A generalization of the Clifford index and determinantal equations for curves and their secant varieties

Introduction

When Brendan Hassett was a post doc at the University of Chicago I spoke to him a lot about a paper of Griffiths . which involved the concept of Shiffer variations. These are infinitesimal deformations of a curve, which have rank one when viewed (via cup product) as homomorphisms from $H^{0}(K_{C})\longrightarrow H^{1}(\mathcal{O}_{C})$. They are in fact parametrized by points of the curve. Initially I was trying to understand on which curves $C$, every rank $j$ infinitesimal deformations is just the sum of $j$ rank one Shiffer variations. Theorem 2.1 shows this holds only for $j<\operatorname{Cliff}(C)$

The arrangement of the thesis is as follows. In the first section we review the definition of the Clifford index and its relationship to the geometry of a curve $C$. Shiffer variations are defined and then generalized to Shiffer variations supported on a divisor $D$ . We then prove a theorem (2.5 ) which explains how Shiffer variations are related to the Clifford index. The main result of this section is a geometric characterization of the Clifford index . Namely we show that in the bicanonical embedding, $\mbox{Sec}^{j}(C)$ is in fact set theoretically the locus of all infinitesimal deformations or rank $j+1$ if and only if $j<c-1$. Recall that points in $\mbox{Sec}^{j}(C)$ are the linear combinations of $j+1$ points of $C$ and hence are the sum of $j+1$ Shiffer variations. Since the sum of $j+1$ rank one matrices is of rank at most $j+1$ , $\mbox{Sec}^{j}(C)$ consists of deformations of rank at most $j+1$. Scheme theoretic equality which implies that $\mbox{Sec}^{j}(C)$ is defined by equations of degree $j+2$ for $j<(\operatorname{Cliff}(C)-1)$ is true and is proven in section 6. This implies that $\mbox{Sec}^{j}(C)$ is determinantally defined . The methods used are different and deferring the proof until later allows us to prove a more general result.

Up until this point the results have been on embeddings in $L^{\otimes 2}$ and it’s obvious factorization as $L\otimes L$. In section 7 we take up the question of what happens when $L$ factors as $L_{1}\otimes L_{2}$. All the machinery developed in sections 3 and 4 generalize and one can prove a theorem stating that for $j<\operatorname{Cliff}(C,L_{1},L_{2})$ the secant varieties are determinantally defined. Rarely is it the case that a divisor is ’special’ for both $L_{1}$ and $L_{2}$ and hence one gets improved bounds on the circumstances under which one can say that the secant varieties are determinantal. In particular for $j=0$ that is to say for the case of the curve $C$ itself, one recovers the main result of in the case of smooth curves, which is if $\deg(L_{i})\geq 2g+1$ and $L_{1}\neq L_{2}$ if $\deg(L_{1})=\deg(L_{2})=2g+1$ then for $L=L_{1}\otimes L_{2}$, $C$ is determinantally defined in $L$.

In section 8 we discuss the results that occur when $h^{1}\left({L}\right)=1$ . First as an application of the ideas of Clifford index we give another proof of the theorem of Green and Lazarsfeld giving a bound in terms of $\deg(L)$ and $\operatorname{Cliff}(C)$ as to when an imbedding by a very ample line bundle is quadratically normal. In essence, a very ample line bundle satisfying $\operatorname{Cliff}(C,L)>0$ is quadratically normal, and one can only realize a line bundle of clifford index zero as the projection from a linear space of dimension $p-1$ of a line bundle of clifford index $p$. A computation finishes the proof. We then take up the question of bounding when $\mbox{Sec}^{j}(C)$ is determinantally defined. The same bounds hold as in the case $h^{1}\left({L}\right)=0$ but it is much harder to prove, that for $j>\operatorname{Cliff}(C,L)$ that the two varieties, the rank locus and the secant variety differ. Ironically this means that for special line bundles one is far more likely to have a stronger result than for line bundles with $h^{1}\left({L}\right)=0$.

