Higher ramification and varieties of secant divisors on the generic curve

TL;DR

This paper investigates the enumerative geometry of secant divisors on generic curves, clarifying classical formulas' validity, computing dimensions of secant cycles, and analyzing ramification points of line bundles.

Contribution

It provides a complete characterization of when secant cycles are non-empty, computes their dimensions, and studies ramification distributions on generic curves.

Findings

Conditions for non-emptiness of secant cycles

Explicit dimension formulas for secant cycles

Distribution patterns of ramification points

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 completely clarify this problem in the case of the generic curve C of given genus. Using degeneration techniques and a few facts about the birational geometry of moduli spaces of stable pointed curves 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. In a different direction, in the last part of the paper we study the distribution of ramification points of the powers of a line bundle on C having prescribed ramification at a given point.

Equations

$$G_d^r(C):=\{(L,V):L\in \text{mbox}{Pic}^d(C),V\in G(r+1,H^0(L))\}.$$

$$V_e^{e-f}(l):=\{D\in C_e:\text{mbox}diml(-D)\ge r-e+f\}$$

$$\dim \{l\in G_d^r(C):V_e^{e-f}(l)\ne \varnothing \}\le \rho (g,r,d)-f(r+1-e+f)+e.$$

$$\mbox{dim}\bigl{\{}(D,l)\in C_{e}\times G^{r}_{d}(C):D\in V_{e}^{e-f}(l)\bigr{\}}\leq\rho(g,r,d)-f(r+1-e+f)+e,$$

$$\rho (g,r,d)-f(r+1-e+f)+e\ge 0\text{mbox}and\rho (g,r-e+f,d-e)\ge 0.$$

$$(i)\text{mbox}2f\le e-1,(ii)\text{mbox}e=2r-2\text{mbox}andf=r-1,\text{mbox}(iii)\text{mbox}e<2(r+1-e+f),\text{mbox}or$$

$$(iv)\text{mbox}\rho (g,r,d)\ge f(r+1-e+f)-(g-d+r).$$

$$\dim \{(D,l)\in C_e\times G_d^r(C):D\in V_e^{e-f}(l)\}=\rho (g,r,d)-f(r+1-e+f)+e.$$

$$C(d,g,r)=\sum _{i=0}^{r-1}\frac{(-1{)}^i}{r-i}\left(\genfrac{}{}{0ex}{}{d-r-i+1}{r-1-i}\right)\left(\genfrac{}{}{0ex}{}{d-r-i}{r-1-i}\right)\left(\genfrac{}{}{0ex}{}{g}{i}\right).$$

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

$$a^l(p):a_0^l(p)<\dots <a_r^l(p)\le d$$

$$G_d^r(C,p,\overline{\alpha }):=\{l\in G_d^r(C):a_i^l(p)\ge {\alpha }_i+i\text{mbox}fori=0\dots r\}$$

$$\text{mbox}dim{G}_d^r(C,p,\overline{\alpha })=\rho (g,r,d,\overline{\alpha })\text{mbox}\text{mbox}(cf.\text{cite}[cite][\text{@@bibref}EH2]Theorem1.1).$$

$$a<nd-\rho(g,r,d,\overline{\alpha})-g-\bigl{[}\frac{g}{m}\bigr{]},$$

$$a<\mathrm{max}\{2d+2-2g-\rho(g,r,d,\overline{\alpha})+\bigl{[}\frac{g-1}{2}\bigr{]},\ \mbox{ }2d+2-2g-2\rho(g,r,d,\overline{\alpha})+2\bigl{[}\frac{g}{3}\bigr{]}\},$$

$$a\geq nd-g-\rho(g,r,d,\overline{\alpha})-\bigl{[}\frac{g}{m}\bigr{]},\ \ \mbox{ when }n\geq 3,$$

$$a\geq 2d+2-2g-\rho(g,r,d,\overline{\alpha})+\bigl{[}\frac{g-1}{2}\bigr{]},\ \$$

$$a\geq 2d+2-2g-2\rho(g,r,d,\overline{\alpha})+2\bigl{[}\frac{g}{3}\bigr{]},\ \ \mbox{when }n=2.$$

$$\rho (g,r,d,\overline{\alpha })\ge \rho (l_{E_0},p_0,p_1)+\rho (l_{E_1},p_1,p_2)+\cdots +\rho (l_{E_{g-2}},p_{g-2},p_{g-1})+\rho (l_{E_{g-1}},p_{g-1}),$$

$${\mathcal{M}}_i:={\omega }_{\pi }\otimes {\tilde{\mathcal{L}}}_i^{\otimes (-n)}\otimes {\mathcal{O}}_{\mathcal{X}}(a\sigma (B))\in \text{mbox}Pic(\mathcal{X}).$$

$${\rho }_i:={\tilde{\rho }}_{i|E_i}\in H^0(E_i,{\mathcal{M}}_{i|E_i}).$$

$$\text{mbox}{ord}_{p_i}({\rho }_i)+\text{mbox}ord{p}_i({\rho }_{i-1})\ge 2g-2-nd+a=\text{mbox}deg({\mathcal{M}}_{i|E_i}).$$

$$0\le \text{mbox}{ord}_{p_0}({\rho }_0)\le \text{mbox}{ord}_{p_1}({\rho }_1)\le \dots \le \text{mbox}{ord}_{p_i}({\rho }_i)\le \dots \le \text{mbox}{ord}_{p_{g-1}}({\rho }_{g-1}).$$

$$a_j^{l_{E_i}}(p_i)+a_{r-j}^{l_{E_i}}(p_{i+1})\ge d-1\text{mbox}forall0\le j\le r,$$

$$L_{E_i}^{\otimes n}={\mathcal{O}}_{E_i}((a+2i-b)\cdot p_i+(nd-a+b-2i)\cdot p_{i+1})\in \text{mbox}{Pic}^{nd}(E_i).$$

$$L_{E_{i}}=\mathcal{O}_{E_{i}}\Bigl{(}\frac{a+2i-b}{n}\cdot p_{i}+\frac{nd-a+b-2i}{n}\cdot p_{i+1}\Bigr{)}.$$

$${\mathrm{ord}}_{p_i}({\rho }_i)+R(i)\ge b(m-1).$$

$${\mathrm{ord}}_{p_{i+m}}({\rho }_{i+m})-{\mathrm{ord}}_{p_i}({\rho }_i)+R(i+m)-R(i)\ge m-1.$$

$$(ii)\text{mbox}{ord}_{p_{i+l}}({\rho }_{i+l})=\text{mbox}{ord}_{p_{i+l+1}}({\rho }_{i+l+1}):=b,\text{mbox}\text{mbox}{ord}_{p_{i+j}}({\rho }_{i+j})=\text{mbox}{ord}_{p_{i+j+1}}({\rho }_{i+j+1}):=c.$$

$$L_{E_{i+l}}=\mathcal{O}_{E_{i+l}}\bigl{(}\frac{a+2i+2l-b}{n}\cdot p_{i+l}+\frac{nd-a+b-2i-2l}{n}\cdot p_{i+l+1}\bigr{)},\ \mbox{ and }$$

Full text

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

ABSTRACT

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

[TABLE]

The main result of Brill-Noether theory states that if $[C]\in\mathcal{M}_{g}$ is a general curve then $G^{r}_{d}(C)$ is a smooth variety of dimension equal to $\rho(g,r,d):=g-(r+1)(g-d+r)$. For a linear series $l=(L,V)\in G^{r}_{d}(C)$ and an effective divisor $D\in C_{e}$, using the natural inclusion $H^{0}(C,L\otimes\mathcal{O}_{C}(-D))\subset H^{0}(C,L)$, we can define a new linear series l(-D):=\bigl{(}L\otimes\mathcal{O}_{C}(-D),V\cap H^{0}(L\otimes\mathcal{O}_{C}(-D))\bigr{)}. We fix integers $0\leq f<e$ and introduce the determinantal cycle

[TABLE]

of effective divisors of degree $e$ which impose at most $e-f$ independent conditions on $l$. If $l$ is very ample and we view $C\stackrel{{\scriptstyle l}}{{\hookrightarrow}}{\textbf{P}}^{r}$ 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 $[V_{e}^{e-f}(l)]^{{virt}}\in A^{f(r+1-e+f)}(C_{e})$ 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 ${\textbf{P}}^{r}$ which are $(2r-2)$-secant to $C$ (cf. [Ca]).

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 ${\textbf{P}}^{3}$). The aim of this paper is to clarify this problem for a general curve $[C]\in\mathcal{M}_{g}$. For every linear series $l\in G^{r}_{d}(C)$ we determine precisely under which conditions the cycle $V_{e}^{e-f}(l)$ is non-empty and has the expected dimension. Then having fixed $[C]\in\mathcal{M}_{g}$, we determine the dimension of the family of linear series $l\in G^{r}_{d}(C)$ 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 $\overline{\mathcal{M}}_{0,g}$ to prove the following:

[TABLE]

In particular, if $\rho(g,r,d)-f(r+1-e+f)+e<0$, then $V_{e}^{e-f}(l)=\emptyset$, for every  $l\in G^{r}_{d}(C)$.

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

[TABLE]

which obviously implies Theorem 0.1. This result generalizes the Brill-Noether theorem. Indeed, when $l=K_{C}$, then $V_{e}^{e-f}(K_{C})=C_{e}^{f}:=\{D\in C_{e}:h^{0}(\mathcal{O}_{C}(D))\geq f+1\}$. Since the fibres of the Abel-Jacobi map $C_{e}^{f}\rightarrow W_{e}^{f}(C)$ are at least $f$-dimensional, clearly $G_{e}^{f}(C)\neq\emptyset$ implies that $\mbox{dim }C_{e}^{f}\geq f$. Our result reads $G_{e}^{f}(C)=\emptyset$ when $\rho(g,f,e)<0$, which is the non-existence part of the classical Brill-Noether theorem (cf. [EH1]). More generally, we have the following result in the case $\rho(g,r,d)=0$:

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 $f=1$):

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 $l\in G^{r}_{d}(C)$ is said to be $(e-1)$-very ample if $\mbox{dim }l(-p_{1}-\cdots-p_{e})=r-e$, for any choice of (not necessarily distinct) $e$ points $p_{1},\ldots,p_{e}\in C$. Thus [math]-very ampleness is equivalent to generation by global sections and $1$-very ampleness reduces to the classical notion of very ampleness.

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 $\mathfrak{g}^{r}_{d}$ on an arbitrary smooth curve of genus $g$.

