊗ B^{∨}, where B ∈ W_{e−f+1}^{1}(Y) is a pencil such that h^{0}(Y, B ⊗ O_{Y} (−(e − 2f + 1) · p)) ≥ 1 (There are finitely many such B ∈ W_{e−f+1}^{1}(Y) for a generic choice of [Y, p] ∈ M_{e,1}). Applying the base-point-free pencil trick, (18) is equivalent to the injectivity of the multiplication map

μ_{B,M} : H^{0}(Y, B) ⊗ H^{0}(Y, M) → H^{0}(Y, B ⊗ M),

where M := K_{Y} ⊗ A^{∨} ⊗ O_{Y} ((d − f − e + 2) · p) ∈ W_{2e−f}^{e−f}(Y) is a complete linear series with vanishing sequence at p equal to

(23) a^{M}(p) = (0, 1, …, e − f − a − 1, e − f − a + c, r − a + 2, r − a + 3, …, r, r + 1).

Here we have set a := [e/(r+1−e+f)], hence we can write e = a·(r+1−e+f)+c, where 0 ≤ c ≤ r − e + f. By assumption we have that e − 2a > c and clearly ρ(M, α^{M}(p)) = 0, that is, there are finitely many M ∈ W_{2e−f}^{e−f}(Y) satisfying (23).

To prove that μ_{B,M} is injective, we degenerate [Y, p] ∈ M_{e,1} to a particular stable curve: [Y_{0}, p_{0}] := [E_{0} ∪_{p_{1}} E_{1} ∪ … ∪ E_{e−2a−1} ∪_{p_{e−2a}} T, p_{0}], where E_{0}, …, E_{e−2a−1} are elliptic curves, [T = E_{e−2a}, p_{e−2a}] ∈ M_{2a,1} is a Petri general smooth pointed curve and the points p_{i}, p_{i+1} ∈ E_{i} are such that p_{i+1} − p_{i} ∈ Pic^{0}(E_{i}) is not a torsion class for 0 ≤ i ≤ e − 2a − 1. Note that p_{0} lies on the first component E_{0}. By contradiction, we assume that μ_{B,M} is not injective for every [Y, p] ∈ M_{e,1} and for each of the finitely many linear series M ∈ W_{2e−f}^{e−f}(Y) satisfying (23) and each B ∈ G_{e−f+1}^{1}(Y, p, (0, e−2f)).

We construct a limit 𝔤_{2e−f}^{e−f} on [Y_{0}, p_{0}], say 𝔪 = {(M_{E_{i}}, V_{i}) ∈ G_{2e−f}^{e−f}(E_{i})}_{i=0}^{e−2a}, which satisfies condition (23) with respect to p_{0}, by specifying the vanishing sequences a^{𝔪_{E_{i}}} (p_{i}) for 0 ≤ i ≤ e − 2a. For 0 ≤ i ≤ c − 1, the sequence a^{𝔪_{E_{i+1}}} (p_{i+1}) is obtained from a^{𝔪_{E_{i}}} (p_{i}) by raising all entries by 1, except for the term

a_{e−f−a}^{𝔪_{E_{i+1}}}(p_{i+1}) = a_{e−f−a}^{𝔪_{E_{i}}}(p_{i}) = e − f − a + c.

After c steps we arrive at the following vanishing sequence on E_{c} with respect to p_{c}:

a^{𝔪_{Ec}} (p_{c}) = (c, c+1, …, e−f −a+c−1, e−f −a+c, r−a+2+c, r−a+3+c, …, r+c+1).

For an index c ≤ i ≤ e−2a−1 which we write as i = c+a·β +j, with 0 ≤ j ≤ a−1 and 0 ≤ β ≤ r − 2 − e + f, we choose a^{𝔪_{E_{i+1}}} (p_{i+1}) to be obtained from a^{𝔪_{E_{i}}} (p_{i}) by raising all entries by 1, except for the term

a_{e−f−a+j+1}^{𝔪_{E_{i+1}}}(p_{i+1}) = a_{e−f−a+j+1}^{𝔪_{E_{i}}}(p_{i}) = r − a + 2 + c + (a − 1) · β + 2j.

