Lower bounds for Clifford indices in rank three

TL;DR

This paper establishes lower bounds for the Clifford indices of rank 3 vector bundles on algebraic curves and demonstrates the existence of non-generated bundles computing these indices on certain plane curves.

Contribution

It provides new lower bounds for rank 3 Clifford indices and shows the existence of non-generated bundles computing these indices on smooth plane curves of degree at least 10.

Findings

Lower bounds for Clifford indices in rank 3 bundles are established.

Existence of non-generated bundles computing Clifford indices on plane curves of degree ≥10.

Results extend understanding of vector bundle properties on algebraic curves.

Abstract

Clifford indices for semistable vector bundles on a smooth projective curve of genus at least 4 were defined in previous papers of the authors. In the present paper, we establish lower bounds for the Clifford indices for rank 3 bundles. As a consequence we show that, on smooth plane curves of degree at least 10, there exist non-generated bundles of rank 3 computing one of the Clifford indices.

Equations

$$\gamma (E):=\frac{1}{n}\left(d-2(h^0(E)-n)\right)=\mu (E)-2\frac{h^0(E)}{n}+2,$$

$$\gamma_{n}:=\min_{E}\left\{\gamma(E)\left|\begin{array}[]{c}E\;\mbox{semistable of rank}\;n\\ h^{0}(E)\geq n+1,\;\mu(E)\leq g-1\end{array}\right\}\right.$$

$$\gamma_{n}^{\prime}:=\min_{E}\left\{\gamma(E)\;\left|\begin{array}[]{c}E\;\mbox{semistable of rank}\;n\\ h^{0}(E)\geq 2n,\;\mu(E)\leq g-1\end{array}\right\}.\right.$$

$$d_r:=\min \{d_L|L\text{mbox}alinebundleonC\text{mbox}with{h}^0(L)\ge r+1\}.$$

$${\gamma }_3^{\prime }\ge \min \left\{\frac{d_9}{3}-2,\frac{2{\gamma }_2^{\prime }+1}{3}\right\}.$$

$$d_r<d_{r+1}\text{mbox}forallr;d_r\ge \min \{{\gamma }_1+2r,g+r-1\}$$

$$\gamma (E)\ge \frac{d_9}{3}-2.$$

$$\gamma (E)\ge \frac{d_9}{3}+s-3-\frac{2s}{3}\ge \frac{d_9}{3}-2.$$

$$\gamma (E)\ge \min \left\{\frac{2{\gamma }_1+1}{3},\frac{1}{3}\left(4{\gamma }_1+2g+2-d\right)\right\}.$$

$$\gamma (E)\ge d_t-\frac{2s}{3}\ge {\gamma }_1$$

$$d_{E/F}\ge \underset{1\le u\le v-1}{\min }\left\{d_{2v},d_v+\frac{d}{3},d_u+d_{v-u}\right\}.$$

$$d\ge d_t+d_{2v}\ge {\gamma }_1+2t+d_6+2v-6\ge {\gamma }_1+2s+d_6-6,$$

$$\gamma (E)\ge \frac{{\gamma }_1}{3}+\frac{d_6}{3}-2\ge \frac{{\gamma }_1}{3}+\frac{{\gamma }_1+7}{3}-2=\frac{2{\gamma }_1+1}{3}.$$

$$2\gamma (E)\ge 2{\gamma }_1+2t+2v-\frac{4s}{3}\ge 2{\gamma }_1+\frac{2s}{3}\ge 2{\gamma }_1+2.$$

$$\frac{2d}{3}\ge d_t+d_v\ge {\gamma }_1+2t+g+v-1,$$

$$2\gamma (E)$$

$$d\ge 3{\gamma }_1+2t+2u+2(v-u)\ge 3{\gamma }_1+2s.$$

$$d\ge 2{\gamma }_1+2t+g+u-1+2(v-u)\ge 2{\gamma }_1+s+t+v-u+g-1$$

$$\gamma (E)$$

$$\frac{1}{2}\gamma (E)$$

$$\gamma (E)\ge \frac{4{\gamma }_1}{3}+\frac{2g-2-d}{3}+\frac{2}{3}(t+v-u)\ge \frac{1}{3}(4{\gamma }_1+2g+2-d).$$