[TABLE]

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

[TABLE]

[TABLE]

Then there exists a linear series $l\in G^{r}_{d}(C)$ such that $V_{e}^{e-f}(l)\neq\emptyset$. Moreover, one has that the following dimensionality statement:

[TABLE]

The inequalities $\rho(g,r-e+f,d-e)\geq 0$ and $\rho(g,r,d)+e-f(r+1-e+f)\geq 0$ are obvious necessary conditions for the existence of $l\in G^{r}_{d}(C)$ with $V_{e}^{e-f}(l)\neq\emptyset$ on a general curve $[C]\in\mathcal{M}_{g}$. To give an example, an elliptic quartic curve $C\subset{\textbf{P}}^{3}$ has no $3$-secant lines even though $\rho(g,r,d)+e-f(r+1-e+f)>0$ (note that $e=3$ and $f=1$ in this case). Theorem 0.5 is stated in the range $f(r+1-e+f)\geq e$, corresponding to the case when linear series $l\in G^{r}_{d}(C)$ with $V_{e}^{e-f}(l)\neq\emptyset$ are expected to be special in the Brill-Noether cycle $G^{r}_{d}(C)$. It is clear though that the methods of this paper can be applied to the case $e\geq f(r+1-e+f)$ as well. In that range however, when one expects existence of $e$-secant $(e-f-1)$-planes for every $l\in G^{r}_{d}(C)$, 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 $[C]\in\mathcal{M}_{g}$ and $l\in G^{r}_{d}(C)$, assuming that $d\geq 2e-1$ and  $e-f(r+1-e+f)\geq r-e+f$, one knows that $V_{e}^{e-f}(l)\neq\emptyset$ (cf. [CM1], Theorem 1.2). For $l\in G^{r}_{d}(C)$ such that $g-d+r\leq 1$ (e.g. when $l$ is non-special), if we keep the assumption  $e-f(r+1-e+f)\geq 0$, it is known that $V_{e}^{e-f}(l)\neq\emptyset$ if and only if $\rho(g,r-e+f,d-e)\geq 0$ (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 $[Y,p]\in\mathcal{M}_{e,1}$ 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 $u\geq 1$, and we obtain the following result concerning the classical problem of existence of $uf$-secant secant $(uf-f-1)$-planes to curves in ${\textbf{P}}^{r}$:

If $[C]\in\mathcal{M}_{g}$ 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 $C\subset{\textbf{P}}^{r}$ have a precise enumerative meaning:

[TABLE]

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 $C\subset{\textbf{P}}^{3}$ of degree $d$ (cf. [C]):

[TABLE]

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 ${\textbf{P}}^{3}$, provided that $C(d,g,3)$ is defined as the degree of a certain [math]-cycle $\mbox{Sec}_{4}(C)$ in ${\textbf{G}}(1,3)$ (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 ${\textbf{P}}^{3}$ (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 $[C,p]\in\mathcal{M}_{g,1}$ and a linear series $l=(L,V)\in G^{r}_{d}(C)$, the vanishing sequence of $l$ at $p$

[TABLE]

is obtained by ordering the set $\{\mbox{ord}_{p}(\sigma)\}_{\sigma\in V}$. The weight of $p$ with respect to $l$ is defined as $w^{l}(p):=\sum_{i=0}^{r}(a_{i}^{l}(p)-i)$. One says that $p$ is a ramification point of $l$ if $w^{l}(p)\geq 1$ and we denote by $R(l)$ the finite set of ramification points of $l$. If $[C,p]\in\mathcal{M}_{g,1}$ and $\overline{\alpha}:0\leq\alpha_{0}\leq\ldots\leq\alpha_{r}\leq d-r$ is a Schubert index of type $(r,d)$, the cycle

[TABLE]

can be realized as a generalized determinantal variety inside $G^{r}_{d}(C)$ having virtual dimension $\rho(g,r,d,\overline{\alpha}):=\rho(g,r,d)-\sum_{j=0}^{r}\alpha_{j}$. For a general pointed curve $[C,p]\in\mathcal{M}_{g,1}$, it is known that the virtual dimension equals the actual dimension, that is,

[TABLE]

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

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

[TABLE]

we have that $h^{0}(C,L^{\otimes n}(-ap))=h^{0}(C,L^{\otimes n})-a=nd+1-g-a$. In other words, $a_{i}^{L^{\otimes n}}(p)=i$ for $0\leq i\leq a-1$.

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

[TABLE]

we have that $h^{0}(C,L^{\otimes 2}(-ap))=h^{0}(C,L^{\otimes 2})-a=2d+1-g-a$.

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

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

(1) All the ramification points of the powers $L^{\otimes n}$ are ordinary, that is, $w^{L^{\otimes n}}(p)\leq 1$ for all $p\in C$ and $n\geq 1$.

(2) $R(L^{\otimes a})\cap R(L^{\otimes b})=\emptyset$ for $a\neq b$, that is, a point $p\in C$ 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 $\overline{\alpha}:0\leq\alpha_{0}\leq\ldots\leq\alpha_{r}\leq d-r$ and two integers $a\geq 0,n\geq 2$. We also set $m:=[(n+1)/2]$.

We assume that for every $[C,p]\in\mathcal{M}_{g,1}$ there exists a linear series $l=(L,V)\in G^{r}_{d}(C,p,\overline{\alpha})$ such that $H^{0}(K_{C}\otimes L^{\otimes(-n)}\otimes\mathcal{O}_{C}(ap))\neq 0$. By a degeneration argument we are going to show that this implies the inequalities

[TABLE]

[TABLE]

and

[TABLE]

This will prove both Theorems 0.8 and 0.9.

We degenerate $[C,p]$ to a stable curve $[X_{0}:=E_{0}\cup_{p_{1}}E_{1}\cup_{p_{2}}\ldots\cup_{p_{g-1}}E_{g-1},p_{0}]$, where $E_{i}$ is a general elliptic curve, $p_{i},p_{i+1}\in E_{i}$ are points such that $p_{i+1}-p_{i}\in\mbox{Pic}^{0}(E_{i})$ is not a torsion class and moreover $E_{i}\cap E_{i+1}=\{p_{i+1}\}$ for $0\leq i\leq g-2$. 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 $\pi:\mathcal{X}\rightarrow B$ together with a section $\sigma:B\rightarrow\mathcal{X}$, such that $B=\mbox{Spec}(R)$ with $R$ being a discrete valuation ring having uniformizing parameter $t$. We assume that $\mathcal{X}$ is a smooth surface and that there exists an isomorphism between $X_{0}$ and $\pi^{-1}(0)$. Under this isomorphism we also assume that $\sigma(0)=p_{0}\in X_{0}$. Here $0\in B$ is the point corresponding to the maximal ideal of $R$ and we denote by $\eta$ and $\overline{\eta}$ the generic and geometric generic point of $B$ respectively. By assumption, there exists a linear series $l_{\overline{\eta}}=(L_{\overline{\eta}},V_{\overline{\eta}})\in G^{r}_{d}(X_{\overline{\eta}},\sigma(\overline{\eta}),\overline{\alpha}),$ such that $H^{0}(X_{\overline{\eta}},\omega_{X_{\overline{\eta}}}\otimes L_{X_{\overline{\eta}}}^{\otimes(-n)}\otimes\mathcal{O}_{X_{\overline{\eta}}}(a\sigma(\overline{\eta})))\neq 0$. By possibly blowing up $\mathcal{X}$ 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 $p_{1},\ldots,p_{g-1}$, we may assume that $l_{\overline{\eta}}$ comes from a linear series $l_{\eta}=(L_{\eta},V_{\eta})\in G^{r}_{d}(X_{\eta},\sigma(\eta),\overline{\alpha})$ on the generic fibre $X_{\eta}$.

We denote by $l_{E_{i}}=(L_{E_{i}},V_{E_{i}})\in G^{r}_{d}(E_{i})$ the $E_{i}$-aspect of the limit linear series on $X$ induced by $l_{\eta}$: Precisely, if $\mathcal{L}$ is a line bundle on $\mathcal{X}$ extending $L_{\eta}$, then $L_{E_{i}}\in\mbox{Pic}^{d}(E_{i})$ is the restriction to $E_{i}$ of the unique twist $\mathcal{L}_{E_{i}}$ of $\mathcal{L}$ along components of $\pi^{-1}(0)$ such that $\mbox{deg}_{Z}(\mathcal{L}_{i|Z})=0$ for any irreducible component $Z\neq E_{i}$ of $\pi^{-1}(0)$ (see also [EH1], p. 348). Since we gave ourselves the freedom of blowing-up $\mathcal{X}$ at the nodes of $\pi^{-1}(0)$, we can also assume that $\{l_{E_{i}}\}_{i=0}^{g-1}$ constitutes a limit $\mathfrak{g}^{r}_{d}$ on $X_{0}$ which is obtained from a refined limit $\mathfrak{g}^{r}_{d}$ 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:

[TABLE]

where $\rho(l_{E_{i}},p_{i},p_{i+1}):=\rho(1,r,d)-w^{l_{E_{i}}}(p_{i})-w^{l_{E_{i}}}(p_{i+1})$. By assumption, there exists a non-zero section \rho_{\eta}\in H^{0}\bigl{(}X_{\eta},\omega_{X_{\eta}}\otimes\mathcal{L}_{\eta}^{\otimes(-n)}\otimes\mathcal{O}_{X_{\eta}}(a\sigma(\eta))\bigr{)}. This implies that if we denote by $\tilde{\mathcal{L}}_{i}$ the unique line bundle on the surface $\mathcal{X}$ such that (1) $\tilde{\mathcal{L}}_{i|X_{\eta}}=L_{\eta}$, and (2) $\mbox{deg}_{Z}(\omega_{X}\otimes\tilde{\mathcal{L}_{i}}^{\otimes(-n)}\otimes\mathcal{O}_{X}(ap_{0}))=0$, for every component $Z$ of $X$ such that $Z\neq E_{i}$, then $H^{0}(E_{i},\omega_{X}\otimes\tilde{\mathcal{L}}_{i}^{\otimes(-n)}\otimes\mathcal{O}_{X}(ap_{0})\otimes\mathcal{O}_{E_{i}})\neq 0$. We set

[TABLE]

Then \mathcal{M}_{i|E_{i}}=\mathcal{O}_{E_{i}}\bigl{(}(a+2i)\cdot p_{i}+(2g-2-2i)\cdot p_{i+1}\otimes L_{E_{i}}^{\otimes(-n)}\bigr{)} for all $0\leq i\leq g-1$. For each such $i$ we denote by $n_{i}$ the smallest integer such that $\tilde{\rho}_{i}:=t^{n_{i}}\rho_{\eta}\in\pi_{*}(\mathcal{M}_{i})$ and we set

[TABLE]

Thus $0\neq\rho_{i}\in H^{0}(E_{i},\mathcal{O}_{E_{i}}((a+2i)\cdot p_{i}+(2g-2-2i)\cdot p_{i+1}\otimes L_{E_{i}}^{\otimes(-n)}))$ and in a way similar to [EH1] Proposition 2.2, we can prove that

[TABLE]

One also has the inequalities $\mbox{ord}_{p_{i}}(\rho_{i})+\mbox{ord}_{p_{i+1}}(\rho_{i})\leq 2g-2-nd+a$ (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

[TABLE]

Since $\rho_{g-1}$ is a non-zero section of a line bundle of degree $2g-2-nd+a$ on $E_{g-1}$, we must have that $\mbox{ord}_{p_{g-1}}(\rho_{g-1})\leq 2g-2-nd+a$. 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 $\rho(l_{E_{i}},p_{i},p_{i+1})=0$. By counting dimensions, we see that for every $0\leq j\leq r$ there exists a section $u_{j}\in V_{E_{i}}$ such that $\mbox{div}(u_{j})\geq a_{j}^{l_{E_{i}}}(p_{i})\cdot p_{i}+a_{r-j}^{l_{E_{i}}}(p_{i+1})\cdot p_{i+1}$. In particular, we have that $a_{j}^{l_{E_{i}}}(p_{i})+a_{r-j}^{l_{E_{i}}}(p_{i+1})\leq d$. Since $p_{i+1}-p_{i}\in\mbox{Pic}^{0}(E_{i})$ 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 $0\leq j\leq r$. Because $\rho(l_{E_{i}},p_{i},p_{i+1})=0$, this implies that

[TABLE]

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 $\mbox{div}(u_{j})=a_{j}^{l_{E_{i}}}(p_{i})\cdot p_{i}+a_{r-j}^{l_{E_{i}}}(p_{i+1})\cdot p_{i+1}$, and for degree reasons we must have that $L_{E_{i}}=\mathcal{O}_{E_{i}}(a_{j}^{l_{E_{i}}}(p_{i})\cdot p_{i}+a_{r-j}^{l_{E_{i}}}(p_{i+1})\cdot p_{i+1})\in\mbox{Pic}^{d}(E_{i}).$

To summarize, if $\rho(l_{E_{i}},p_{i},p_{i+1})=0$, then the vanishing sequence $a^{l_{E_{i+1}}}(p_{i+1})$ of the $E_{i+1}$-aspect of the limit $\mathfrak{g}^{r}_{d}$ 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 $0\leq j\leq r$ and $a_{k}^{l_{E_{i+1}}}(p_{i+1})=a_{k}^{l_{E_{i}}}(p_{i})+1$ for $k\neq j$.