In this way 𝔪 ∈ G̃_{2e−f}^{e−f}(Y_{0}) becomes a (refined) limit linear series which smooths to a complete linear series M ∈ G_{2e−f}^{e−f}(Y) on every smooth pointed curve [Y, p] ∈ M_{e,1} such that the ramification condition (23) with respect to p is satisfied.

Next we construct a limit 𝔤_{e−f+1}^{1} on [Y_{0}, p_{0}], say 𝔟 = {(B_{E_{i}}, W_{i}) ∈ G_{e−f+1}^{1}(E_{i})}_{i=0}^{e−2a} such that a^{𝔟}(p_{0}) = (0, e − 2f + 1). For 0 ≤ i ≤ e − 2f we set a^{𝔟_{E_{i}}} (p_{i}) = (i, e − 2f + 1). For an index of type i = e − 2f + 2k − 1 where 0 ≤ k ≤ f − a, we choose a^{𝔟_{E_{i}}} (p_{i}) = (e − 2f + k − 1, e − 2f + k + 1). If i = e − 2f + 2k, we choose the sequence a^{𝔟_{E_{i}}} (p_{i}) = (e − 2f + k, e − 2f + k + 1). It is clear that each sequence a^{𝔟_{E_{i}}} (p_{i}) is obtained from a^{𝔟_{E_{i−1}}} (p_{i−1}) by raising one entry by 1 while keeping the other fixed, hence 𝔟 is a limit 𝔤_{e−f+1}^{1} which smooths to a pencil B ∈ G_{e−f+1}^{1}(Y, p, (0, e − 2f)) on every nearby smooth curve [Y, p]. For each 0 ≤ i ≤ e − 2a − 1, there exists a section (unique up to scaling) σ_{i} ∈ W_{i} such that ord_{p_{i}}(σ_{i}) + ord_{p_{i+1}}(σ_{i}) = deg(B_{E_{i}}). We denote by σ_{i}^{c} ∈ W_{i} a

complementary section such that {ord_{p_{i}}(σ_{i}), ord_{p_{i}}(σ_{i}^{c})} = {a_{0}^{𝔟_{E_{i}}} (p_{i}), a_{1}^{𝔟_{E_{i}}} (p_{i})}.

Using the set-up developed in [EH3] and [F2] for studying degenerations of multiplication maps, we find that the assumption that μ_{B,M} is not injective implies the existence elements 0 ≠ ρ_{i} ∈ Ker{W_{i} ⊗ V_{i} → H^{0}(E_{i}, B_{E_{i}} ⊗ M_{E_{i}})} for each 0 ≤ i ≤ e − 2a, satisfying the property that ord_{p_{i+1}}(ρ_{i+1}) ≥ ord_{p_{i}}(ρ_{i})+1, for all i (see e.g. [F2] Section 4, for an explanation of how to obtain the ρ_{i}’s). Moreover, if ord_{p_{i+1}}(ρ_{i+1}) = ord_{p_{i}}(ρ_{i}) + 1, then if τ_{i} ∈ V_{i} is the section (unique up to scaling) such that ord_{p_{i}}(τ_{i}) + ord_{p_{i+1}}(τ_{i}) = deg(M_{E_{i}}), then we must have that

ord_{p_{i}}(ρ_{i}) = ord_{p_{i}}(τ_{i}) + ord_{p_{i}}(σ_{i}^{c}) = ord_{p_{i}}(σ_{i}) + ord_{p_{i}}(τ′_{i}),

where τ′_{i} ∈ V_{i} is another section such that ord_{p_{i}}(τ′_{i}) ≠ ord_{p_{i}}(τ_{i}). In particular, since we have explicitly described all the sequences a^{𝔟_{E_{i}}} (p_{i}) and a^{𝔪_{E_{i}}} (p_{i}), the assumption that ord_{p_{i+1}}(ρ_{i+1}) ≤ ord_{p_{i}}(ρ_{i}) + 1 uniquely determines ord_{p_{i}}(ρ_{i}).

Since a^{𝔟_{E_{0}}} (p_{0}) = (0, e − 2f + 1) and μ_{B_{E_{0}},M_{E_{0}}} (ρ_{0}) = 0, the non-zero section ρ_{0} must involve both sections σ_{0} and σ_{0}^{c} and then clearly ord_{p_{0}}(ρ_{0}) ≥ e − 2f + 1. We prove inductively that for all integers 0 ≤ i ≤ e − 2a we have the inequality

(24)  ord_{p_{i}}(ρ_{i}) ≥ e − 2f + 1 + 2i.