$$d\ge {\gamma }_1+2t+g+u-1+g+v-u-1\ge {\gamma }_1+s+t+2g-2.$$

$$\gamma (E)\ge \frac{{\gamma }_1}{3}+\frac{t+2g-2}{3}-\frac{s}{3}=\frac{{\gamma }_1}{3}+\frac{t+2g-2}{3}-\frac{1}{2}\left(\frac{d}{3}-\gamma (E)\right)$$

$$\frac{\gamma (E)}{2}$$

$$\gamma (E)\ge \frac{2{\gamma }_1+1}{3}.$$

$$\gamma (E)\ge \min \left\{{\gamma }_2^{\prime },\frac{2{\gamma }_1+1}{3},\frac{1}{3}(2{\gamma }_1+2g+4-d)\right\}.$$

$$1\le t\le s.$$

$$d_F\ge d_{2t}.$$

$$\gamma (E)>{\gamma }_2^{\prime }.$$

$$d_{E/F}\ge d_u.$$

Full text

Both authors are members of the research group VBAC (Vector Bundles on Algebraic Curves). The second author would like to thank the Department Mathematik der Universität Erlangen-Nürnberg for its hospitality

1. Introduction

Let $C$ be a smooth projective curve of genus $g\geq 4$ defined over an algebraically closed field of characteristic zero. In [4], we generalised the classical Clifford index $\gamma_{1}$ of $C$ to semistable bundles in the following way. First we define, for any vector bundle $E$ of rank $n$ and degree $d$,

[TABLE]

where $\mu(E)=\frac{d}{n}$. Then the Clifford indices $\gamma_{n}$ and $\gamma_{n}^{\prime}$ are defined by

[TABLE]

and

[TABLE]

We say that a bundle $E$ contributes to $\gamma_{n}$ if it is semistable of rank $n$ with $\mu(E)\leq g-1$ and $h^{0}(E)\geq n+1$ and that $E$ computes $\gamma_{n}$ if in addition $\gamma(E)=\gamma_{n}$. Similar definitions are made for $\gamma^{\prime}_{n}$.

In [4] we obtained a number of results for the Clifford indices, including a useful lower bound for $\gamma_{2}^{\prime}$ and a formula for $\gamma_{2}$. In a second paper [5] we showed that, under certain conditions, any bundle computing $\gamma_{n}$ or $\gamma_{n}^{\prime}$ is generated (a feature assumed in some earlier definitions of E. Ballico [1]). These conditions hold when $n=2$ but could fail for $n\geq 3$; however, for $n=3$, we were able to show that bundles computing either index are generically generated.

Our main object in this paper is to obtain a useful lower bound for $\gamma_{3}^{\prime}$; this is almost certainly not best possible, but it does enable us to find a lower bound for $\gamma_{3}$ and in some cases to compute $\gamma_{3}$. To state our results, we recall that the gonality sequence $d_{1},d_{2},\ldots,d_{r},\ldots$ of $C$ is defined by

[TABLE]

Our main theorem is

Theorem 4.1.

• Suppose $\gamma_{1}\geq 3$. Then*

[TABLE]

(For $\gamma_{1}\leq 2$, we know the precise values of all the Clifford indices (see Remark 4.2).)

As a consequence of this theorem, we obtain a lower bound for $\gamma_{3}$ when $\frac{d_{2}}{2}\geq\frac{d_{3}}{3}$ and precise values for $\gamma_{3}$ for general curves of genus $g\geq 7$ (Proposition 4.5), for plane curves of degree $\delta\geq 10$ (Proposition 4.6) and for curves of Clifford dimension at least 3 (see Proposition 4.9).

The proof of the theorem involves two different methods for obtaining a lower bound for $\gamma_{3}^{\prime}$. The first method (in section 2) is similar to arguments in [4] and involves considering the dimension of $h^{0}(F)$ for suitably chosen subbundles $F$ of $E$; this works well for $d\leq 2g+3$. The second method (in section 3) involves taking a proper subbundle $F$ of maximal slope and estimating the Clifford indices of $F$ and $E/F$; this works best when $d\geq 2g+4$. The combination of the methods gives the theorem.