We now study what happens to the non-decreasing sequence (6) as we pass through a component $E_{i}$ with $\rho(l_{E_{i}},p_{i},p_{i+1})=0$. Assume that $\mbox{ord}_{p_{i}}(\rho_{i})=\mbox{ord}_{p_{i+1}}(\rho_{i+1}):=b$. This implies that $\mbox{ord}_{p_{i+1}}(\rho_{i})=2g-2-nd+a-b$ and

[TABLE]

Because $\rho(l_{E_{i}},p_{i},p_{i+1})=0$, 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}}=\mathcal{O}_{E_{i}}\bigl{(}a_{j}^{l_{E_{i}}}(p_{i})\cdot p_{i}+a_{r-j}^{l_{E_{i}}}(p_{i+1})\cdot p_{i+1}\bigr{)} for a unique $0\leq j\leq r$. 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

[TABLE]

In particular, we have that $(a+2i-b)/n\in\mathbb{Z}$ and $a_{j}^{l_{E_{i}}}(p_{i})=(a+2i-b)/n$.

We consider a connected subcurve $Y\subset X$ 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 $bm\leq i\leq g-1$. We denote by $R(i):=\#\{0\leq l\leq i-1:\rho(l_{E_{l}},p_{l},p_{l+1})\geq 1\}$. Then the following inequality holds:

[TABLE]

Proof.

We proceed by induction on $b$. For $b=0$ there is nothing to prove. We set $b\geq 1$, $i:=(b-1)m$ and we assume that $\mbox{ord}_{p_{i}}(\rho_{i})+R(i)\geq(b-1)(m-1)$. We are going to prove that the following inequality holds:

[TABLE]

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

[TABLE]

Using (7) this implies that

[TABLE]

[TABLE]

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

For each integer $0\leq e\leq l-1$ we have that either $\rho(l_{E_{i+e}},p_{i+e},p_{i+e+1})\geq 1$ or $\mbox{ord}_{p_{i+e+1}}(\rho_{i+e+1})>\mbox{ord}_{p_{i+e}}(\rho_{i+e})$. This fact leads to the inequality

[TABLE]

Similarly, by studying the subcurve of $Y$ containing $E_{i+j+1},\ldots,E_{i+m-1}$, we find that

[TABLE]

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

[TABLE]

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 $2b\leq i\leq g-1$. Then $\mathrm{ord}_{p_{i}}(\rho_{i})+R(i)\geq b$.

(2) We fix $0\leq i\leq g-4$ 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,*

[TABLE]

then we have the inequality $\mathrm{ord}_{p_{i+3}}(\rho_{i+3})\geq\mathrm{ord}_{p_{i}}(\rho_{i})+2.$

Proof.

We only prove (2), the remaining statement being analogous to Lemma 1.1. We may assume that $\mbox{ord}_{p_{i}}(\rho_{i})=\mbox{ord}_{p_{i+1}}(\rho_{i+1}):=b$. Hence $(a+2i-b)/2\in\mathbb{Z}$ and there exists an index $0\leq j\leq r$ such that

[TABLE]

If $\mbox{ord}_{p_{i+2}}(\rho_{i+2})=\mbox{ord}_{p_{i+1}}(\rho_{i+1})=b$, 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 $\mbox{ord}_{p_{i+2}}(\rho_{i+2})\geq b+1$. Next, if $\mbox{ord}_{p_{i+3}}(\rho_{i+3})=b+1$, this implies that $\mbox{ord}_{p_{i+3}}(\rho_{i+3})=\mbox{ord}_{p_{i+2}}(\rho_{i+2})=b+1$, hence again \bigl{(}a+2(i+2)-(b+1)\bigr{)}/2\in\mathbb{Z}, which is not possible for parity reasons. Thus we must have that $\mbox{ord}_{p_{i+3}}(\rho_{i+3})\geq b+2$. ∎

Proof of Theorem 0.8. We complete the proof of our result in the case $n\geq 3$. We write $g=bm+c$ with $0\leq c\leq m-1$ and we set $i:=bm$. From Lemma 1.1 we obtain that $\mbox{ord}_{p_{i}}(\rho_{i})+R(i)\geq b(m-1)$. Using the reasoning of Lemma 1.1 for the connected subcurve of $X$ which contains $E_{i},E_{i+1},\ldots,E_{i+c-1}=E_{g-1}$, we get that

[TABLE]

Using (12), together with the inequality $R(g-1)\leq\rho(g,r,d,\overline{\alpha})$, we can write that

[TABLE]

which finishes the proof of Theorem 0.8. $\Box$

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

[TABLE]

Since $R(g-1)\leq\rho(g,r,d,\overline{\alpha})$, this leads to the inequality $a\geq 2d+2-2g+[(g-1)/2]-\rho(g,r,d,\overline{\alpha})$. To prove (3) we divide $X$ into $e:=[g/3]+1$ connected subcurves $Y_{1},\ldots,Y_{e}$ such that $Y_{1},\ldots,Y_{e-1}$ each contain three elliptic components, $\#(Y_{i}\cap Y_{i+1})=1$ for all $1\leq i\leq e-2$ and $Y_{e}:=\overline{(\cup_{i=1}^{e-1}Y_{i})^{c}}$. The curves $Y_{i}$ fall into two categories: those for which there exists an elliptic component $E_{l}\subset Y_{i}$ such that $\rho(l_{E_{l}},p_{l},p_{l+1})\geq 1$ (and there are at most $\rho(g,r,d,\overline{\alpha})$ such $Y_{i}$’s), and those for which $\rho(l_{E_{l}},p_{l},p_{l+1})=0$ for each elliptic component $E_{l}\subset Y_{i}$. Lemma 1.2 part (2) gives that $\mbox{ord}_{p_{g-1}}(\rho_{g-1})\geq 2([g/3]-\rho(g,r,d,\overline{\alpha}))$. This proves (2) and finishes the proof of Theorem 0.9. $\Box$

Remark 1.3**.**

It is natural to ask how close to being optimal are the bounds we obtained above. For $\rho(g,r,d,\overline{\alpha})$ relatively small, when any $L\in G^{r}_{d}(C,p,\overline{\alpha})$ 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 $\rho(g,r,d,\overline{\alpha})=0$. Thus we look at $\mathfrak{g}^{3}_{6}$’s on a general $[C,p]\in\mathcal{M}_{3,1}$ having ramification at $p$ equal to $(0\leq\alpha_{0}\leq\alpha_{1}\leq\alpha_{2}\leq\alpha_{3}\leq 3)$, where $\sum_{i=0}^{3}\alpha_{i}=3$. Theorem 0.9 gives us that $H^{0}(K_{C}\otimes L^{\otimes(-2)}\otimes\mathcal{O}_{C}(a\cdot p))=0$ for every integer $a\leq 9$. We show that this is optimal by noting that when $a=10$ and $\overline{\alpha}=(0,0,1,2)$, we have that

[TABLE]

Indeed, any such linear series is of the form $L=K_{C}\otimes A^{\vee}\otimes\mathcal{O}_{C}(5p)\in W^{3}_{6}(C)$, where $A\in W^{1}_{3}(C)$ is such that $h^{0}(A(-2p))\geq 1$. A non-hyperelliptic curve of genus $3$ has two such $\mathfrak{g}^{1}_{3}$’s. Precisely, if $z,t\in C$ are the two points the tangent line at $p$ to $C\stackrel{{\scriptstyle|K_{C}|}}{{\hookrightarrow}}{\textbf{P}}^{2}$ meets $C$ again, then $A=\mathcal{O}_{C}(2p+z)$ or $A=\mathcal{O}_{C}(2p+t)$. Say, we choose $A=\mathcal{O}_{C}(2p+z)$. By direct calculation we obtain that $L^{\otimes 2}\otimes\mathcal{O}_{C}(-10p)=K_{C}^{\otimes 2}\otimes A^{\otimes(-2)}=\mathcal{O}_{C}(2t)$, hence $h^{0}(K_{C}\otimes L^{\otimes(-2)}\otimes\mathcal{O}_{C}(10p))=1$.

2. Varieties of secant planes to the general curve

We fix a smooth curve $[C]\in\mathcal{M}_{g}$ and two integers $0\leq f<e$. In this section we study the varieties $V_{e}^{e-f}(l)$ of $e$-secant $(e-f-1)$-planes corresponding to a linear series $l\in G^{r}_{d}(C)$. We first define the correspondence

[TABLE]

and denote by $\pi_{1}:\Sigma_{C}\rightarrow C_{e}$ and $\pi_{2}:\Sigma_{C}\rightarrow G^{r}_{d}(C)$ the two projections. We assume that $\Sigma_{C}\neq\emptyset$ for the general curve $[C]\in\mathcal{M}_{g}$. Under this assumption, we show that

[TABLE]

(We recall that the dimension of a scheme is the maximum of the dimensions of its irreducible components). Since $\Sigma_{C}$ is a determinantal subvariety of $C_{e}\times G^{r}_{d}(C)$, it follows that for a general $[C]\in\mathcal{M}_{g}$, if non-empty, the scheme $\Sigma_{C}$ is equidimensional and $\mbox{dim}(\Sigma_{C})=\rho(g,r,d)-f(r+1-e+f)+e$. Note that this result does not establish the non-emptiness of $\Sigma_{C}$ which is an issue that we will deal with in Section 3. In any event, (13) implies the dimensional estimate

[TABLE]

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

We start by setting some notation. We denote by $j:\overline{\mathcal{M}}_{0,g}\rightarrow\overline{\mathcal{M}}_{g}$ the “flag” map obtaining by attaching to each stable curve $[R,x_{1},\ldots,x_{g}]\in\overline{\mathcal{M}}_{0,g}$ fixed elliptic tails $E_{1},\ldots,E_{g}$ at the points $x_{1},\ldots,x_{g}$ respectively. Thus $j([R,x_{1},\ldots,x_{g}]):=[\tilde{R}]=[R\cup_{x_{1}}E_{1}\cup\ldots\cup_{x_{g}}E_{g}]$ and for such a curve, we denote by $p_{R}:\tilde{R}\rightarrow R$ the projection onto $R$, that is, $p_{R}(E_{i})=\{x_{i}\}$ for $1\leq i\leq g$. We denote by $\overline{\mathcal{C}}_{g,n}=\overline{\mathcal{M}}_{g,n+1}$ the universal curve and by $\pi:\overline{\mathcal{C}}_{g,n}\rightarrow\overline{\mathcal{M}}_{g,n}$ the morphism forgetting the $(n+1)$-st marked point. We write $\pi_{e}:\overline{\mathcal{C}}_{g,n}^{e}\rightarrow\overline{\mathcal{M}}_{g,n}$ for the $e$-fold fibre product of $\overline{\mathcal{C}}_{g,n}$ over $\overline{\mathcal{M}}_{g,n}$ and we introduce a map $\chi:\overline{\mathcal{M}}_{0,g}\times_{\overline{\mathcal{M}}_{g}}\overline{\mathcal{C}}_{g}^{e}\rightarrow\overline{\mathcal{C}}_{0,g}^{e}$ which collapses the elliptic tails. Thus $\chi$ is defined by

[TABLE]

for points $y_{1},\ldots,y_{e}\in\tilde{R}$. Let $W\subset\overline{\mathcal{C}}_{g}^{e}$ be the closure of the locus

[TABLE]

By assumption $\pi_{e}(W)=\overline{\mathcal{M}}_{g}$ and we define the locus U:=\chi\bigl{(}W\cap(\overline{\mathcal{M}}_{0,g}\times_{\overline{\mathcal{M}}_{g}}\overline{\mathcal{C}}_{g}^{e})\bigr{)}. Then $\pi_{e}(U)=\overline{\mathcal{M}}_{0,g}$ and we denote by $e-m$ the minimal fibre dimension of the map $\pi_{e|U}:U\rightarrow\overline{\mathcal{M}}_{0,g}$. Thus $0\leq m\leq e$ and $\mbox{dim}(U\cap\pi_{e}^{-1}[R,x_{1},\ldots,x_{g}])\geq e-m$, for every $[R,x_{1},\ldots,x_{g}]$, with equality for a general point $[R,x_{1},\ldots,x_{g}]\in\overline{\mathcal{M}}_{0,g}$.

