BRILL-NOETHER WITH RAMIFICATION AT UNASSIGNED POINTS

GAVRIL FARKAS

Abstract. We discuss how via limit linear series and standard facts about divisors on moduli spaces of pointed curves, one can establish a non-existence Brill-Noether results for linear series with prescribed ramification at unassigned points.

In the course of developing their theory of limit linear series, among many other applications, Eisenbud and Harris [EH1], [EH2] also considered the Brill-Noether problem with prescribed ramification at assigned points. For a smooth curve C of genus g, a point p ∈ C and a linear series ℓ = (L, V) ∈ G_{d}^{r}(C), one denotes by

α^{ℓ}(p) : 0 ≤ α_{0}^{ℓ}(p) ≤ … ≤ α_{r}^{ℓ}(p) ≤ d − r

the ramification sequence of ℓ at p.

Having fixed points p_{1}, …, p_{n} ∈ C, Schubert indices ᾱ^{j} : 0 ≤ α_{0}^{j} ≤ … ≤ α_{r}^{j} ≤ d − r of type (r, d) for j = 1, …, n, the locus of linear series on C having prescribed ramification at p_{1}, …, p_{n} ∈ C, that is,

G_{d}^{r}(C, (p_{j}, ᾱ^{j})) := {ℓ ∈ G_{d}^{r}(C) : α^{ℓ}(p_{j}) ≥ ᾱ^{j} for j = 1, …, n}

is a generalized determinantal variety of expected dimension

ρ(g, r, d, ᾱ^{1}, …, ᾱ^{n}) := ρ(g, r, d) − ∑_{j=1}^{n} ∑_{i=0}^{r} α_{i}^{j},

where ρ(g, r, d) := g−(r+1)(g−d+r) is the Brill-Noether number. It is proved in [EH2] Theorem 1.1, that for a general pointed curve [C, p_{1}, …, p_{n}] ∈ M_{g,n}, each component of G_{d}^{r}(C, (p_{j}, ᾱ^{j})) has dimension ρ(g, r, d, ᾱ^{1}, …, ᾱ^{n}). For n = 1, a necessary and sufficient condition for existence is given. Denoting the positive part of an integer n ∈ ℤ by (n)_{+} := max{n, 0}, the pointed curve [C, p] ∈ M_{g,1} carries a linear series ℓ ∈ G_{d}^{r}(C) with ramification α^{ℓ}(p) ≥ ᾱ if and only if

(1)  ∑_{i=0}^{r}(α_{i} + g − d + r)_{+} ≤ g.

Setting ᾱ = (0, …, 0), one recovers the Brill-Noether theorem. In this note we explain how the methods of [EH2], [EH3] coupled with basic facts about Pic(\overline{M}_{g,n}) provide a solution to the Brill-Noether problem with prescribed ramification at unassigned points.

Theorem 0.1. Let C be a general curve of genus g ≥ 2 and ᾱ : 0 ≤ α_{0} ≤ … ≤ α_{r} ≤ d − r a Schubert ramification index. If

ρ(g, r, d) − ∑_{i=0}^{r} α_{i} < −1,

then C carries no linear series ℓ ∈ G_{d}^{r}(C) having ramification α^{ℓ}(p) ≥ ᾱ at some point p ∈ C.

As expected, since on an elliptic curve, due to the existence of translations, one cannot speak of an unassigned point, Theorem 0.1 is true for every smooth curve of genus 1. The inequality ρ(1, r, d, α^{ℓ}(p)) ≥ 0 holds for every elliptic curve [E, p] ∈ M_{1,1} and every linear series ℓ ∈ G_{d}^{r}(E), and lies at the heart of the proof in [EH1] of the non-existence part of the Brill-Noether theorem by degeneration to a flag curve with g elliptic tails. We note that some partial results in the direction of Theorem 0.1 were obtained in [L]. The previous result can be, to some extent, generalized to multiple points:

Theorem 0.2. We fix integers r, d ≥ 1 and g ≥ 2 as well as Schubert indices ᾱ^{1}, …, ᾱ^{n}, with

ρ(g, r, d, ᾱ^{1}, …, ᾱ^{n}) < −1.