Throughout the paper (except in Remark 4.2), $C$ will denote a smooth projective curve of genus $g$ and Clifford index $\gamma_{1}\geq 3$ (hence also $g\geq 7$) defined over an algebraically closed field of characteristic zero. For a vector bundle $G$ on $C$, the rank and degree of $G$ will be denoted by $r_{G}$ and $d_{G}$ respectively. We shall make use of the following facts about the gonality sequence:

[TABLE]

(see [4, Lemmas 4.2(a) and 4.6]). The following lemma plays an important role in section 2.

Lemma of Paranjape-Ramanan. Let $E$ be a vector bundle on $C$ of rank $n\geq 2$ with $h^{0}(E)=n+s$ ($s\geq 1$) such that $E$ possesses no proper subbundle $F$ with $h^{0}(F)>r_{F}$. Then $d_{E}\geq d_{ns}$.

This is a restatement of [7, Lemma 3.9] (see [4, Lemma 4.8]).

2. Lower bound for $\gamma^{\prime}_{3}$, first method

Let $E$ be a bundle of degree $d$ computing $\gamma^{\prime}_{3}$. According to [5, Theorem 3.3], $E$ is generically generated. Suppose $h^{0}(E)=3+s$ for some $s\geq 3$.

Lemma 2.1**.**

If $E$ has no proper subbundle $F$ with $h^{0}(F)\geq r_{F}+1$, then

[TABLE]

Proof.

By the Lemma of Paranjape-Ramanan, $d\geq d_{3s}\geq d_{9}+3s-9$. This gives

[TABLE]

∎

Lemma 2.2**.**

If $E$ has a line subbundle $F$ with $h^{0}(F)\geq 2$, then

[TABLE]

Proof.

Write $h^{0}(F)=t+1,\;t\geq 1$. Then $d_{F}\geq d_{t}$. So $d\geq 3d_{t}$ which gives

[TABLE]

if $t\geq\frac{s}{3}$.

If $t<\frac{s}{3}$, then $h^{0}(E/F)=2+v$ with $v>\frac{2s}{3}\geq 2$. According to [4, Lemma 7.2] we have

[TABLE]

There are 3 possibilities:

(i) $d_{E/F}\geq d_{2v}$. This implies

[TABLE]

since $v\geq 3$ and $t+v\geq s$, and hence

[TABLE]

(ii) $d_{E/F}\geq d_{v}+\frac{d}{3}$. So $d\geq d_{t}+d_{v}+\frac{d}{3}$, which is equivalent to $\frac{2d}{3}\geq d_{t}+d_{v}$. We have $d_{t}\geq\gamma_{1}+2t$, since $d_{t}\leq g-1$.

If $d_{v}<g+v-1$, then $d_{v}\geq\gamma_{1}+2v$ and hence

[TABLE]

If $d_{v}\geq g+v-1$, then we have

[TABLE]

which gives

[TABLE]

(Here we use Clifford’s Theorem for $E$ (see [2, Theorem 2.1]) which implies $s\leq\frac{d}{2}\leq\frac{3(g-1)}{2}$.) In both cases, $\gamma(E)>\gamma_{1}$ contradicting the fact that $E$ computes $\gamma^{\prime}_{3}$.

(iii) $d_{E/F}\geq d_{u}+d_{v-u}$. So $d\geq d_{t}+d_{u}+d_{v-u}$.

If $d_{u}<g+u-1$ and $d_{v-u}<g+v-u-1$, then

[TABLE]

This gives $\gamma(E)\geq\gamma_{1}$.

If $d_{u}\geq g+u-1$ and $d_{v-u}<g+v-u-1$, then

[TABLE]

and hence, using $\gamma(E)=\frac{d-2s}{3}$,

[TABLE]

This gives

[TABLE]

So

[TABLE]

If $d_{u}<g+u-1$ and $d_{v-u}\geq g+v-u-1$, we obtain the same inequality just by interchanging $u$ and $v-u$ in the above argument.

If $d_{u}\geq g+u-1$ and $d_{v-u}\geq g+v-u-1$, then

[TABLE]

Hence, using $\gamma(E)=\frac{d-2s}{3}$,

[TABLE]

which gives

[TABLE]

and hence, since $\gamma_{1}\leq\frac{g-1}{2}$,

[TABLE]

∎

Lemma 2.3**.**

If $E$ has a subbundle $F$ of rank $2$ with $h^{0}(F)\geq 3$ and no line subbundle with $h^{0}\geq 2$, then

[TABLE]

Proof.