We recall that for every choice of $4$ marked points $\{i,j,k,l\}\subset\{1,\ldots,g\}$, one has a fibration $\pi_{ijkl}:\overline{\mathcal{M}}_{0,g}\rightarrow\overline{\mathcal{M}}_{0,4}$ 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 $\infty$, in this way we obtain a birational map $\pi_{123}=(\pi_{1234},\ldots,\pi_{123i},\ldots,\pi_{123g}):\overline{\mathcal{M}}_{0,g}\rightarrow\overline{\mathcal{M}}_{0,4}^{g-3}=({\textbf{P}}^{1})^{g-3}$ defined by

[TABLE]

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

[TABLE]

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

[TABLE]

Finally, for $2\leq k\leq e$ we define the diagonal loci $\Delta_{k}\subset({\textbf{P}}^{1})^{g-3+e}$ as consisting of those points $(x_{4},\ldots,x_{g},y_{1},\ldots,y_{e})$ for which at least $k$ of the points $y_{1},\ldots,y_{e}$ coincide. We need the following result concerning existence of sublinear limit linear series of a fixed limit $\mathfrak{g}^{r}_{d}$, having prescribed vanishing sequence at a given point:

Lemma 2.1**.**

Let $X$ be a curve of compact type, $Y\subset X$ an irreducible component and let $p\in Y$ be a smooth point of $X$. Assume that $l$ is a (refined) limit $\mathfrak{g}^{r}_{d}$ on $X$ and let $(a_{0}<a_{1}<\ldots<a_{r})$ be the vanishing sequence $a^{l}(p)$. We fix a subsequence $(a_{j_{0}}<a_{j_{1}}<\ldots<a_{j_{b}})$ of $a^{l}(p)$, where $0\leq b\leq r$. Then there exists a limit $\mathfrak{g}_{d}^{b}$ on $X$, say $l^{\prime}\subset l$, such that $a^{l^{\prime}}(p)=(a_{j_{0}},\ldots,a_{j_{b}})$.

Proof.

Let us denote by $l:=\{l_{Z}=(L_{Z},V_{Z})\}_{Z\subset X}$ the original limit $\mathfrak{g}^{r}_{d}$ on $X$. For each integer $0\leq k\leq b$ there exists a section $\sigma_{j_{k}}\in V_{Y}$ such that $\mbox{ord}_{p}(\sigma_{j_{k}})=a_{j_{k}}$. We consider the subspace $W_{Y}:=<\sigma_{j_{0}},\ldots,\sigma_{j_{b}}>\subset V_{Y}$. Since $\#\{\mbox{ord}_{p}(\sigma)\}_{\sigma\in W_{Y}}=b+1$, we obtain that $\mbox{dim}(W_{Y})=b+1$ and we set $l^{\prime}_{Y}:=(L_{Y},W_{Y})\in G^{b}_{d}(Y)$. Suppose now that $Z$ is a component of $X$ meeting $Y$ in a point $q$. We denote by $(c_{j_{0}}<c_{j_{1}}<\ldots<c_{j_{b}})$ the vanishing sequence $a^{l^{\prime}_{Y}}(q)$. Let $(e_{j_{0}}<e_{j_{1}}<\ldots<e_{j_{b}})$ be the complementary sequence, that is, $e_{j_{k}}=d-c_{j_{b-k}}$ for each $0\leq k\leq b$. Then we can choose a section $\tau_{k}\in V_{Z}$ such that $\mbox{ord}_{q}(\tau_{k})=e_{j_{k}}$. We define $W_{Z}:=<\tau_{0},\ldots,\tau_{b}>\subset V_{Z}$. Because all the entries $(e_{j_{k}})_{k=0}^{b}$ are distinct, we get that $\mbox{dim}(W_{Z})=b+1$ and then set $l^{\prime}_{Z}:=(L_{Z},W_{Z})\in G^{b}_{d}(Z)$. We continue inductively, and for each irreducible component $Z^{\prime}\subset X$ we obtain an aspect $l_{Z^{\prime}}^{\prime}=(L_{Z^{\prime}},W_{Z^{\prime}})\in G^{b}_{d}(Z^{\prime})$. The collection $\{l_{Z}^{\prime}\}_{Z\subset X}$ is the desired limit $\mathfrak{g}_{d}^{b}$ on $X$. ∎

Next we explain how the assumption that for every $[C]\in\mathcal{M}_{g}$ there exists a linear series $l\in G^{r}_{d}(C)$ with $V^{e-f}_{e}(l)\neq\emptyset$, can be used to construct a flag curve $\tilde{R}\in j(\overline{\mathcal{M}}_{0,g})$ such that all the $e$ points coming from the limit of an effective divisor $D\in V_{e}^{e-f}(l)$ specialize to a connected subcurve of $\tilde{R}$ having arithmetic genus at most $\mbox{min}\{g,e\}$.

Proposition 2.2**.**

Let $U\subset\overline{\mathcal{C}}_{0,g}^{e}$ be an irreducible component of the closure of the locus of limits of $e$-secant divisors with respect to linear series $\mathfrak{g}^{r}_{d}$ on flag curves from $\overline{\mathcal{M}}_{g}$. Assuming that $\mathrm{dim}(U)=g-3+e-m$   with $0\leq m\leq e$, there exists a point $([R,x_{1},\ldots,x_{g}],\tilde{y}_{1},\ldots,\tilde{y}_{e})\in W\cap(\overline{\mathcal{M}}_{0,g}\times_{\overline{\mathcal{M}}_{g}}\overline{\mathcal{C}}_{g}^{e})$ corresponding to a genus $g$ flag curve

[TABLE]

such that either (i)  $\tilde{y}_{1}=\cdots=\tilde{y}_{e}\in R-\{x_{1},\ldots,x_{g}\}$, or else, (ii)  all the points $\tilde{y}_{1},\ldots,\tilde{y}_{e}$ lie on a connected subcurve $Y\subset\tilde{R}$ satisfying $p_{a}(Y)\leq\mathrm{min}\{m,g\}$ and $\#(Y\cap(\overline{\tilde{R}-Y}))\leq 1$.

Proof.

We start by noting that if $m=0$ then $U=\overline{\mathcal{C}}_{0,g}^{e}$ and possibility (i) is satisfied. Thus we may assume that $m\geq 1$. First, we claim that $\mbox{dim }f(U)=\mbox{dim }U=g-3+e-m$. Indeed, since $\pi_{e}(U)=\overline{\mathcal{M}}_{0,g}$ it follows that $p_{1}(f(U))=({\textbf{P}}^{1})^{g-3}$ and we choose a general point $t=(x_{4},\ldots,x_{g})\in({\textbf{P}}^{1}-\{0,1,\infty\})^{g-3}$, such that $x_{i}\neq x_{j}$ for $i\neq j$. Then $\pi_{e}^{-1}(t)=({\textbf{P}}^{1})^{e}$ and $f_{|\pi_{e}^{-1}(t)}$ is an isomorphism onto its image, hence $f_{|U}$ is birational onto its image as well. Obviously, when $m\geq g$ we can take $Y=\tilde{R}$. From now on we shall assume that $1\leq m\leq g-1$.

Let us assume first that $f(U)\cap\Delta_{e}\neq\emptyset$. Then \mbox{dim}\bigl{(}f(U)\cap\Delta_{e}\bigr{)}\geq g-m-2. For dimension reasons, there must exist a point $z=(x_{4},\ldots,x_{g},y_{1},\ldots,y_{1})\in f(U)\cap\Delta_{e}$ such that either (i) at least $g-m-3$ of the points $x_{j}$ with $4\leq j\leq g$ are mutually distinct and belong to the set ${\textbf{P}}^{1}-\{0,1,\infty,y_{1}\}$ and $y_{1}\in{\textbf{P}}^{1}-\{0,1,\infty\}$, or (ii) at least $g-m-2$ of the $x_{j}$’s ($4\leq j\leq g$) are mutually distinct and belong to the set ${\textbf{P}}^{1}-\{0,1,\infty,y_{1}\}$ and then $y_{1}\in{\textbf{P}}^{1}$ may, or may not be equal to one of the points $0,1$ or $\infty$. Suppose we are in situation (i), the remaining case being similar. We fix a point $([R,x_{1},\ldots,x_{g}],y_{1},\ldots,y_{e})\in f^{-1}(z)$, hence $y_{1},\ldots,y_{e}\in R$. If $Z\subset R$ denotes the minimal connected subcurve of $R$ containing all the points $y_{1},\ldots,y_{e}$, then $x_{1},x_{2},x_{3}\in R-Z$, unless $y_{1}=\cdots=y_{e}$. (In the latter case either $y_{1}\in R-\{x_{1},\ldots,x_{g}\}$ which corresponds to the situation when all the points $\tilde{y}_{i}=y_{i}$ specialize to the same smooth point of $\tilde{R}$ lying on the rational spine, or else, if $y_{1}=x_{j}$ for some $4\leq j\leq g$, then we can find a connected subcurve of $\tilde{R}$ of genus $1$ containing $\tilde{y}_{1},\ldots,\tilde{y}_{e}$, where $p_{R}(\tilde{y}_{i})=y_{i}$ for $1\leq i\leq e$). Since at least $g-m=3+(g-m-3)$ of the points $x_{1},\ldots,x_{g}$ lie on $Z^{c}$, it follows that $\tilde{y}_{1},\ldots,\tilde{y}_{e}$ lie on a connected subcurve of $\tilde{R}$ of genus $\leq m$, which completes the proof in this case.

We are left with the possibility $f(U)\cap\Delta_{e}=\emptyset$ and we denote by $k\leq e-1$ the largest integer for which $f(U)\cap\Delta_{k}\neq\emptyset$ and by $L$ an irreducible component of $f(U)\cap\Delta_{k}$. Since by definition $f(U)\cap\Delta_{k+1}=\emptyset$, it follows that there exists a point $t_{0}=(p_{1},\ldots,p_{e})\in({\textbf{P}}^{1})^{e}$ such that $L\subset({\textbf{P}}^{1})^{g-3}\times\{t_{0}\}$. In particular, the projection map $p_{1|L}:L\rightarrow p_{1}(L)$ is $1:1$ and then $\mbox{dim }p_{1}(L)=\mbox{dim}(L)\geq g-m+(e-k-2)\geq g-m$, unless $k=e-1$, when $\mbox{dim }p_{1}(L)\geq g-m-1$. In the first case it follows that there exists a point $(x_{4},\ldots,x_{g},p_{1},\ldots,p_{e})\in f(U)\cap\Delta_{k}$ such that at least $g-m$ of the points $x_{4},\ldots,x_{g}$ are equal to a fixed point $r\in{\textbf{P}}^{1}-\{p_{1},\ldots,p_{e}\}$. In the second case, that is, when $k=e-1$, since $\#\{p_{i}\}_{i=1}^{e}=2$, one of the points $0,1$ or $\infty$, say [math], does not appear among the $p_{i}$’s. Then we can find a point $(x_{4},\ldots,x_{g},p_{1},\ldots,p_{e})\in f(U)\cap\Delta_{e-1}$ with at least $g-m$ of the $x_{j}$’s equal to [math].

The conclusion in both cases is that there exists a point \bigl{(}[R,x_{1},\ldots,x_{g}],y_{1},\ldots,y_{e}\bigr{)}\in W\cap(\overline{\mathcal{M}}_{0,g}\times_{\overline{\mathcal{M}}_{g}}\overline{\mathcal{C}}_{g}^{e}) corresponding to the flag curve $\tilde{R}=R\cup_{x_{1}}E_{1}\cup\ldots\cup_{x_{g}}E_{g}$, such that the points $y_{1},\ldots,y_{e}$ lie on a connected subcurve $Y\subset\tilde{R}$ where $\#(Y\cap(\overline{\tilde{R}-Y}))\leq 1$ and $p_{a}(Y)\leq m\leq e$. ∎

Proof of Theorem 0.1. We choose $\tilde{R}=R\cup_{x_{1}}E_{1}\cup\ldots\cup_{x_{g}}E_{g}$ as above and denote by $Y\subset\tilde{R}$ a connected subcurve onto which the points $y_{1},\ldots,y_{e}$ specialize. We know that either (a)  $p_{a}(Y)=m\leq\mathrm{min}\{e,g\}$, or (b)  $y_{1}=\cdots=y_{e}\in R-\{x_{1},\ldots,x_{g}\}$.