Then the locus {[C, p_{1}, …, p_{n}] ∈ M_{g,n} : G_{d}^{r}(C, (p_{j}, ᾱ^{j})) ≠ ∅} has codimension at least two in moduli.

Setting n = 1, one recovers Theorem 0.1. Contrary to our first result, Theorem 0.2 is not true in genus 1 and fails already on M_{1,2}. Indeed, for an integer a ≥ 2, we fix an elliptic curve E and distinct points p_{1}, p_{2} ∈ E such that p_{1} − p_{2} ∈ Pic^{0}(E)[a]. Choose a rational function f ∈ ℂ(E) with div(f) = a · p_{2} − a · p_{1} and consider the linear series ℓ := (O_{E}(2a · p_{1}), V) ∈ G_{2a}^{2}(E), where V := ⟨1, f, f^{2}⟩. Clearly α^{ℓ}(p_{j}) = (0, a − 1, 2a − 2) for j = 1, 2, hence one calculates ρ(1, 2, 2a, α^{ℓ}(p_{1}), α^{ℓ}(p_{2})) = −2. On the other, the locus D_{a} := {[E, p_{1}, p_{2}] ∈ M_{1,2} : p_{1} − p_{2} ∈ Pic^{0}(E)[a]} is obviously a divisor in the moduli space. Considering higher dimensional linear series, the same idea produce counterexamples with even more negative Brill-Noether number.

Straightforward generalizations of Theorem 0.2 to lower Brill-Noether numbers and higher codimension in moduli are false when r ≥ 2, as the following example illustrates. Let C be a curve of genus 2 and L = K_{C}^{⊗2} ∈ G_{4}^{2}(C). If p ∈ C is a Weierstrass point, then α^{L}(p) = (0, 1, 2). In particular, if both p_{1}, p_{2} ∈ C are Weierstrass points, then ρ(2, 2, 4, α^{L}(p_{1}), α^{L}(p_{2})) = −4. On the other hand, the codimension of the locus

{[C, p_{1}, p_{2}] ∈ M_{2,2} : G_{4}^{2}(C, (p_{j}, (0, 1, 2))) ≠ ∅}

is two, in particular it projects onto M_{2}. Similar examples can be constructed for arbitrary genus and, in this sense, Theorem 0.2 is optimal. Even though these counterexamples invite caution before formulating new predictions, we believe the following statement should be true:

Conjecture 0.3. Let g, r, d be positive integers and ᾱ a Schubert index of type (r, d) with ρ(g, r, d, ᾱ) < −2. Then the locus of curves [C] ∈ M_{g} carrying a linear series ℓ ∈ G_{d}^{r}(C) with α^{ℓ}(p) ≥ ᾱ for some point p ∈ C, has codimension at least two in moduli.

I would like to thank the referee for a careful reading and pertinent comments which improved the presentation.

1. The proofs

The proof of Theorem 0.1 is an adaptation of an idea that appears in [EH3]. First, by explicit methods, we prove Theorem 0.1 in genus 3. Then, assuming by contradiction that a general curve [C] ∈ M_{g} possesses a point p ∈ C with G_{d}^{r}(C, (p, ᾱ)) ≠ ∅, the locus of such pairs [C, p] in the universal curve M_{g,1} is an effective divisor. By using the solution to the problem in genus 3 and basic facts about the Picard group of \overline{M}_{g,1} (that have already been used in [EH2], [EH3]), we show that this effective divisor must be empty, thus establishing Theorem 0.1. Theorem 0.2 is proved by induction and relies on the already established case n = 1.