Write $h^{0}(F)=2+t,\;t\geq 1$. Since $E$ is generically generated, we have $h^{0}(F)\leq 2+s$. So

[TABLE]

The quotient $E/F$ is a line bundle and $h^{0}(E/F)=1+u$ with $u\geq s-t$. The Lemma of Paranjape-Ramanan applied to $F$ gives

[TABLE]

If $\frac{t}{2}>\frac{s}{3}$, [5, Lemma 2.1] gives

[TABLE]

If $\frac{t}{2}\leq\frac{s}{3}$, then $u\geq s-t\geq\frac{s}{3}$. This implies $u\geq 1$ and hence

[TABLE]

This gives

[TABLE]

We distinguish 4 cases:

If $d_{2t}<2t+g-1$ and $d_{u}<u+g-1$, then

[TABLE]

So

[TABLE]

If $d_{2t}\geq 2t+g-1$ and $d_{u}\geq u+g-1$, then

[TABLE]

This implies, using $\gamma(E)=\frac{d-2s}{3}$,

[TABLE]

Hence, using $d\leq 3g-3$,

[TABLE]

This gives

[TABLE]

If $d_{2t}\geq 2t+g-1$ and $d_{u}<u+g-1$, then

[TABLE]

and we get as above,

[TABLE]

Finally, if $d_{2t}<2t+g-1$ and $d_{u}\geq u+g-1$, then

[TABLE]

This gives

[TABLE]

So

[TABLE]

and hence

[TABLE]

Combining (2.2), (2.3) and (2.4) we get the result. ∎

Proposition 2.4**.**

Let $E$ be a semistable bundle of degree $d$ computing $\gamma^{\prime}_{3}$. Then

[TABLE]

Proof.

This is a consequence of Lemmas 2.1, 2.2 amd 2.3. ∎

3. Lower bound for $\gamma^{\prime}_{3}$, second method

Let $E$ be a bundle of degree $d$ computing $\gamma^{\prime}_{3}$. Then $d\leq 3g-3$ and $h^{0}(E)=3+s$ for some $s\geq 3$.

Let $F$ be a proper subbundle of $E$ of maximal slope. By [6] we have

[TABLE]

Lemma 3.1**.**

If $F$ has rank $1$, then

[TABLE]

Proof.

By (3.1) we have $d_{E/F}-2d_{F}\leq 2g$. Since $d_{E/F}+d_{F}=d$, this gives $d_{F}\geq\frac{d-2g}{3}$. The semistability of $E$ implies $d_{F}\leq\frac{d}{3}$. So altogether we get

[TABLE]

If $h^{0}(F)\geq 2$, then $F$ contributes to $\gamma_{1}$ and

[TABLE]

If $h^{0}(F)\leq 1$, then by definition of $\gamma$ and (3.2) we get

[TABLE]

If $E/F$ is not semistable, it has a line subbundle $L$ of degree $>\frac{d}{3}$. Pulling back $L$ to $E$, we obtain a rank-2 subbundle of $E$ with slope $>d_{F}$ contradicting the maximality of $\mu(F)$. So $E/F$ is semistable.

Suppose $d_{E/F}\leq 2g-2$. If $h^{0}(E/F)\geq 4$, then $E/F$ contributes to $\gamma^{\prime}_{2}$ which gives

[TABLE]

If $h^{0}(E/F)\leq 3$, the definition of $\gamma$ and (3.2) give

[TABLE]

If $d_{E/F}>2g-2$, apply the same argument to $(E/F)^{*}\otimes K$ to get

[TABLE]

if $h^{1}(E/F)\geq 4$ and, for $h^{1}(E/F)\leq 3$,

[TABLE]

Here we use the inequality $\gamma_{1}\leq\frac{g-1}{2}$. Since $\gamma_{1}\geq\gamma^{\prime}_{2}$ and $\gamma(E/F)$ is a half-integer for $\gamma_{1}=3$, this implies (3.5).

We have proved that, if $h^{0}(E/F)\geq 4$ or $d_{E/F}>2g-2$, then (3.5) is valid. If $d_{E/F}\leq 2g-2$ and $h^{1}(E/F)\leq 3$, then (3.6) is valid.