We first deal with case (a) and dispose of (b) at the end using [EH2]. If $m<g$ we set $Z:=\overline{\tilde{R}-Y}$ and $\{p\}:=Y\cap Z$ and we denote by $Y^{\prime}$ and $Z^{\prime}$ the components of $Y$ and $Z$ respectively, containing the point $p$. When $m=g$, then necessarily $e\geq g$ and $Y:=\tilde{R},Z=\emptyset$ and $p\in\tilde{R}$ is a general (smooth) point. By assumption, $[\tilde{R},y_{1},\ldots,y_{e}]\in W$, hence there exists a proper flat morphism $\phi:\mathcal{X}\rightarrow B$ satisfying the following properties:

$\bullet$ $\mathcal{X}$ is a smooth surface, $B$ is a smooth affine curve, $0\in B$ is a point such that $\phi^{-1}(0)$ is a curve stably equivalent to $\tilde{R}$ and $X_{t}=\phi^{-1}(t)$ is a smooth projective curve of genus $g$ for $t\neq 0$. Moreover, there are $e$ sections $\sigma_{i}:B\rightarrow\mathcal{X}$ of $\phi$ satisfying the condition $\sigma_{i}(0)=y_{i}\in\phi^{-1}(0)_{reg}$ for all $1\leq i\leq e$.

$\bullet$ If $X_{\eta}:=\mathcal{X}-\phi^{-1}(0)$, then there exists a line bundle $L_{\eta}\in\mbox{Pic}(X_{\eta})$ of relative degree $d$ and a subvector bundle $V_{\eta}\subset\phi_{*}(L_{\eta})$ having rank $r+1$, such that for $t\neq 0$ we have that

[TABLE]

After possibly making a finite base change and resolving the resulting singularities, the pair $(L_{\eta},V_{\eta})$ induces a (refined) limit $\mathfrak{g}^{r}_{d}$ on $\tilde{R}$, which we denote by $\mathfrak{l}$. The vector bundle V_{\eta}\cap\phi_{*}\bigl{(}L_{\eta}\otimes\mathcal{O}_{X_{\eta}}(-\sum_{j=1}^{e}\sigma_{j}(B-\{0\}))\bigr{)} induces a limit linear series $\mathfrak{g}^{r-e+f}_{d-e}$ on $\phi^{-1}(0)$ which we denote by $\mathfrak{m}$. For a component $A$ of $\phi^{-1}(0)$, if $(L_{A},V_{A})\in G^{r}_{d}(A)$ denotes the $A$-aspect of $\mathfrak{l}$, then there exists a unique effective divisor $D_{A}\in A_{e}$ supported only at the points from $(A\cap\bigcup_{j=1}^{e}\sigma_{j}(B))\bigcup(A\cap\overline{\phi^{-1}(0)-A})$ such that the $A$-aspect of $\mathfrak{m}$ is of the form

[TABLE]

The collection $\mathfrak{m}_{Y}:=\{\mathfrak{m}_{A}\}_{A\subset Y}$ forms a limit $\mathfrak{g}^{r-e+f}_{d-e}$ on $Y$. We denote by $(a_{0}<\ldots<a_{r})$ the vanishing sequence of $\mathfrak{l}_{Y^{\prime}}$ at $p$, thus $\{a_{i}\}_{i=0}^{r}=\{\mbox{ord}_{p}(\sigma)\}_{\sigma\in V_{Y^{\prime}}}$ and we denote by $(b_{0}<\ldots<b_{r})$ the vanishing sequence $a^{\mathfrak{l}_{Z^{\prime}}}(p)$. By ordering the set $\{\mbox{ord}_{p}(\sigma)\}_{\sigma\in W_{Y^{\prime}}}$ we obtain a subsequence $(a_{i_{0}}<\ldots<a_{i_{r-e+f}})$ of $a^{\mathfrak{l}_{Y^{\prime}}}(p)$. When we order the entries in $\{a_{i}\}_{i=0}^{r}-\{a_{i_{k}}\}_{k=0}^{r-e+f}$ we obtain a new sequence $(a_{j_{0}}<a_{j_{1}}<\ldots<a_{j_{e-f-1}})$. Using Lemma 2.1, we find that there exists a limit linear series $\mathfrak{l}^{\prime}_{Y}$ of type $\mathfrak{g}^{e-f-1}_{d}$ on $Y$ with the property that $a^{\mathfrak{l}^{\prime}_{Y}}(p)=(a_{j_{0}},a_{j_{1}},\ldots,a_{j_{e-f-1}})$.

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

[TABLE]

since $\sum_{k=0}^{r-e+f}a_{i_{k}}+\sum_{k=0}^{e-f-1}a_{j_{k}}\geq{r+1\choose 2}$. Thus we obtain the same numerical conclusion as in the case $m<g$.

Assume now that we are in the case (b) when $y_{1}=\cdots=y_{e}\in R-\{x_{1},\ldots,x_{g}\}$. Then reasoning as above, we find a limit $\mathfrak{g}^{r}_{d}$ on $\tilde{R}$ having vanishing sequence at $y_{1}$ at least $(0,1,\ldots,e-f-1,e,e+1,\ldots,r+f-1,r+f)$. Using once more [EH2], Theorem 1.1, we obtain the inequality

[TABLE]

Using the semicontinuity of the dimension of the fibres, it follows that for a general curve $[C]\in\mathcal{M}_{g}$, if $\pi_{1}:\Sigma_{C}\rightarrow C_{e}$ is the first projection, then the minimal fibre dimension of $\pi_{1}$ cannot exceed the dimension of the space of pairs of limit linear series $\mathfrak{l}\supset\mathfrak{m}$ consisting of a $\mathfrak{g}^{r}_{d}\supset\mathfrak{g}_{d-e}^{r-e+f}$ on the flag curve $\phi^{-1}(0)$ such that $\mathfrak{m}=\mathfrak{l}(-D_{e})$, where $D_{e}$ is a degree $e$ effective divisor on $\phi^{-1}(0)$ with the property that $\mbox{supp}(D_{e})\subset Y\cap\phi^{-1}(0)_{reg}$. Since the map $(\mathfrak{l}\supset\mathfrak{m},\mathfrak{m}_{Y},\mathfrak{l}_{Y}^{\prime})\mapsto(\mathfrak{m}_{Y},\mathfrak{l}_{Y}^{\prime},\mathfrak{l}_{Z})\in\tilde{G}_{d-e}^{r-e+f}(Y)\times\tilde{G}_{d}^{e-f-1}(Y)\times\tilde{G}_{d}^{r}(Z)$ is injective, it follows that for a general divisor $D_{gen}\in\pi_{1}(\Sigma_{C})$ we have the estimate

[TABLE]

hence $\mbox{dim}(\Sigma_{C})=\mbox{dim }\pi_{1}^{-1}(D_{gen})+e-m\leq\rho(g,r,d)-f(r+1-e+f)+e$. This finishes the proof of Theorem 0.1. $\Box$

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 $\mathfrak{g}^{r}_{d}$ on a curve of compact type $[Y\cup_{p}Z]\in\overline{\mathcal{M}}_{g}$, where $(Y,p)$ and $(Z,p)$ are suitably general smooth pointed curves of genus $e$ and $g-e$ respectively. These limit $\mathfrak{g}^{r}_{d}$’s will carry a sublinear series $\mathfrak{g}_{d-e}^{r-e+f}=\mathfrak{g}^{r}_{d}(-D_{e})$, where $D_{e}$ is a degree $e$ effective divisor on $Y$. Like in the proof of Theorem 0.1, such $\mathfrak{g}^{r}_{d}$’s are determined by their $Z$-aspect and by a pair of linear series $(\mathfrak{g}_{d-e}^{r-e+f},\mathfrak{g}_{d}^{e-f-1})$ on $Y$. We determine the dimension of the space of such pairs, which will enable us to show that the original pair $(\mathfrak{g}_{d-e}^{r-e+f},\mathfrak{g}_{d}^{e-f-1})$ on $Y\cup_{p}Z$ 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 $[Y,p]\in\mathcal{M}_{e,1}$ and $[Z,p]\in\mathcal{M}_{g-e,1}$ 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 $\overline{\alpha}:0\leq\alpha_{0}\leq\ldots\leq\alpha_{r}\leq d-r$ is a Schubert index of type $(r,d)$, then $(Y,p)$ possesses a $\mathfrak{g}^{r}_{d}$ with ramification sequence $\geq\overline{\alpha}$ at the point $p$, if and only if

[TABLE]

In case this inequality is satisfied, then $\mbox{dim }G^{r}_{d}(Y,p,\overline{\alpha})=\rho(g,r,d,\overline{\alpha})$ (One obviously has a similar statement for $[Z,p]$).

We denote by $\pi:\mathcal{X}\rightarrow(T,0)$ the versal deformation space of the stable curve $\pi^{-1}(0)=X_{0}:=Y\cup_{p}Z$. Let $\Delta\subset T$ be the boundary divisor corresponding to singular curves, and we write $\pi^{-1}(\Delta)=\Delta_{e}+\Delta_{g-e}$, where $\Delta_{e}$ (resp. $\Delta_{g-e}$) 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 $\mathcal{U}:=(\mathcal{X}-\Delta_{g-e})\times_{T}\cdots\times_{T}(\mathcal{X}-\Delta_{g-e})$, the projection $\phi:\mathcal{U}\rightarrow T$ and the induced curve $p_{2}:\mathcal{X}\times_{T}\mathcal{U}\rightarrow\mathcal{U}$. Then we introduce the stack of limit linear series of type $\mathfrak{g}^{r}_{d}$ over $\mathcal{U}$

[TABLE]

and we write $\tau:=\phi\circ\sigma:\widetilde{\mathfrak{G}}_{d}^{r}(p_{2})\rightarrow T$ (see [EH1] Theorem 3.4, for details on the construction of $\widetilde{\mathfrak{G}}^{r}_{d}(\pi)$). The fibre $\tau^{-1}(t)$ corresponding to a point $t\in\Delta$ (in which case one can write $\pi^{-1}(t)=Y_{t}\cup Z_{t}$, with $g(Y_{t})=e,g(Z_{t})=g-e$), parameterizes limit $\mathfrak{g}^{r}_{d}$’s on $Y_{t}\cup Z_{t}$ together with $e$-tuples $(x_{1},\ldots,x_{e})\in(Y_{t}-Y_{t}\cap Z_{t})^{e}$. Let us denote by $\mathcal{L}_{Y}$ a degree $d$ Poincaré bundle on $\pi_{2}:\mathcal{X}\times_{T}\widetilde{\mathfrak{G}}_{d}^{r}(p_{2})\rightarrow\widetilde{\mathfrak{G}}_{d}^{r}(p_{2})$ characterized by the property that its restriction to curves of type $Y_{t}\cup Z_{t}$ are line bundles of bidegree $(d,0)$. We also write $\mathcal{V}_{Y}\subset(\pi_{2})_{*}(\mathcal{L}_{Y})$ for the rank $r+1$ tautological bundle whose fibres correspond to the global sections of the genus $e$-aspect of each limit $\mathfrak{g}^{r}_{d}$. Finally, for $1\leq j\leq e$, we denote by $D_{j}\subset\mathcal{X}\times_{T}\widetilde{\mathfrak{G}}^{r}_{d}(p_{2})$ the diagonal divisor corresponding to pulling back the diagonal under the map $\mathcal{X}\times_{T}\widetilde{\mathfrak{G}}^{r}_{d}(p_{2})\rightarrow\mathcal{X}\times_{T}\mathcal{X}$ which projects onto the $j$-th factor, that is, $(x,l,x_{1},\ldots,x_{e})\mapsto(x,x_{j})$ where $x,x_{1},\ldots,x_{e}\in\pi^{-1}(t)$. There exists an evaluation vector bundle morphism over $\widetilde{\mathfrak{G}}^{r}_{d}(p_{2})$

[TABLE]

and we denote by $\mathcal{H}$ the rank $e-f$ degeneracy locus of the map $\chi$. Set-theoretically, $\mathcal{H}$ consists of those points $(t,l,x_{1},\ldots,x_{e})$ with $\phi(x_{1},\ldots,x_{e})=t\in T$ and $l\in\widetilde{G}^{r}_{d}(\pi^{-1}(t))$, satisfying the condition that $\mbox{dim }l(-x_{1}-\cdots-x_{e})\geq r+1-e+f$. The dimension of every irreducible component of $\mathcal{H}$ is at least $\rho(g,r,d)+\mbox{dim }T+e-f(r+1-e+f)$.