Assuming (24) for i ≤ e − 2a − 1, since ord_{p_{i+1}}(ρ_{i+1}) ≥ ord_{p_{i}}(ρ_{i}) + 1, the only way (24) can fail for i + 1 is when ord_{p_{i}}(ρ_{i}) = e − 2f + 2i + 1 and ord_{p_{i+1}}(ρ_{i+1}) = ord_{p_{i}}(ρ_{i}) + 1. As explained above, this implies that ord_{p_{i}}(ρ_{i}) = ord_{p_{i}}(τ_{i}) + ord_{p_{i}}(σ_{i}^{c}).

Writing i = c + a · β + j as above, then ord_{p_{i}}(τ_{i}) = r − a + 2 + c + (a − 1) · β + 2j if i ≥ c, while ord_{p_{i}}(τ_{i}) = e − f − a + c, for 0 ≤ i ≤ c − 1. We deal only with the case i ≥ c, the case 0 ≤ i ≤ c − 1 being analogous. To determine ord_{p_{i}}(σ_{i}^{c}) we must distinguish between two cases: When i = e−2f +2k −1 with k ≥ 1, then ord_{p_{i}}(σ_{i}^{c}) = e−2f +k −1. Otherwise, we write i = e − 2f + 2k in which case ord_{p_{i}}(σ_{i}^{c}) = e − 2f + k + 1. Suppose we are in the former case. Then we obtain the equality

e − 2f + 2i + 1 = ord_{p_{i}}(ρ_{i}) = (r − a + 2 + c + (a − 1) · β + 2j) + (e − 2f + k − 1),

which ultimately leads to the relation (a + 2)(r − e + f − β) = a − j − 1. But j ≤ a − 1 and β ≤ r − e + f − 1, hence we have reached a contradiction. The case when one can write i = e − 2f + 2k is dealt with similarly. All in all, we may assume that we have proved the inequality ord_{p_{e−2a}}(ρ_{e−2a}) ≥ e−2f +1+2(e−2a). We note that on the curve [T, q] = [E_{e−2a}, p_{e−2a}] we have that a^{𝔟_{T}} (p_{e−2a}) = (e − f − a, e − f − a + 1), while

a^{𝔪_{T}} (p_{e−2a}) = (e − 2a, e − 2a + 1, …, 2e − f − 3a, 2e − f − 3a + 3, …, 2e − f − 2a + 2).

Equivalently 𝔟_{T} = |B|+(e−f −a)·q, where B ∈ W_{a+1}^{1}(T), while 𝔪_{T} = (e−2a)·q +|N|, where N ∈ Pic^{e−f+2a}(T) has the property that h^{0}(T, N(−(e − f − a + 3) · q)) ≥ a. Remembering that ord_{q}(ρ_{e−2a}) ≥ (e − 2f + 1) + 2(e − 2a), after subtracting the base locus supported at q, we find an element

0 ≠ ρ_{T} ∈ Ker{H^{0}(B) ⊗ H^{0}(N) → H^{0}(B ⊗ N)}

such that ord_{q}(ρ_{T}) ≥ e − f − a + 1. Equivalently, the multiplication map

μ_{B,N} : H^{0}(B) ⊗ H^{0}(N(−(e − f − a + 3) · q)) → H^{0}(B ⊗ N(−(e − f − a + 3) · q))

is not injective. By using Riemann-Roch we find that N(−(e − f − a + 3) · q) = K_{T} ⊗ B̃^{∨}, where B̃ ∈ W_{a+1}^{1}(T). Choosing B̃ = B ∈ W_{a+1}^{1}(T), we notice that μ_{B,N} can be identified with the Petri map H^{0}(B) ⊗ H^{0}(K_{T} ⊗ B^{∨}) → H^{0}(K_{T}) which is injective because [T] ∈ M_{2a} was chosen to be Petri general. Thus we have reached a contradiction by reducing

(18) to the Gieseker-Petri theorem which completes the proof in the case (i).