Now note that

[TABLE]

So, combining these results with (3.3) and (3.4) and noting that, if $h^{0}(F)<2$, then $h^{0}(E/F)\geq 4$, we get the result. ∎

Lemma 3.2**.**

Let the notations be as at the beginning of the section and suppose $F$ has rank $2$. Then

[TABLE]

Proof.

For $F$ of rank 2, inequality (3.1) says $2d_{E/F}-d_{F}\leq 2g$. Since $d_{E/F}+d_{F}=d$ and by semistability of $E$, $d_{F}\leq\frac{2d}{3}$, we get

[TABLE]

Certainly $F$ is semistable. So if $h^{0}(F)\geq 4$, then $F$ contributes to $\gamma^{\prime}_{2}$, which gives

[TABLE]

If $h^{0}(F)\leq 3$, then by definition of $\gamma$ and (3.7) we get

[TABLE]

If $d_{E/F}\leq g-1$ and $h^{0}(E/F)\geq 2$, then $E/F$ contributes to $\gamma_{1}$, which gives

[TABLE]

If $d_{E/F}\leq g-1$ and $h^{0}(E/F)\leq 1$, then by definition of $\gamma$ and (3.7),

[TABLE]

If $d_{E/F}>g-1$, we apply the same argument to $(E/F)^{*}\otimes K$ to give:

[TABLE]

if $h^{1}(E/F)\geq 2$ and, if $h^{1}(E/F)\leq 1$,

[TABLE]

Now we have

[TABLE]

Note that if $h^{0}(F)\leq 3$, then $h^{0}(E/F)\geq 3$. But it could still happen in this case that $h^{1}(E/F)\leq 1$. So combining the above results, we obtain the lemma. ∎

Combining Lemmas 3.1 and 3.2 we get

Proposition 3.3**.**

[TABLE]

Proof.

Let $E$ be a semistable bundle of rank 3 and degree $d$ computing $\gamma^{\prime}_{3}$. Combining Lemmas 3.1 and 3.2 we see that $\gamma(E)$ is greater than or equal to the minimum of the 5 numbers

[TABLE]

Now, since $d\leq 3g-3$,

[TABLE]

Moreover, by [4, Theorem 5.2], $\gamma^{\prime}_{2}\geq\min\{\gamma_{1},\frac{d_{4}}{2}-2\}$. If $\gamma^{\prime}_{2}<\gamma_{1}$, then $\gamma^{\prime}_{2}\geq\frac{d_{4}}{2}-2$. Now $d_{4}\geq d_{1}+3\geq\gamma_{1}+5$. So $2\gamma^{\prime}_{2}\geq d_{4}-4\geq\gamma_{1}+1$. So

[TABLE]

This inequality holds also if $\gamma^{\prime}_{2}=\gamma_{1}$. The result follows. ∎

4. The main theorem

Theorem 4.1**.**

Suppose $\gamma_{1}\geq 3$. Then

[TABLE]

Proof.

Suppose that $E$ is a semistable bundle of rank 3 and degree $d$ computing $\gamma^{\prime}_{3}$. If $d\leq 2g+3$, the result follows from Proposition 2.4.

If $d\geq 2g+4$, we use Proposition 3.3 and show that the numbers occurring in the statement are $\geq\frac{2\gamma^{\prime}_{2}+1}{3}$. For the first and second ones this is obvious and we have, using $\gamma_{1}\leq\frac{g-1}{2}$,

[TABLE]

Also, if $\gamma_{1}\geq 4$,

[TABLE]

If $\gamma_{1}=3$ and $d\geq 2g+5$,

[TABLE]

If $\gamma_{1}=3$ and $d=2g+4$, then $\frac{d+2g-12}{9}=\frac{4g-8}{9}$, while $\frac{2\gamma_{1}+1}{3}=\frac{7}{3}$. For $g=7$, this does not give the required inequality. However in this case we get $\gamma(E)\geq\frac{4g-8}{9}=\frac{20}{9}$. Hence $\gamma(E)\geq\frac{7}{3}$, since $3\gamma(E)$ is an integer. ∎

Remark 4.2**.**

Theorem 4.1 is trivially true for $\gamma_{1}=1$ or 2, since then $\gamma^{\prime}_{2}=\gamma^{\prime}_{3}=\gamma_{1}$. For $\gamma_{1}=0$ the result is false. For $\gamma_{1}=2$, the inequality is strict. **

Corollary 4.3**.**

Suppose $\frac{d_{2}}{2}\geq\frac{d_{3}}{3}$. Then

[TABLE]

Proof.