We work throughout over the complex numbers and we assume familiarity with the theory of limit linear series [EH1]. We begin with some preliminaries on moduli spaces of curves and recall that for g ≥ 3 and n ≥ 1, the rational Picard group Pic(\overline{M}_{g,n}) is freely generated by the Hodge class λ, the relative cotangent classes ψ_{1}, …, ψ_{n}, the boundary divisor class δ_{irr} := [Δ_{irr}] of irreducible n-pointed stable curves of curves g and by the classes δ_{i:S} := [Δ_{i:S}], where for each i ≥ 0 and S ⊂ {1, …, n}, the general point of the boundary divisor Δ_{i:S} corresponds to a transverse union of two smooth curves of genus i and g − i respectively, meeting in one point, the marked points lying on the genus i component being precisely those labeled by S. Obviously δ_{i:S} = δ_{g−i:S^{c}}. When n = 0, we write as usually δ_{i} := δ_{i:∅} ∈ Pic(\overline{M}_{g}) for i = 0, …, ⌊\frac{g}{2}⌋. Similarly, for n = 1, we write δ_{i} := δ_{i:\{1\}} ∈ Pic(\overline{M}_{g,1}), for i = 1, …, g − 1. The following well-known fact will be used in the course of the proof of Theorem 0.2:

Lemma 1.1. Let g ≥ 3 and D an effective divisor on M_{g,n} such that all the λ, ψ_{1}, …, ψ_{n} and \{δ_{0:S}\}_{|S|=2}-coefficients in the expansion of the class [\overline{D}] ∈ Pic(\overline{M}_{g,n}) of its closure in \overline{M}_{g,n} are equal to zero. Then D = 0.

Proof. Let π : \overline{M}_{g,n} → \overline{M}_{g} be the forgetful morphism. If π(\overline{D}) = \overline{M}_{g}, then there exists an index 1 ≤ i ≤ n such that \overline{D} intersects non-trivially the general fibre F_{i} of the morphism π_{i} : \overline{M}_{g,n} → \overline{M}_{g,n−1} forgetting the i-th marked point. Note that F_{i} · ψ_{i} = 2g − 3 + n, as well as F_{i} · ψ_{j} = 1 and F_{i} · δ_{0:ji} = 1 for j ∈ {i}^{c}, whereas the intersection numbers with all other generators of Pic(\overline{M}_{g,n}) are zero. We obtain that F_{i} · \overline{D} = 0, a contradiction. Therefore, D = π^{∗}(D′), where D′ is an effective divisor on M_{g}. Since the Hodge class on \overline{M}_{g,n} is pulled back via π via the Hodge class on \overline{M}_{g} and the map π^{∗} : Pic(\overline{M}_{g}) → Pic(\overline{M}_{g,n}) is injective, it follows that the λ-coefficient in the expansion of [\overline{D}′] ∈ Pic(\overline{M}_{g}) is equal to 0 as well. But M_{g} admits a compactification, namely the Satake compactification \overline{M}_{g}^{s}, having boundary \overline{M}_{g}^{s} − M_{g} of codimension 2; in fact, there is a regular Torelli map t : \overline{M}_{g} → \overline{M}_{g}^{s}, assigning to a stable curve [C] ∈ \overline{M}_{g} the product of the degree zero Jacobian varieties of the components of its normalization C̃ of C, and under this map t_{∗}(Δ_{i}) = 0, for i = 0, …, ⌊\frac{g}{2}⌋. A consequence of the existence of \overline{M}_{g}^{s} is that there exists no effective divisor D′ on M_{g} whose λ-coefficient is zero, hence D′ = 0, and thus D = 0. □

Following [EH2], we consider the clutching map φ_{g,n} : \overline{M}_{0,g+n} → \overline{M}_{g,n}, given by
φ_{g,n}([R, x_{1}, …, x_{g}, p_{1}, …, p_{n}]) := [R ∪_{x_{1}} E_{1} ∪ … ∪_{x_{g}} E_{g}, p_{1}, …, p_{n}],

obtained by gluing fixed elliptic curves E_{1}, …, E_{g} at the first g marked points of a curve R of arithmetic genus 0. The marked points p_{1}, …, p_{n} of the resulting genus g stable curve lie on the rational spine R. We consider the action of the symmetric group 𝔖_{g} on \overline{M}_{0,g+n} by permuting the marked points labeled by x_{1}, …, x_{g}. For a subset S ⊂ {p_{1}, …, p_{n}} and an integer 0 ≤ i ≤ g with 2 ≤ |S| + i ≤ g + n − 2, we define the 𝔖_{g}-invariant boundary divisor