Next we turn to case (iii) when [e/(r + 1 − e + f)] < 2. Since the argument is similar to the one for (i), we only outline the main steps. If e ≤ r − e + f, that is, when α_{0} = d − r − f − e, we can easily determine a linear series (A, W_{A}) ∈ G_{d−e}^{r−e+f}(Y, p, \overline{α}). Precisely, one can see that A = K_{Y} ⊗ O_{Y} ((d − 3e + 2) · p) and

|W_{A}| = (d − r − f − e) · p + |K_{Y} ⊗ O_{Y} ((r + f − 2e + 2) · p)|.

In this case we have that |G_{d−e}^{r−e+f}(Y, p, \overline{α})| = 1. Condition (18) translates into saying that for a generic (L, W_{L}) ∈ G_{2e−f−1}^{e−f−1}(Y, p, \overline{β}) we have the vanishing statement

(25) H^{0}(Y, L ⊗ K_{Y}^{∨}((e + f − 2) · p)) = 0 ⇔ H^{0}(Y, K_{Y}^{⊗2} ⊗ L^{∨}(−(e + f − 2) · p)) = 0.

One can prove (25) by degenerating Y to a generic string of elliptic curves and we skip the details. Finally, if [e/(r + 1 − e + f)] = 1, then c = 2e − r − f − 1 and condition (18) boils down to showing that one can find a pencil B ∈ G_{e−c+1}^{1}(Y, p, (0, r − e + f − c + 1)) and a linear series L ∈ G_{2e−f−1}^{e−f−1}(Y, p, \overline{β}), such that the multiplication map

H^{0}(B) ⊗ H^{0}(K_{Y}^{⊗2} ⊗ L^{∨}(−(2e − 4 − r) · p))→ H^{0}(K_{Y}^{⊗2} ⊗ B ⊗ L^{∨}(−(2e − 4 − r) · p))

is injective. This situation is handled along the lines of (i) and we omit the details. □

Finally, we prove Theorem 0.5 assuming that condition (22) is satisfied. This case is not covered by cases (i) − (iii) above:

Proposition 3.3. Let [Y, p] ∈ M_{e,1} be a general pointed curve. Then there exists a linear series (A, W_{A}) ∈ G_{d−e}^{r−e+f}(Y, p, \overline{α}) such that H^{0}(Y, O_{Y} ((d − 1) · p ⊗ A^{∨})) = 0.

Proof. By contradiction, we assume that H^{0}(O_{Y} ((d − 1) · p) ⊗ A^{∨}) ≠ 0 for every [Y, p] ∈ M_{e,1} and for every linear series (A, V_{A}) ∈ G_{d−e}^{r−e+f}(Y, p, \overline{α}). We let [Y, p] degenerate to the stable curve [Y_{0} := E_{0} ∪_{p_{1}} E_{1} ∪_{p_{2}} … ∪_{p_{e−3}} E_{e−3} ∪_{p_{e−2}} B, p_{0}], where E_{0}, …, E_{e−3} are elliptic curves, the points p_{i}, p_{i+1} ∈ E_{i} are such that p_{i} − p_{i+1} ∈ Pic^{0}(E_{i}) is not a torsion class, and [B, p_{e−2}] ∈ M_{2,1} is such that p_{e−2} ∈ B is not a Weierstrass point. For all integers 0 ≤ i ≤ e − 3 we find that there exist sections

0 ≠ τ_{i} ∈ H^{0}(O_{E_{i}}((d − 1) · p_{i}) ⊗ A_{E_{i}}^{∨}) and 0 ≠ τ_{B} = τ_{e−2} ∈ H^{0}(O_{B}((d − 1) · p_{e−2}) ⊗ A_{B}^{∨})

such that

0 ≤ ord_{p_{0}}(τ_{0}) ≤ ord_{p_{1}}(τ_{1}) ≤ … ≤ ord_{p_{e−3}}(τ_{e−3}) ≤ ord_{p_{e−2}}(τ_{B}).