By [4, Theorem 6.1] we have $\gamma_{3}=\min\{\gamma^{\prime}_{3},\frac{d_{3}-2}{3}\}$. According to [4, Theorem 5.2], $\gamma^{\prime}_{2}\geq\min\{\gamma_{1},\frac{d_{4}}{2}-2\}$. Now by the theorem and using $\frac{d_{9}}{3}-2\geq\frac{d_{3}}{3}>\frac{d_{3}-2}{3}$ we get the result, noting that $\frac{d_{4}-3}{3}\geq\frac{d_{3}-2}{3}$. ∎

Remark 4.4**.**

If in addition $d_{3}\neq 3d_{1}$, then there always exist semistable bundles $E$ of rank 3 contributing to $\gamma_{3}$ with $\gamma(E)=\frac{d_{3}-2}{3}$ (see [4, Theorem 4.15 (a)]). If $d_{3}=3d_{1}$, there exist semistable bundles $E$ of rank 3 and degree $d_{3}$ which contribute to $\gamma^{\prime}_{3}$; hence $\gamma^{\prime}_{3}<\frac{d_{3}-2}{3}$. **

Proposition 4.5**.**

For a general curve of genus $g\geq 7$,

[TABLE]

and this value is attained by the bundles $E_{L}$ given by

[TABLE]

where $L$ is a line bundle of degree $d_{3}$ with $h^{0}(L)=4$.

Proof.

In view of Corollary 4.3 and Remark 4.4 it is sufficient to show that $\frac{2\gamma_{1}+1}{3}\geq\frac{d_{3}-2}{3}$. This is a straightforward computation. ∎

Proposition 4.6**.**

For a smooth plane curve of degree $\delta\geq 7$ we have

[TABLE]

and

[TABLE]

Proof.

For a smooth plane curve of degree $\delta$, we have $d_{9}=3\delta$ and $\gamma^{\prime}_{2}=\gamma_{1}=\delta-4$ by [4, Proposition 8.1]. So Theorem 4.1 implies

[TABLE]

By [4, Proposition 8.3],

[TABLE]

Now the result follows if we have

[TABLE]

which is valid for $\delta\geq 10$. ∎

Remark 4.7**.**

For $7\leq\delta\leq 9$ the estimate on $\gamma^{\prime}_{3}$ is not sufficient to give the result for $\gamma_{3}$. For $\delta=5$ (and similarly for $\delta=6$) the result for $\gamma_{3}$ is false, since $\gamma_{3}=\gamma^{\prime}_{3}=1$ (see [4, Proposition 2.6 and Theorem 3.6(e)]). **

We are now in a position to justify the claim made in [5, Remark 3.8].

Corollary 4.8**.**

For a smooth plane curve of degree $\delta\geq 10$, $\gamma_{3}>\gamma_{2}$. Moreover, there exists a non-generated bundle computing $\gamma_{3}$.

Proof.

We have formulae for both $\gamma_{2}$ and $\gamma_{3}$; a straightforward computation shows that $\gamma_{3}>\gamma_{2}$. The existence statement follows from [5, Proposition 3.5 and Remark 3.6]. ∎

Finally we consider curves of Clifford dimension at least 3.

Proposition 4.9**.**

Let $C$ be a curve for which neither $d_{1}$ nor $d_{2}$ computes $\gamma_{1}$. Then

[TABLE]

Proof.

Suppose $d_{r}$ computes $\gamma_{1}$, $r\geq 3$. By [3, Corollary 3.5], $d_{r}\geq 4r-3\geq 3r$. So

[TABLE]

Now $d_{3}\leq d_{3}-r+3$ and, since $d_{2}$ does not compute $\gamma_{1}$,

[TABLE]

hence $\frac{d_{2}}{2}>\frac{d_{3}}{3}$ and by Corollary 4.3 we have

[TABLE]

This value is attained by $E_{L}$ where $L$ is a line bundle of degree $d_{3}$ with $h^{0}(L)=4$. ∎