B_{i}^{S} := ∑_{\begin{matrix}T ⊂\{x_{1},...,x_{g}\} \\ |T |=i \end{matrix}} δ_{0:S∪T} ∈ Pic(\overline{M}_{0,g+n})^{𝔖_{g}}.

Clearly φ_{g,n}^{∗}(δ_{i:S}) = B_{i}^{S}, whenever i + |S| ≥ 2, whereas φ_{g,n}^{∗}(ψ_{i}) = ψ_{p_{i}}. In order to distinguish between cotangent classes on \overline{M}_{0,g+n} and \overline{M}_{g,n}, we label the corresponding marked point by p_{i} on the genus zero curve and by i ∈ {1, …, n} on the genus g curve.

Lemma 1.2. The 𝔖_{g}-invariant rational Picard group Pic(\overline{M}_{0,g+2})^{𝔖_{g}} is freely generated by the boundary classes \{B_{i}^{p_{1}}\}_{i=1}^{g−1} and \{B_{j}^{p_{1}p_{2}}\}_{j=1}^{g−2}. In particular, dim_{ℚ} Pic(\overline{M}_{0,g+2})^{𝔖_{g}} = 2g − 3.

Proof. This follows for instance from [FG] Proposition 1. It is shown in loc. cit. that there is precisely one relation between the 𝔖_{g}-invariant boundary divisor classes, which expresses B_{0}^{p_{1}p_{2}} = δ_{0:p_{1}p_{2}} in terms of all the other invariant boundary classes. The cotangent classes are also expressible in this basis, see [FG] Lemma 1:

ψ_{p_{1}} = ∑_{j=1}^{g−1} \frac{(g + 1 − j)(g − j)}{(g + 1)g}(B_{j}^{p_{1}} + B_{j−1}^{p_{1}p_{2}}).

□

We fix a general pointed curve [C, p′, q] ∈ M_{g−3,2} and consider another clutching map

j : \overline{M}_{3,1} → \overline{M}_{g,1}, j([B, p]) := [B ∪_{p∼p′} C, q].

The following formulas are easy to prove, see for instance the proof of Lemma 4.3 in

[EH3] or Lemma 3.3 in [AC]:

j^{∗}(λ) = λ, j^{∗}(ψ) = 0, j^{∗}(δ_{0}) = 0, j^{∗}(δ_{g−3}) = −ψ, j^{∗}(δ_{g−2}) = δ_{1}, j^{∗}(δ_{g−1}) = δ_{2},

and j^{∗}(δ_{i}) = 0, for i = 1, …, g − 4.

The next result establishes Theorem 0.1 for g = 3. In genus 2, it is shown in [EH3] Lemma 3.3 that the inequality ρ(2, r, d, α^{ℓ}(p)) ≥ −1 holds for any pointed curve [C, p] ∈ M_{2,1} and any linear series ℓ ∈ G_{d}^{r}(C), with equality if and only if p ∈ C is a Weierstrass point and ℓ = |(r + 2) · p| + (d − r − 2) · p. This establishes Theorem 0.1 in genus 2. The genus 3 analogue of this statement will be the starting step in our induction argument.