Moreover, we have that ord_{p_{i}}(τ_{i}) ≥ i for 0 ≤ i ≤ e−2. In particular, ord_{p_{e−2}}(τ_{B}) ≥ e−2. Since ρ(e, r − e + f, d − e, \overline{α}) = 0, limit 𝔤_{d−e}^{r−e+f} on E_{0} ∪ … ∪ E_{e−3} ∪ B are smoothable to every curve of genus g. These finitely many limit 𝔤_{d−e}^{r−e+f} are in bijective correspondence with possibilities of choosing the vanishing sequences {a^{l_{E_{i}}} (p_{i})}_{0≤i≤e−3} and a^{l_{B}}(p_{e−2}) in such a way that for all 0 ≤ i ≤ e − 3, the sequence a^{l_{E_{i+1}}} (p_{i+1}) is obtained from a^{l_{E_{i}}} (p_{i}) by raising all entries by 1 except a single entry which remains unchanged. To finish the proof it suffices to exhibit a single limit 𝔤_{d−e}^{r−e+f} on E_{0}∪…∪E_{e−3}∪B having the property that if (A_{B}, V_{B}) denotes its B-aspect, then H^{0}(O_{B}((d − e + 1) · p_{e−2}) ⊗ A_{B}^{∨}) = 0.

We describe such a 𝔤_{d−e}^{r−e+f} explicitly by specifying the sequences {α^{l_{E_{i}}} (p_{i})}_{0≤i≤e−3} and α^{l_{B}}(p_{e−2}). Clearly, α^{l_{E_{0}}} (p_{0}) equals (α_{0}, …, α_{0}, α_{r−e+f+1−c}^{l_{E_{0}}}(p_{0}) = α_{0} +1, …, α_{0} +1). For 1 ≤ i ≤ c, α^{l_{E_{i}}} (p_{i}) is obtained from α^{l_{E_{i−1}}} (p_{i−1}) by increasing all entries by 1, except for α_{r−e+f+i−c}^{l_{E_{i}}}(p_{i}) = α_{r−e+f+i−c}^{l_{E_{i−1}}}(p_{i−1}). Thus α^{l_{Ec}} (p_{c}) = (α_{0} + c, …, α_{0} + c). Next, for an index i such that c + β(r + 1 − e + f) < i ≤ c + (β + 1)(r + 1 − e + f), where 0 ≤ β ≤ [e/(r + 1 − e + f)], if we write i ≡ j + c mod r + 1 − e + f, with 1 ≤ j ≤ r − e + f, the sequence α^{l_{E_{i}}} (p_{i}) is obtained from α^{l_{E_{i−1}}} (p_{i−1}) by raising all entries by 1, except for α_{j−1}^{l_{E_{i}}}(p_{i}) = α_{j−1}^{l_{E_{i−1}}} (p_{i−1}). Switching from ramification to vanishing sequences we obtain

a^{l_{B}}(p_{e−2}) = (d − r − f − 2, d − r − f − 3, …, d − e − 5, d − e − 4, d − e − 2, d − e − 1),

that is, A_{B} = O_{B}((d − e − 2) · p_{e−2}) ⊗ 𝔤_{2}^{1}, and then

H^{0}(O_{B}((d − e + 1) · p_{e−2}) ⊗ A_{B}^{∨}) = H^{0}(O_{B}(3 · p_{e−2}) ⊗ (𝔤_{2}^{1})^{∨}) = 0.

This contradicts the fact ord_{p_{e−2}}(τ_{B}) ≥ e − 2 which completes the proof. □

4. Higher ramification points of a general line bundle

In this section we prove Theorem 0.10. We fix an arbitrary smooth curve C of genus g and for n ≥ 1 we denote by [n]_{C} : Pic^{d}(C) → Pic^{nd}(C) the multiplication by n map, [n]_{C}(L) := L^{⊗n}. It is an immediate consequence of Riemann-Roch that for a general L ∈ Pic^{d}(C), we have that h^{0}(L^{⊗n}) = max{nd + 1 − g, 0}.

First we show that for a very general L ∈ Pic^{d}(C) we have that w^{L^{⊗n}}(p) ≤ 1 for all p ∈ C and n ≥ 1. Indeed, let us assume that w^{L^{⊗n}}(p) ≥ 2, where n is chosen such that nd ≥ g, so that h^{0}(C, L^{⊗n}) = nd + 1 − g. Then there are two possibilities:

(i) h^{0}(C, L^{⊗n}(−(nd + 2 − g) · p)) ≥ 1 or (ii) h^{0}(C, L^{⊗n}(−(nd − g) · p)) ≥ 2.