In order to show that $\tau:\mathcal{H}\rightarrow T$ is dominant, it suffices to prove that $\tau^{-1}(0)$ has at least one irreducible component of dimension $\rho(g,r,d)+e-f(r+1-e+f)$. This in fact will prove the stronger statement that $\Sigma_{C}\neq\emptyset$ for every $[C]\in\mathcal{M}_{g}$. Indeed, even though $\tau:\widetilde{\mathfrak{G}}^{r}_{d}(p_{2})\rightarrow T$ is not a proper morphism, the restriction $\tau_{\tau^{-1}(T-\Delta)}:\tau^{-1}(T-\Delta)\rightarrow T-\Delta$ is proper, hence there exists an irreducible component of $\mathcal{H}$ which maps onto $T-\Delta$. Since $\pi:\mathcal{X}\rightarrow(T,0)$ can be chosen in such a way that there exists a point $t\in T$ with $\pi^{-1}(t)\cong C$, this proves our contention. We set the integer

[TABLE]

thus we can write $\rho(e,r-e+f,d-e)=\alpha_{0}\cdot(r+1-e+f)+c$, where $0\leq c\leq r-e+f$. Then there exists a unique Schubert index of type $(r-e+f,d-e)$,

[TABLE]

with $\alpha_{r-e+f}-\alpha_{0}\leq 1$, such that $\sum_{j=0}^{r-e+f}\alpha_{j}=\rho(e,r-e+f,d-e)$. We have that $\alpha_{j}=\alpha_{0}$ for $0\leq j\leq r-e+f-c$ and $\alpha_{j}=\alpha_{0}+1$ for $r-e+f-c+1\leq j\leq r-e+f$. Note that since $\alpha_{0}+g(Y)-(d-e)+r-e+f=[e/(r+1-e+f)]\geq 0$, condition (17) is verified and the variety $G^{r-e+f}_{d-e}(Y,p,\overline{\alpha})$ is non-empty of dimension $\rho(e,r-e+f,d-e)-\sum_{j=0}^{r-e+f}\alpha_{j}=0$.

Next we set $\beta_{0}:=[e/(e-f)]$ and write  $e=\beta_{0}\cdot(e-f)+\tilde{c}$, where $0\leq\tilde{c}\leq e-f-1.$ Then there exists a unique Schubert index of type $(e-f-1,2e-f-1)$

[TABLE]

such that $\beta_{e-f+1}-\beta_{0}\leq 1$ and $\sum_{j=0}^{e-f-1}\beta_{j}=e$. Precisely, $\beta_{j}=\beta_{0}$ for $0\leq j\leq e-f-\tilde{c}-1$ and $\beta_{j}=\beta_{0}+1$ for $e-f-\tilde{c}\leq j\leq e-f-1$. By (17), the variety $G^{e-f-1}_{2e-f-1}(Y,p,\overline{\beta})$ is non-empty and of dimension $e-\sum_{j=0}^{e-f-1}\beta_{j}=0$.

First we are going to prove Theorem 0.5 under the assumption that there exist two linear series $(A,W_{A})\in G^{r-e+f}_{d-e}(Y,p,\overline{\alpha})$ and $(L,W_{L})\in G^{e-f-1}_{2e-f-1}(Y,p,\overline{\beta})$ satisfying the condition

[TABLE]

Note that \mbox{deg}\bigl{(}L\otimes A^{\vee}\otimes\mathcal{O}_{Y}((d+f-2e)\cdot p\bigr{)}=g(Y)-1, and (18) states that a suitable translate of at least one of the finitely many line bundles of type $L\otimes A^{\vee}$ 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 $(A,W_{A})\in G_{d-e}^{r-e+f}(Y,p,\overline{\alpha})$ and $(L,W_{L})\in G_{2e-f-1}^{e-f-1}(Y,p,\overline{\beta})$. For instance, in the case $e=2r-2$ and $f=r-1$, corresponding to $(2r-2)$-secant $(r-2)$-planes which every curve $Y\subset{\textbf{P}}^{r}$ is expected to possess in finite number, we obtain that $A=B\otimes\mathcal{O}_{Y}((d-3r+2)\cdot p)$, where $B\in W^{1}_{r}(Y)$ and $L\otimes\mathcal{O}_{Y}(-2p)\in W^{r-2}_{3r-6}(Y)$. By Riemann-Roch, we can write that $L=K_{Y}\otimes\mathcal{O}_{Y}(2\cdot p)\otimes\tilde{B}^{\vee}$, where $\tilde{B}\in W^{1}_{r}(Y)$ and then (18) translates into the vanishing statement $H^{0}(Y,B\otimes\tilde{B}\otimes\mathcal{O}_{Y}(-3\cdot p))=0$. The curve $Y$ has $\frac{(2r-2)!}{r!(r-1)!}$ pencils $\mathfrak{g}^{1}_{r}$. If we choose $B\neq\tilde{B}\in W^{1}_{r}(Y)$, then $h^{0}(Y,B\otimes\tilde{B})\geq 4$ and (18) has no chance of being satisfied. If $B=\tilde{B}$, then the Gieseker-Petri theorem implies that the map $H^{0}(Y,B)\otimes H^{0}(Y,K_{Y}\otimes B^{\vee})\rightarrow H^{0}(Y,K_{Y})$ is an isomorphism, whence $h^{0}(Y,B^{\otimes 2})=3$. Choosing $p\in Y$ outside the set of ramification points of the finitely many line bundles $B^{\otimes 2}$ where $B\in W^{1}_{r}(Y)$, we obtain that $H^{0}(B^{\otimes 2}\otimes\mathcal{O}_{Y}(-3\cdot p))=0$. Therefore in this case, condition $(\ref{assumption2})$ 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}\bigl{(}Y,L\otimes A^{\vee}\otimes\mathcal{O}_{Y}((d+f-2e+1)\cdot p)\bigr{)}=1. Assuming that $(A,W_{A})\in G^{r-e+f}_{d-e}(Y,p,\overline{\alpha})$ and $(L,W_{L})\in G^{e-f-1}_{2e-f-1}(Y,p,\overline{\beta})$ satisfy (18), it follows from Riemann-Roch that there exists a unique effective divisor of degree $e$

[TABLE]

and moreover $p\notin\mbox{supp}(D)$. We introduce the space of sections

[TABLE]

[TABLE]

We claim that $\mbox{dim}(V_{Y})=r+1$, hence $\mathfrak{l}_{Y}=(L\otimes\mathcal{O}_{Y}((d-2e+f+1)\cdot p),V_{Y})\in G^{r}_{d}(Y)$. Moreover, $\mathfrak{l}_{Y}$ has the following vanishing sequence at $p$:

[TABLE]

Indeed, our original assumption $f(r+1-e+f)\geq e$ is equivalent with the inequality $\alpha_{r-e+f}+r-e+f<d-2e+f+1$, which shows that the sequence (19) contains $r+1$ distinct entries. Since $p\notin\mbox{supp}(D)$, we obtain that the vanishing orders of the sections from $W_{A}\subset H^{0}(L\otimes\mathcal{O}_{Y}((d-2e+f+1)\cdot p))$ are precisely

[TABLE]

while those of the sections from $W_{L}\subset H^{0}(L\otimes\mathcal{O}_{Y}((d-2e+f+1)\cdot p))$ are precisely

[TABLE]

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

Next we choose a linear series $\mathfrak{l}_{Z}\in G^{r}_{d}(Z,p)$ such that $\{\mathfrak{l}_{Y},\mathfrak{l}_{Z}\}$ is a refined limit $\mathfrak{g}^{r}_{d}$. Then the ramification sequence of $\mathfrak{l}_{Z}$ at the point $p$ must be equal to

[TABLE]

We claim that condition (17) is satisfied for $Z$ and that the variety $G^{r}_{d}(Z,p,\overline{\gamma})$ is non-empty and of dimension $\rho(g-e,r,d,\overline{\gamma})=\rho(g,r,d)+e-f(r+1-e+f)$. For this to happen, one has to check that the following inequality holds:

[TABLE]

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

[TABLE]

hence $\alpha^{\mathfrak{l}_{Z}}_{j}(p)+g-e-d+r\geq 0$ for all $e-f\leq j\leq r$. Second, that since $0\leq\beta_{e-f-1}-\beta_{0}\leq 1$, in order to estimate the sum of the first $e-f$ terms in the sum (20), there are two cases to consider. Either $\alpha^{\mathfrak{l}_{Z}}_{0}(p)+g-e-d+r\geq 0$, in which case we find that

[TABLE]

[TABLE]

Else, if $\alpha^{\mathfrak{l}_{Z}}_{0}(p)+g-e-d+r\leq-1$, then also $\alpha^{\mathfrak{l}_{Z}}_{j}(p)+g-e-d+r\leq 0$ for $0\leq j\leq e-f-1$ and the left hand side of (20) equals

[TABLE]

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

Since the chosen $(A,W_{A})\in G^{r-e+f}_{d-e}(Y,p,\overline{\alpha})$ and $(L,W_{L})\in G^{e-f-1}_{2e-f-1}(Y,p,\overline{\beta})$ are isolated points in their corresponding varieties of linear series on $Y$, it follows that limit $\mathfrak{g}^{r}_{d}$’s on $X_{0}$ constructed in the way we just described, fill-up a component of $\tau^{-1}(0)\subset\mathcal{H}$.

Indeed, suppose $(\mathfrak{n}_{Y},\mathfrak{n}_{Z},\tilde{D})\in\mathcal{H}$ is a point lying in the same irreducible component of $\tau^{-1}(0)$ as $(\mathfrak{l}_{Y},\mathfrak{l}_{Z},D)$. Here, $\mathfrak{n}_{Y}\in G^{r}_{d}(Y),\ \mathfrak{n}_{Z}\in G^{r}_{d}(Z,p,\overline{\gamma})$ and $\tilde{D}\in Y_{e}$ is a divisor such that $p\notin\mbox{supp}(\tilde{D})$. Then $a^{\mathfrak{n}_{Y}}(p)=a^{\mathfrak{l}_{Y}}(p)$ which is given by (19), therefore $\mathfrak{n}_{Y}(-(d-2e+f+1)\cdot p)\in G^{2e-f-1}_{e-f-1}(Y,p,\overline{\beta})$ which is a reduced [math]-dimensional variety. This implies that $\mathfrak{n}_{Y}(-(d-2e+f+1)\cdot p)=(L,W_{L})$. Next, we consider the linear series $\mathfrak{n}_{Y}(-\tilde{D})\in G^{r-e+f}_{d-e}(Y)$. Since $p\notin\mbox{supp}(\tilde{D})$, the vanishing sequence of this linear series is a subsequence of length $r+1-e+f$ of $a^{\mathfrak{l}_{Y}}(p)$. Necessarily, $\alpha^{\mathfrak{n}_{Y}(-\tilde{D})}(p)\geq\overline{\alpha}$ and because $\rho(e,r-e+f,d-e,\overline{\alpha})=0$, we must have that $\mathfrak{n}_{Y}(-\tilde{D})\in G^{r-e+f}_{d-e}(Y,p,\overline{\alpha})$ which is a discrete set, hence $\mathfrak{n}_{Y}(-\tilde{D})=(A,W_{A})$ and $\tilde{D}=D\in Y_{e}$. This shows that $\mathfrak{n}_{Y}=\mathfrak{l}_{Y}$ and every point of this component of $\tau^{-1}(0)$ is determined by the $\mathfrak{n}_{Z}$. The dimension of this component is thus equal to

[TABLE]

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 $(A,W_{A})\in G^{r-e+f}_{d-e}(Y,p,\overline{\alpha})$ and assume that the following condition holds:

[TABLE]