Proposition 1.3. Let [C, p] ∈ M_{3,1} be a pointed curve of genus 3 and ℓ ∈ G_{d}^{r}(C). If ρ(3, r, d, α^{ℓ}(p)) ≤ −2, then either p ∈ C is a hyperflex, that is, K_{C} = O_{C}(4p), or else, C is hyperelliptic and p ∈ C is a Weierstrass point. In particular, for any Schubert index ᾱ : α_{0} ≤ … ≤ α_{r} ≤ d − r such that ρ(3, r, d, ᾱ) ≤ −2, each component of the locus

{[C, p] ∈ M_{3,1} : G_{d}^{r}(C, (p, ᾱ)) ≠ ∅}

has codimension at least 2.

Proof. From Riemann-Roch it follows that if Λ ∈ G_{d}^{r}(C) is a linear series with r ≥ 1, then the inequality d ≥ 3 + r always holds except when r = 2 and Λ = |K_{C}|, or when r = 1. Then d = 3 and Λ = |K_{C}(−p)| ∈ W_{3}^{1}(C) when C is non-hyperelliptic, or d ∈ {2, 3} when C is hyperelliptic respectively.

We fix now a linear series ℓ ∈ G_{d}^{r}(C) and let a_{i} := a_{i}^{ℓ}(p) = α_{i}^{ℓ}(p) + i be the i-th entry in the vanishing sequence of ℓ at p. By definition, then ℓ(−a_{i} · p) ∈ G_{d−a_{i}}^{r−i}(C), for each i = 0, …, r. Assume first that C is non-hyperelliptic. Applying the previous observation, we write the following inequalities:

(2)  d − a_{r−i} ≥ i + 3 for i = 0, …, r − 3, and

(3)  d − a_{r−2} ≥ 4, d − a_{r−1} ≥ 3, d − a_{r} ≥ 0.

Adding these inequalities up, we obtain that (r + 1)(d − r) − ∑_{i=0}^{r} α_{i}^{ℓ}(p) ≥ 3r − 2, or equivalently, ρ(3, r, d, α^{ℓ}(p)) ≥ −2. Equality holds only if the three inequalities appearing in (3) are all equalities. When this happens, then (a_{r−2}, a_{r−1}, a_{r}) = (d − 4, d − 3, d) and we find that h^{0}(C, O_{C}(4p)) = 3, that is, p ∈ C is a hyperflex. This condition defines a codimension 2 subvariety of \overline{M}_{3,1}. Thus, when p ∈ C is not a hyperflex, then ρ(g, r, d, α^{ℓ}(p)) ≥ −1.

Assume now that C is hyperelliptic. The inequalities in (2) still hold, as well as, d − a_{r−2} ≥ 4 and d − a_{r} ≥ 0. Assuming that a_{r−1} = 2, one finds that h^{0}(C, O_{C}(2p)) = 2, that is, p ∈ C is a Weierstrass point. □

Proof of Theorem 0.1. We fix a Schubert index ᾱ : α_{0} ≤ … ≤ α_{i} ≤ d − r, such that ρ(g, r, d, ᾱ) < −1, and assume that for a general curve C, there exist a point p ∈ C with G_{d}^{r}(C, p, ᾱ) ≠ ∅. The locus D := {[C, p] ∈ M_{g,1} : G_{d}^{r}(C, p, ᾱ) ≠ ∅} is then a divisor in M_{g,1} (by using the result of Eisenbud and Harris [EH2] mentioned in the introduction, the possibility D = M_{g,1} can be ruled out). We express the class of its closure \overline{D} in \overline{M}_{g,1} in terms of the generators of Pic(\overline{M}_{g,1}):

[\overline{D}] = aλ + cψ − b_{irr}δ_{irr} − ∑_{i=1}^{g−1} b_{i}δ_{i}.