In case (i) we consider the map C × C_{g−2} → Pic^{nd}(C), (p, E) ↦ O_{C}((nd + 2 − g) · p + E) and we denote by Σ_{n} its image which is a divisor on Pic^{nd}(C). Then (i) is equivalent to L ∈ [n]_{C}^{∗}(Σ_{n}) which is a divisorial condition on Pic^{d}(C) for each n.

In case (ii) we look at the map C × C_{g}^{1} → Pic^{nd}(C), (p, E) ↦ O_{C}((nd − g) · p + E) and we denote by V_{n} its image. Since C_{g}^{1} is generically a P^{1}-bundle over C_{g−2}, it follows that V_{n} is a divisor on Pic^{nd}(C) and then possibility (ii) is equivalent to L ∈ [n]_{C}^{∗}(V_{n}). Thus we see that for L ∈ Pic^{d}(C) − ⋃_{n≥1}[n]_{C}^{∗}(Σ_{n} + V_{n}) all the ramification points of all powers L^{⊗n} with n ≥ 1, are ordinary. This proves the first part of Theorem 0.10. To prove the second part we start with the following:

Proposition 4.1. We fix a point p ∈ C and integers n and d such that nd ≥ g. Then the locus

D_{n} := {L ∈ Pic^{d}(C) : h^{0}(C, L^{⊗n}(−(nd + 1 − g) · p)) ≥ 1}

is an irreducible divisor on Pic^{d}(C) and [D_{n}] = n^{2}θ.

Proof. We set a := max{0, 2g − 1 − nd} and define two vector bundles E_{n} and F_{n} on Pic^{d}(C) of the same rank and having fibres E_{n}(L) = H^{0}(C, L^{⊗n} ⊗ O_{C}(a · p)) and F_{n}(L) = H^{0}(C, L^{⊗n} ⊗ O_{(a+nd+1−g)·p}(a · p)) over each point L ∈ Pic^{d}(C). Then D_{n} is the degeneracy locus of the morphism E_{n} → F_{n} obtained by evaluation sections of L^{⊗n} ⊗ O_{C}(a · p) along (a + nd + 1 − g) · p. The Picard bundle E_{n} is negative (i.e. E_{n}^{∨} is ample), because E_{n} is the pull-back under the finite map [n]_{C} of a negative bundle on Pic^{d}(C) (cf. [ACGH], pg. 310). Moreover, F_{n} is algebraically equivalent to a trivial bundle, hence E_{n}^{∨} ⊗ F_{n} is ample too. Applying the Fulton-Lazarsfeld connectedness theorem (see [FL] or [ACGH] pg. 311), we conclude that D_{n} is connected. Since D_{n} is also smooth in codimension 2 we obtain that D_{n} must be irreducible. Finally, [D_{n}] = c_{1}(F_{n} − E_{n}) = [n]_{C}^{∗}(θ) = n^{2}θ. □

End of the proof of Theorem 0.10. We fix integers 1 ≤ a < b and consider the variety Σ_{ab} := {(p, L) ∈ C×Pic^{d}(C) : p ∈ R(L^{⊗a})∩R(L^{⊗b})} and we denote by ϕ_{1} : Σ_{ab} → C and ϕ_{2} : Σ_{ab} → Pic^{d}(C) the two projections. For a fixed p ∈ C, the fibre ϕ_{1}^{−1}(p) is identified with the intersection of the two irreducible divisors D_{a} and D_{b}. Since [D_{a}] ≠ [D_{b}] for a ≠ b, it follows that D_{a} ∩ D_{b} is of pure codimension 2 inside Pic^{d}(C), therefore dim(Σ_{ab}) = g − 1. We obtain that a line bundle L ∈ Pic^{d}(C) − ⋃_{a<b} ϕ_{2}(Σ_{ab}) will enjoy the property that R(L^{⊗a}) ∩ R(L^{⊗b}) = ∅ for a < b. □

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Department of Mathematics, University of Texas, Austin, TX 78712

      E-mail address: gfarkas@math.utexas.edu