There exists a unique divisor $D\in|\mathcal{O}_{Y}(d\cdot p)\otimes A^{\vee})|$ and (21) guarantees that $p\notin\mbox{supp}(D)$. We define the space of sections

[TABLE]

Reasoning along the same lines as in the previous case, since $p\notin\mbox{supp}(D)$ we find that $\mbox{dim}(V_{Y})=r+1$, hence $\mathfrak{l}_{Y}=(\mathcal{O}_{Y}(d\cdot p),V_{Y})\in G^{r}_{d}(Y)$. Moreover, we can check that

[TABLE]

Like in the previous situation, we choose a linear series $\mathfrak{l}_{Z}\in G^{r}_{d}(Z,p)$ such that $\{\mathfrak{l}_{Y},\mathfrak{l}_{Z}\}$ is a refined limit $\mathfrak{g}^{r}_{d}$. Thus we must have the following ramification sequence at $p$:

[TABLE]

Condition (17) which guarantees the existence of $\mathfrak{l}_{Z}$ is satisfied if and only if

[TABLE]

and

[TABLE]

Since we are always working under the hypothesis $\rho(g,r,d)-f(r+1-e+f)+e\geq 0$, we see that the previous condition holds whenever $g-d+r\geq e$, and that, in general, $\mathfrak{l}_{Z}\in G^{r}_{d}(Z,p,\overline{\gamma})$ exists if and only if

[TABLE]

Assuming (22), the variety $G^{r}_{d}(Z,p,\overline{\gamma})$ is non-empty of dimension $\rho(g-e,d,r,\overline{\gamma})=\rho(g,r,d)-f(r+1-e+f)+e$. The same argument as before shows that limit $\mathfrak{g}^{r}_{d}$’s on $X_{0}$ constructed in such a way, fill-up a component of $\tau^{-1}(0)\subset\mathcal{H}$ of expected dimension $\rho(g,r,d)-f(r+1-e+f)+e$, 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 $(A,W_{A})\in G_{d-e}^{r-e+f}(Y,p,\overline{\alpha})$ and $(L,W_{L})\in G_{2e-f-1}^{e-f-1}(Y,p,\overline{\beta})$ satisfying (18) whenever one of the following conditions is satisfied:

[TABLE]

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

We now treat case $(i)$ when $\beta_{0}=1$ and $\tilde{c}=f\leq e-f-1$. By Riemann-Roch we find that $L=K_{Y}\otimes\mathcal{O}_{Y}((e-2f+2)\cdot p)\otimes B^{\vee}$, where $B\in W^{1}_{e-f+1}(Y)$ is a pencil such that h^{0}\bigl{(}Y,B\otimes\mathcal{O}_{Y}(-(e-2f+1)\cdot p)\bigr{)}\geq 1 (There are finitely many such $B\in W^{1}_{e-f+1}(Y)$ for a generic choice of $[Y,p]\in\mathcal{M}_{e,1}$). Applying the base-point-free pencil trick, (18) is equivalent to the injectivity of the multiplication map

[TABLE]

where $M:=K_{Y}\otimes A^{\vee}\otimes\mathcal{O}_{Y}((d-f-e+2)\cdot p)\in W_{2e-f}^{e-f}(Y)$ is a complete linear series with vanishing sequence at $p$ equal to

[TABLE]

Here we have set $a:=[e/(r+1-e+f)]$, hence we can write $e=a\cdot(r+1-e+f)+c$, where $0\leq c\leq r-e+f$. By assumption we have that $e-2a>c$ and clearly $\rho(M,\alpha^{M}(p))=0$, that is, there are finitely many $M\in W^{e-f}_{2e-f}(Y)$ satisfying (23).

To prove that $\mu_{B,M}$ is injective, we degenerate $[Y,p]\in\mathcal{M}_{e,1}$ to a particular stable curve: $[Y_{0},p_{0}]:=[E_{0}\cup_{p_{1}}E_{1}\cup\ldots\cup E_{e-2a-1}\cup_{p_{e-2a}}T,p_{0}]$, where $E_{0},\ldots,E_{e-2a-1}$ are elliptic curves, $[T=E_{e-2a},p_{e-2a}]\in\mathcal{M}_{2a,1}$ is a Petri general smooth pointed curve and the points $p_{i},p_{i+1}\in E_{i}$ are such that $p_{i+1}-p_{i}\in\mbox{Pic}^{0}(E_{i})$ is not a torsion class for $0\leq i\leq e-2a-1$. Note that $p_{0}$ lies on the first component $E_{0}$. By contradiction, we assume that $\mu_{B,M}$ is not injective for every $[Y,p]\in\mathcal{M}_{e,1}$ and for each of the finitely many linear series $M\in W_{2e-f}^{e-f}(Y)$ satisfying (23) and each B\in G^{1}_{e-f+1}\bigl{(}Y,p,(0,e-2f)\bigr{)}.

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

[TABLE]

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

[TABLE]

For an index $c\leq i\leq e-2a-1$ which we write as $i=c+a\cdot\beta+j$, with $0\leq j\leq a-1$ and $0\leq\beta\leq r-2-e+f$, we choose $a^{\mathfrak{m}_{E_{i+1}}}(p_{i+1})$ to be obtained from $a^{\mathfrak{m}_{E_{i}}}(p_{i})$ by raising all entries by $1$, except for the term

[TABLE]

In this way $\mathfrak{m}\in\tilde{G}^{e-f}_{2e-f}(Y_{0})$ becomes a (refined) limit linear series which smooths to a complete linear series $M\in G^{e-f}_{2e-f}(Y)$ on every smooth pointed curve $[Y,p]\in\mathcal{M}_{e,1}$ such that the ramification condition (23) with respect to $p$ is satisfied.

Next we construct a limit $\mathfrak{g}^{1}_{e-f+1}$ on $[Y_{0},p_{0}]$, say $\mathfrak{b}=\{(B_{E_{i}},W_{i})\in G^{1}_{e-f+1}(E_{i})\}_{i=0}^{e-2a}$ such that $a^{\mathfrak{b}}(p_{0})=(0,e-2f+1)$. For $0\leq i\leq e-2f$ we set $a^{\mathfrak{b}_{E_{i}}}(p_{i})=(i,e-2f+1)$. For an index of type $i=e-2f+2k-1$ where $0\leq k\leq f-a$, we choose $a^{\mathfrak{b}_{E_{i}}}(p_{i})=(e-2f+k-1,e-2f+k+1)$. If $i=e-2f+2k$, we choose the sequence $a^{\mathfrak{b}_{E_{i}}}(p_{i})=(e-2f+k,e-2f+k+1)$. It is clear that each sequence $a^{\mathfrak{b}_{E_{i}}}(p_{i})$ is obtained from $a^{\mathfrak{b}_{E_{i-1}}}(p_{i-1})$ by raising one entry by $1$ while keeping the other fixed, hence $\mathfrak{b}$ is a limit $\mathfrak{g}^{1}_{e-f+1}$ which smooths to a pencil $B\in G^{1}_{e-f+1}(Y,p,(0,e-2f))$ on every nearby smooth curve $[Y,p]$. For each $0\leq i\leq e-2a-1$, there exists a section (unique up to scaling) $\sigma_{i}\in W_{i}$ such that $\mbox{ord}_{p_{i}}(\sigma_{i})+\mbox{ord}_{p_{i+1}}(\sigma_{i})=\mbox{deg}(B_{E_{i}})$. We denote by $\sigma_{i}^{c}\in W_{i}$ a complementary section such that $\{\mbox{ord}_{p_{i}}(\sigma_{i}),\mbox{ord}_{p_{i}}(\sigma_{i}^{c})\}=\{a_{0}^{\mathfrak{b}_{E_{i}}}(p_{i}),a_{1}^{\mathfrak{b}_{E_{i}}}(p_{i})\}$.

Using the set-up developed in [EH3] and [F2] for studying degenerations of multiplication maps, we find that the assumption that $\mu_{B,M}$ is not injective implies the existence elements $0\neq\rho_{i}\in\mathrm{Ker}\{W_{i}\otimes V_{i}\rightarrow H^{0}(E_{i},B_{E_{i}}\otimes M_{E_{i}})\}$ for each $0\leq i\leq e-2a$, satisfying the property that $\mbox{ord}_{p_{i+1}}(\rho_{i+1})\geq\mbox{ord}_{p_{i}}(\rho_{i})+1$, for all $i$ (see e.g. [F2] Section 4, for an explanation of how to obtain the $\rho_{i}$’s). Moreover, if $\mbox{ord}_{p_{i+1}}(\rho_{i+1})=\mbox{ord}_{p_{i}}(\rho_{i})+1$, then if $\tau_{i}\in V_{i}$ is the section (unique up to scaling) such that $\mbox{ord}_{p_{i}}(\tau_{i})+\mbox{ord}_{p_{i+1}}(\tau_{i})=\mbox{deg}(M_{E_{i}})$, then we must have that

[TABLE]

where $\tau_{i}^{\prime}\in V_{i}$ is another section such that $\mbox{ord}_{p_{i}}(\tau_{i}^{\prime})\neq\mbox{ord}_{p_{i}}(\tau_{i})$. In particular, since we have explicitly described all the sequences $a^{\mathfrak{b}_{E_{i}}}(p_{i})$ and $a^{\mathfrak{m}_{E_{i}}}(p_{i})$, the assumption that $\mbox{ord}_{p_{i+1}}(\rho_{i+1})\leq\mbox{ord}_{p_{i}}(\rho_{i})+1$ uniquely determines $\mbox{ord}_{p_{i}}(\rho_{i})$.

Since $a^{\mathfrak{b}_{E_{0}}}(p_{0})=(0,e-2f+1)$ and $\mu_{B_{E_{0}},M_{E_{0}}}(\rho_{0})=0$, the non-zero section $\rho_{0}$ must involve both sections $\sigma_{0}$ and $\sigma_{0}^{c}$ and then clearly $\mbox{ord}_{p_{0}}(\rho_{0})\geq e-2f+1$. We prove inductively that for all integers $0\leq i\leq e-2a$ we have the inequality

[TABLE]

Assuming (24) for $i\leq e-2a-1$, since $\mbox{ord}_{p_{i+1}}(\rho_{i+1})\geq\mbox{ord}_{p_{i}}(\rho_{i})+1$, the only way (24) can fail for $i+1$ is when $\mbox{ord}_{p_{i}}(\rho_{i})=e-2f+2i+1$ and $\mbox{ord}_{p_{i+1}}(\rho_{i+1})=\mbox{ord}_{p_{i}}(\rho_{i})+1$. As explained above, this implies that $\mbox{ord}_{p_{i}}(\rho_{i})=\mbox{ord}_{p_{i}}(\tau_{i})+\mbox{ord}_{p_{i}}(\sigma_{i}^{c})$.