A flag curve X having a rational spine consisting of a tree of P^{1}’s and g elliptic tails satisfies the pointed Brill-Noether theorem with respect to marked points lying on the rational spine, see [EH2] Theorem 1.1; precisely, if p_{1}, …, p_{n} are smooth points of X not lying on any of the elliptic tails, then ρ(g, r, d, α^{ℓ}(p_{1}), …, α^{ℓ}(p_{n})) ≥ 0, for any limit linear series ℓ of type 𝔤_{d}^{r} on X. It thus follows that Im(φ_{g,1}) ∩ \overline{D} = ∅, in particular φ_{g,1}^{∗}([\overline{D}]) = 0. Using the description in [EH3] Lemma 4.2 for the pull-back map at the level of divisors φ_{g,1}^{∗} : Pic(\overline{M}_{g,1}) → Pic(\overline{M}_{0,g+1}), we obtain that for i = 1, …, g − 2 the following relations hold:

(4)  b_{i} = \frac{(g − i)(g − i − 1)}{g(g − 1)}c + \frac{i(g − i)}{g − 1}b_{g−1}.

Next we employ Proposition 1.3. From the additivity of the Brill-Noether number, we find that the pointed curve [B ∪_{p}C, q] ∈ \overline{M}_{g,1} carries no limit linear series ℓ ∈ G_{d}^{r}(C) with ρ(g, r, d, α^{ℓ}(q)) < −1, therefore, [j^{∗}(\overline{D})] = 0, where we recall that j : \overline{M}_{3,1} → \overline{M}_{g,1} was the clutching map attaching a fixed pointed curve of genus g − 3. In terms of coefficients of [\overline{D}] this information is expressed as follows:

0 = [j^{∗}(\overline{D})] = aλ − b_{irr}δ_{irr} + b_{g−3}ψ − b_{g−2}δ_{1} − b_{g−1}δ_{2} ∈ Pic(\overline{M}_{3,1}).

From the independence of the boundary divisor classes on \overline{M}_{3,1}, we find that a = b_{irr} = b_{g−1} = b_{g−2} = b_{g−3} = 0. Using repeatedly (4), we find that [\overline{D}] = 0 ∈ Pic(\overline{M}_{g,1}), that is, D = 0. The contradiction comes from the assumption that D was of codimension one in M_{g,1}. □

Remark 1.4. The argument above is quite close to that in [EH3], where instead of the map j, the clutching map ι : \overline{M}_{2,1} → \overline{M}_{g,1} obtained by attaching a fixed 2-pointed curve of genus g − 2 at the marked point of each stable curve of genus 2 is considered. It is showed in [EH3] Lemma 4.3 that [ι^{∗}(\overline{D})] = 0. However, this does not appear to be enough in order to conclude that [\overline{D}] = 0 and finish the proof. Indeed, in view of Mumford’s relation 10λ = δ_{irr}+2δ_{1} on \overline{M}_{2,1}, there exists a one-dimensional vector space of divisor classes γ ∈ Pic(\overline{M}_{g,1}) satisfying both constraints φ^{∗}(γ) = 0 and ι^{∗}(γ) = 0. To circumvent this problem, we considered instead the map j : \overline{M}_{3,1} → \overline{M}_{g,1}. This also explains why the induction in the proof of Theorem 0.1 starts only from genus 3.

We split the proof of Theorem 0.2 in two parts. Proof of Theorem 0.2, the case n = 2. Let ᾱ^{1} and ᾱ^{2} be Schubert indices of type (r, d) such that ρ(g, r, d, ᾱ^{1}, ᾱ^{2}) < −1 and assume by contradiction that the locus

D := {[C, p_{1}, p_{2}] ∈ M_{g,2} : G_{d}^{r}(C, (p_{1}, ᾱ^{1}), (p_{2}, ᾱ^{2})) ≠ ∅},

is an effective divisor. We express the class of its closure in \overline{M}_{g,2} as a combination

[\overline{D}] = aλ + c_{1}ψ_{1} + c_{2}ψ_{2} − b_{irr}δ_{irr} − ∑_{i=1}^{g−1} b_{i:1} δ_{i:1} − ∑_{i=0}^{g−1} b_{i:12} δ_{i:12} ∈ Pic(\overline{M}_{g,2}).