Writing $i=c+a\cdot\beta+j$ as above, then $\mbox{ord}_{p_{i}}(\tau_{i})=r-a+2+c+(a-1)\cdot\beta+2j$ if $i\geq c$, while $\mbox{ord}_{p_{i}}(\tau_{i})=e-f-a+c$, for $0\leq i\leq c-1$. We deal only with the case $i\geq c$, the case $0\leq i\leq c-1$ being analogous. To determine $\mbox{ord}_{p_{i}}(\sigma_{i}^{c})$ we must distinguish between two cases: When $i=e-2f+2k-1$ with $k\geq 1$, then $\mbox{ord}_{p_{i}}(\sigma_{i}^{c})=e-2f+k-1$. Otherwise, we write $i=e-2f+2k$ in which case $\mbox{ord}_{p_{i}}(\sigma_{i}^{c})=e-2f+k+1$. Suppose we are in the former case. Then we obtain the equality

[TABLE]

which ultimately leads to the relation $(a+2)(r-e+f-\beta)=a-j-1$. But $j\leq a-1$ and $\beta\leq r-e+f-1$, 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 $\mbox{ord}_{p_{e-2a}}(\rho_{e-2a})\geq e-2f+1+2(e-2a)$. We note that on the curve $[T,q]=[E_{e-2a},p_{e-2a}]$ we have that $a^{\mathfrak{b}_{T}}(p_{e-2a})=(e-f-a,e-f-a+1)$, while

[TABLE]

Equivalently $\mathfrak{b}_{T}=|B|+(e-f-a)\cdot q$, where $B\in W_{a+1}^{1}(T)$, while $\mathfrak{m}_{T}=(e-2a)\cdot q+|N|$, where $N\in\mbox{Pic}^{e-f+2a}(T)$ has the property that h^{0}\bigl{(}T,N(-(e-f-a+3)\cdot q)\bigr{)}\geq a. Remembering that $\mbox{ord}_{q}(\rho_{e-2a})\geq(e-2f+1)+2(e-2a)$, after subtracting the base locus supported at $q$, we find an element

[TABLE]

such that $\mbox{ord}_{q}(\rho_{T})\geq e-f-a+1$. Equivalently, the multiplication map

[TABLE]

is not injective. By using Riemann-Roch we find that $N(-(e-f-a+3)\cdot q)=K_{T}\otimes\tilde{B}^{\vee}$, where $\tilde{B}\in W^{1}_{a+1}(T)$. Choosing $\tilde{B}=B\in W^{1}_{a+1}(T)$, we notice that $\mu_{B,N}$ can be identified with the Petri map $H^{0}(B)\otimes H^{0}(K_{T}\otimes B^{\vee})\rightarrow H^{0}(K_{T})$ which is injective because $[T]\in\mathcal{M}_{2a}$ 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 $e\leq r-e+f$, that is, when $\alpha_{0}=d-r-f-e$, we can easily determine a linear series $(A,W_{A})\in G^{r-e+f}_{d-e}(Y,p,\overline{\alpha})$. Precisely, one can see that $A=K_{Y}\otimes\mathcal{O}_{Y}((d-3e+2)\cdot p)$ and

[TABLE]

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

[TABLE]

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\in G^{1}_{e-c+1}\bigl{(}Y,p,(0,r-e+f-c+1)\bigr{)} and a linear series $L\in G_{2e-f-1}^{e-f-1}(Y,p,\overline{\beta})$, such that the multiplication map

[TABLE]

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

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 $[Y,p]\in\mathcal{M}_{e,1}$ be a general pointed curve. Then there exists a linear series $(A,W_{A})\in G^{r-e+f}_{d-e}(Y,p,\overline{\alpha})$ such that H^{0}\bigl{(}Y,\mathcal{O}_{Y}((d-1)\cdot p\otimes A^{\vee})\bigr{)}=0.

Proof.

By contradiction, we assume that $H^{0}(\mathcal{O}_{Y}((d-1)\cdot p)\otimes A^{\vee})\neq 0$ for every $[Y,p]\in\mathcal{M}_{e,1}$ and for every linear series $(A,V_{A})\in G^{r-e+f}_{d-e}(Y,p,\overline{\alpha})$. We let $[Y,p]$ degenerate to the stable curve $[Y_{0}:=E_{0}\cup_{p_{1}}E_{1}\cup_{p_{2}}\ldots\cup_{p_{e-3}}E_{e-3}\cup_{p_{e-2}}B,p_{0}]$, where $E_{0},\ldots,E_{e-3}$ are elliptic curves, the points $p_{i},p_{i+1}\in E_{i}$ are such that $p_{i}-p_{i+1}\in\mbox{Pic}^{0}(E_{i})$ is not a torsion class, and $[B,p_{e-2}]\in\mathcal{M}_{2,1}$ is such that $p_{e-2}\in B$ is not a Weierstrass point. For all integers $0\leq i\leq e-3$ we find that there exist sections

[TABLE]

such that

[TABLE]

Moreover, we have that $\mbox{ord}_{p_{i}}(\tau_{i})\geq i$ for $0\leq i\leq e-2$. In particular, $\mbox{ord}_{p_{e-2}}(\tau_{B})\geq e-2$. Since $\rho(e,r-e+f,d-e,\overline{\alpha})=0$, limit $\mathfrak{g}^{r-e+f}_{d-e}$ on $E_{0}\cup\ldots\cup E_{e-3}\cup B$ are smoothable to every curve of genus $g$. These finitely many limit $\mathfrak{g}^{r-e+f}_{d-e}$ are in bijective correspondence with possibilities of choosing the vanishing sequences $\{a^{l_{E_{i}}}(p_{i})\}_{0\leq i\leq e-3}$ and $a^{l_{B}}(p_{e-2})$ in such a way that for all $0\leq i\leq e-3$, 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 $\mathfrak{g}_{d-e}^{r-e+f}$ on $E_{0}\cup\ldots\cup E_{e-3}\cup B$ having the property that if $(A_{B},V_{B})$ denotes its $B$-aspect, then $H^{0}(\mathcal{O}_{B}((d-e+1)\cdot p_{e-2})\otimes A_{B}^{\vee})=0$.

We describe such a $\mathfrak{g}^{r-e+f}_{d-e}$ explicitly by specifying the sequences $\{\alpha^{l_{E_{i}}}(p_{i})\}_{0\leq i\leq e-3}$ and $\alpha^{l_{B}}(p_{e-2})$. Clearly, $\alpha^{l_{E_{0}}}(p_{0})$ equals $(\alpha_{0},\ldots,\alpha_{0},\alpha^{l_{E_{0}}}_{r-e+f+1-c}(p_{0})=\alpha_{0}+1,\ldots,\alpha_{0}+1)$. For $1\leq i\leq c$, $\alpha^{l_{E_{i}}}(p_{i})$ is obtained from $\alpha^{l_{E_{i-1}}}(p_{i-1})$ by increasing all entries by $1$, except for $\alpha^{l_{E_{i}}}_{r-e+f+i-c}(p_{i})=\alpha^{l_{E_{i-1}}}_{r-e+f+i-c}(p_{i-1})$. Thus $\alpha^{l_{E_{c}}}(p_{c})=(\alpha_{0}+c,\ldots,\alpha_{0}+c)$. Next, for an index $i$ such that $c+\beta(r+1-e+f)<i\leq c+(\beta+1)(r+1-e+f)$, where $0\leq\beta\leq[e/(r+1-e+f)]$, if we write $i\equiv j+c\mbox{ mod }r+1-e+f$, with $1\leq j\leq r-e+f$, the sequence $\alpha^{l_{E_{i}}}(p_{i})$ is obtained from $\alpha^{l_{E_{i-1}}}(p_{i-1})$ by raising all entries by $1$, except for $\alpha^{l_{E_{i}}}_{j-1}(p_{i})=\alpha^{l_{E_{i-1}}}_{j-1}(p_{i-1})$. Switching from ramification to vanishing sequences we obtain

[TABLE]

that is, $A_{B}=\mathcal{O}_{B}((d-e-2)\cdot p_{e-2})\otimes\mathfrak{g}^{1}_{2}$, and then

[TABLE]

This contradicts the fact $\mbox{ord}_{p_{e-2}}(\tau_{B})\geq e-2$ 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 $n\geq 1$ we denote by $[n]_{C}:\mbox{Pic}^{d}(C)\rightarrow\mbox{Pic}^{nd}(C)$ the multiplication by $n$ map, $[n]_{C}(L):=L^{\otimes n}$. It is an immediate consequence of Riemann-Roch that for a general $L\in\mbox{Pic}^{d}(C)$, we have that $h^{0}(L^{\otimes n})=\mbox{max}\{nd+1-g,0\}$.

First we show that for a very general $L\in\mbox{Pic}^{d}(C)$ we have that $w^{L^{\otimes n}}(p)\leq 1$ for all $p\in C$ and $n\geq 1$. Indeed, let us assume that $w^{L^{\otimes n}}(p)\geq 2$, where $n$ is chosen such that $nd\geq g$, so that $h^{0}(C,L^{\otimes n})=nd+1-g$. Then there are two possibilities:

[TABLE]

In case $(i)$ we consider the map C\times C_{g-2}\rightarrow\mbox{Pic}^{nd}(C),\ (p,E)\mapsto\mathcal{O}_{C}\bigl{(}(nd+2-g)\cdot p+E\bigr{)} and we denote by $\Sigma_{n}$ its image which is a divisor on $\mbox{Pic}^{nd}(C)$. Then $(i)$ is equivalent to $L\in[n]_{C}^{*}(\Sigma_{n})$ which is a divisorial condition on $\mbox{Pic}^{d}(C)$ for each $n$.

In case $(ii)$ we look at the map C\times C^{1}_{g}\rightarrow\mbox{Pic}^{nd}(C),(p,E)\mapsto\mathcal{O}_{C}\bigl{(}(nd-g)\cdot p+E\bigr{)} and we denote by $V_{n}$ its image. Since $C^{1}_{g}$ is generically a ${\textbf{P}}^{1}$-bundle over $C_{g-2}$, it follows that $V_{n}$ is a divisor on $\mbox{Pic}^{nd}(C)$ and then possibility $(ii)$ is equivalent to $L\in[n]_{C}^{*}(V_{n})$. Thus we see that for $L\in\mbox{Pic}^{d}(C)-\bigcup_{n\geq 1}[n]_{C}^{*}(\Sigma_{n}+V_{n})$ all the ramification points of all powers $L^{\otimes n}$ with $n\geq 1$, 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 $p\in C$ and integers $n$ and $d$ such that $nd\geq g$. Then the locus

[TABLE]

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

Proof.

We set $a:=\mbox{max}\{0,2g-1-nd\}$ and define two vector bundles $\mathcal{E}_{n}$ and $\mathcal{F}_{n}$ on $\mbox{Pic}^{d}(C)$ of the same rank and having fibres $\mathcal{E}_{n}(L)=H^{0}(C,L^{\otimes n}\otimes\mathcal{O}_{C}(a\cdot p))$ and $\mathcal{F}_{n}(L)=H^{0}(C,L^{\otimes n}\otimes\mathcal{O}_{(a+nd+1-g)\cdot p}(a\cdot p))$ over each point $L\in\mbox{Pic}^{d}(C)$. Then $D_{n}$ is the degeneracy locus of the morphism $\mathcal{E}_{n}\rightarrow\mathcal{F}_{n}$ obtained by evaluation sections of $L^{\otimes n}\otimes\mathcal{O}_{C}(a\cdot p)$ along $(a+nd+1-g)\cdot p$. The Picard bundle $\mathcal{E}_{n}$ is negative (i.e. $\mathcal{E}_{n}^{\vee}$ is ample), because $\mathcal{E}_{n}$ is the pull-back under the finite map $[n]_{C}$ of a negative bundle on $\mbox{Pic}^{d}(C)$ (cf. [ACGH], pg. 310). Moreover, $\mathcal{F}_{n}$ is algebraically equivalent to a trivial bundle, hence $\mathcal{E}_{n}^{\vee}\otimes\mathcal{F}_{n}$ 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, $[D_{n}]=c_{1}(\mathcal{F}_{n}-\mathcal{E}_{n})=[n]_{C}^{*}(\theta)=n^{2}\theta$. ∎

End of the proof of Theorem 0.10. We fix integers $1\leq a<b$ and consider the variety $\Sigma_{ab}:=\{(p,L)\in C\times\mbox{Pic}^{d}(C):p\in R(L^{\otimes a})\cap R(L^{\otimes b})\}$ and we denote by $\phi_{1}:\Sigma_{ab}\rightarrow C$ and $\phi_{2}:\Sigma_{ab}\rightarrow\mbox{Pic}^{d}(C)$ the two projections. For a fixed $p\in C$, the fibre $\phi_{1}^{-1}(p)$ is identified with the intersection of the two irreducible divisors $D_{a}$ and $D_{b}$. Since $[D_{a}]\neq[D_{b}]$ for $a\neq b$, it follows that $D_{a}\cap D_{b}$ is of pure codimension $2$ inside $\mbox{Pic}^{d}(C)$, therefore $\mbox{dim}(\Sigma_{ab})=g-1$. We obtain that a line bundle $L\in\mbox{Pic}^{d}(C)-\bigcup_{a<b}\phi_{2}(\Sigma_{ab})$ will enjoy the property that $R(L^{\otimes a})\cap R(L^{\otimes b})=\emptyset$ for $a<b$. $\Box$