Let π_{2} : \overline{M}_{g,2} → \overline{M}_{g,1} be the morphism forgetting the second marked point. We claim that the divisor (π_{2})_{∗}(\overline{D} · Δ_{0:12}) is trivial. Indeed, suppose this is not the case and we choose a general point [C, p] ∈ (π_{2})_{∗}(\overline{D} · Δ_{0:12}). We insert a smooth rational curve denoted by P^{1} at the point p and view p_{1}, p_{2} as distinct smooth points on P^{1}. Since [C ∪_{p} P^{1}, p_{1}, p_{2}] ∈ \overline{D}, there exists a limit linear series ℓ = \{ℓ_{C}, ℓ_{P}_{1}\} on C ∪_{p} P^{1} with ramification α^{ℓ}(p_{1}) ≥ ᾱ^{1} and α^{ℓ}(p_{2}) ≥ ᾱ^{2}. We write the following series of inequalities:

−1 > ρ(g, r, d, ᾱ^{1}, ᾱ^{2}) ≥ ρ(g, r, d, α^{ℓ}(p_{1}), α^{ℓ}(p_{2})) ≥
ρ(g, r, d, α^{ℓ_{C}}(p)) + ρ(0, r, d, α^{ℓ_{_{P}1}} (p_{1}), α^{ℓ}(p_{1}), α^{ℓ}(p_{2})) ≥ ρ(g, r, d, α^{ℓ_{C}}(p)),

where the third inequality reflects the additivity of the Brill-Noether number that is incorporated in the definition of the limit linear series (see also [EH2] p. 365), whereas the last inequality follows from the Plücker formula (see also [EH2] Theorem 1.1). Thus ρ(g, r, d, α^{ℓ_{C}}(p)) < −1. Theorem 0.1 guarantees that the locus of curves [C, p] ∈ M_{g,1} satisfying such a Brill-Noether condition is a subvariety of M_{g,1} of codimension at least 2. This leads to a contradiction since [C, p] was chosen as being a general point of a divisor on \overline{M}_{g,1}, therefore we conclude (π_{2})_{∗}(\overline{D} · Δ_{0:12}) = 0. At the level of classes, from the formulas

(π_{2})_{∗}(ψ_{1} · δ_{0:12}) = (π_{2})_{∗}(ψ_{2} · δ_{0:12}) = 0, (π_{2})_{∗}(δ_{i:1} · δ_{0:12}) = 0 for i = 1, …, g − 1,

respectively

(π_{2})_{∗}(λ · δ_{0:12}) = λ, (π_{2})_{∗}(δ_{irr} · δ_{0:12}) = δ_{irr}, (π_{2})_{∗}(δ_{0:12}^{2}) = −ψ_{1}, (π_{2})_{∗}(δ_{i:12} · δ_{0:12}) = δ_{i:1}, we obtain the following identity

0 = (π_{2})_{∗}([\overline{D}] · δ_{0:12}) = aλ − b_{irr}δ_{irr} + b_{0:12}ψ_{1} − ∑_{i=1}^{g−1} b_{i:12}δ_{i:1} ∈ Pic(\overline{M}_{g,1}),

hence a = b_{irr} = b_{i:12} = 0, for i = 0, …, g − 1.

To show that the coefficients c_{1} and c_{2} of [\overline{D}] vanish, we note that Im(φ_{g,2})∩\overline{D} = ∅ (cf. [EH2] Theorem 1.1), hence φ_{g,2}^{∗}([\overline{D}]) = 0. We express this pull-back as an element of the Picard group Pic(\overline{M}_{0,g+2})^{𝔖_{g}}, where the symmetric group 𝔖_{g} acts on \overline{M}_{0,g+2} by permuting the first g marked points of each element [R, x_{1}, …, x_{g}, p_{1}, p_{2}]. Using for instance [AC], we write the following formulas:

φ_{g,2}^{∗}(ψ_{1}) = ψ_{p_{1}}, φ_{g,2}^{∗}(ψ_{2}) = ψ_{p_{2}}, φ_{g,2}^{∗}(δ_{i:1}) = B_{i}^{p_{1}}.
Via the description of Pic(\overline{M}_{0,g+2})^{𝔖_{g}} in terms of 𝔖_{g}-invariant boundary classes

as well as the expression of the cotangent classes ψ_{p_{1}} and ψ_{p_{2}} given in Lemma 1.2, it is straightforward to check that the divisor classes ψ_{p_{1}}, ψ_{p_{2}} and {B_{i}^{p_{1}}}_{i=1}^{g−1} are linearly independent. In particular, from the assumption φ_{g,2}^{∗}(\overline{D}) = 0, we conclude that c_{1} = c_{2} = 0. Thus we can apply Lemma 1.1, which establishes Theorem 0.2 for the case of 2 marked points. □

The case n ≥ 3 can be easily reduced to the situation discussed above: Proof of Theorem 0.2, the case n ≥ 3. We assume, by contradiction, that the locus in moduli

D := {[C, p_{1}, …, p_{n}] ∈ M_{g,n} : G_{d}^{r}(C, (p_{j}, ᾱ^{j})) ≠ ∅}

is a divisor. We shall show that the λ, \{ψ_{j}\}_{j=1}^{n} and \{δ_{0:S}\}_{|S|=2}-coefficients of [\overline{D}] are zero, then use Lemma 1.1 to conclude. The inductive hypothesis coupled with [EH2] Theorem 1.1 ensures that D enjoys two geometric properties:

(1) If π_{i} : \overline{M}_{g,n} → \overline{M}_{g,n−1} is the map forgetting the i-th marked point, then for each 1 ≤ i ≤ n and j ∈ \{i\}^{c}, we have that (π_{i})_{∗}([\overline{D}] · δ_{0:ij}) = 0.

(2) If φ_{g,n} : \overline{M}_{0,g+n} → \overline{M}_{g,n} is the flag map, then φ_{g,n}^{∗}([\overline{D}]) = 0.

We claim that these restrictions, together with the assumption that D is effective imply that D = 0. Firstly, condition (1) implies that the λ, δ_{irr}, ψ_{1}, …, ψ_{n}-coefficients of [\overline{D}] are equal to zero. Indeed, the λ, δ_{irr} and ψ_{k}-coefficients of the divisor classes [\overline{D}] ∈ Pic(\overline{M}_{g,n}) and (π_{i})_{∗}([\overline{D}] · δ_{0:ij}) ∈ Pic(\overline{M}_{g,n−1}) respectively, are equal for all k ∈ \{i, j\}^{c} (this is where the assumption n ≥ 3 plays a role). But then via condition (2), the δ_{0:ij}- coefficient of [\overline{D}] is also equal to zero, since φ_{g,n}^{∗}(δ_{0:ij}) = δ_{0:p_{i}p_{j}}. Furthermore, since all the ψ-coefficients in the expression [\overline{D}] are zero, the coefficient of δ_{0:p_{i}p_{j}} in [φ_{g,n}^{∗}(\overline{D})] equals the δ_{0:ij}-coefficient in [\overline{D}], thus showing that the latter coefficient is equal to zero. We now apply Lemma 1.1 and conclude. □

References

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[EH2] D. Eisenbud and J. Harris, The Kodaira dimension of the moduli space of curves of genus ≥ 23 Inventiones Math. 90 (1987), 359–387.

[EH3] D. Eisenbud and J. Harris, Irreducibility of some families of linear series with Brill-Noether number −1, Annales Scientifique École Normale Supérieure 22 (1989), 33-53.

[FG] G. Farkas and A. Gibney, The Mori cones of moduli spaces of curves of small genus, Transactions of the American Math. Society 355 (2002), 1183-1199.

[L] R. Lehmann, Brill-Noether type theorems with a movable ramification point, arXiv:0804.4657.

Humboldt-Universität zu Berlin, Institut Für Mathematik, Unter den Linden 6 10099 Berlin, Germany

      E-mail address: farkas@math.hu-berlin.de