Finally in the last section , we consider the question of the relationship between these results and Green’s conjecture. The story here is incomplete, as one would like to be able to use these results to prove the conjecture. What we can say is that, the classes we produce in Theorem 2.5 for $D$ a base point free divisor, do give rise to non-trivial Koszul cohomology classes. This applies in particular to divisors which calculate the Clifford index of $C$. Further these classes are ’decomposable’ (see section 9 for details) and that there are no such ’decomposable’ Koszual cohomology classes in $K_{p,2}(C,K_{C})$ for $c<\operatorname{Cliff}(C)$. This hardly settles the matter though.

I want to acknowledge the help I have received over the years from numerous people. Firstly, I want to express my deepest thanks to my thesis advisor, Prof. Spencer Bloch. Spencer was unstinting in his time and advice. Spencer’s remark, ‘You know I’m retiring soon, so if you want to write a thesis you should probably get it done now rather than later.’ provided the final encouragement to get things done. Spencer was a great friend and advisor. I also want to thank Professor Brendan Hassett who first brought my attention the question of the relationship between the Clifford index and determinantal embeddings. Brendan spent numerous hours discussing the problem with me and proofreading this thesis. My good friend Professor Mohan Kumar discussed this paper with me and read various version offering numerous helpful suggestions . In addition, over the years we have had so many conversations about so many mathematical topics, it is fair to say that I could/would never have kept my interest in mathematics alive without his help, advice, and information. Nicholas Passell and Mihnea Popa also read preliminary versions of the thesis offering numerous helpful suggestions.

I want to thank my wife Carol Lind for providing an environment that was very conducive to writing this thesis. I want to thank my children David Sidney Ginensky and Katherine Miriam Ginensky for being my children. I want to thank WH Trading for all their consideration in providing me with very flexible working hours while I was writing. I must also thank Mr. Tom Carrideo for his enthusiastic demonstrating of the ’wobbly H’. It helped to provide the inspiration to get this thesis done.

The Clifford Index

(Max Noether) $\Phi_{1}$ is a projectively normal embedding unless $C$ is hyperelliptic.

We wish to rephrase these theorems in terms of the Clifford index of C. Recall that the Clifford index of $C$, Cliff($C$) is defined as follows:

Let L be any line bundle. $\operatorname{Cliff}(L)=\deg(L)-2(h^{0}\left({C,L}\right)-1)$

The Clifford index of C written $\operatorname{Cliff}(C)$ is $\min\{\operatorname{Cliff}(L)|h^{0}\left({L}\right)$ and$\,h^{1}\left({L}\right)\geq 2\}$

In this section we will give a geometric generalization of Thms A and B that characterizes the Clifford index.

To state our theorem we must recall some more notation. Recall that if $\xi\in H^{1}(C,T_{C})$ then via cup product we get a map $\xi:H^{0}(C,K_{C})\longrightarrow H^{1}(C,\mathcal{O}_{C})$ called the Kodaira-Spencer map. This induces a map

Let C be a smooth curve then, $\operatorname{Sec}^{j-1}(C)=R^{j}(C)$ for $j<\mbox{Cliff}(C)$ and $\mbox{Sec}^{j}(C)\subsetneq R^{j}(C)$ for $j\geq\mbox{Cliff}(C)$.

This result is than weaker, but connected to, Green’s conjecture. We will discuss the exact relationship in the last section of this thesis. Notice in the case $\operatorname{Cliff}(C)=0$ our theorem merely states that $\operatorname{Cliff}(C)>0\Leftrightarrow H^{o}(K_{C})\otimes H^{0}(K_{C})\Longrightarrow H^{0}(K_{C}^{\otimes 2})$ is surjective and in the case $\mbox{Cliff}(C)=1$ we get only set theoretic and not scheme theoretic results. The theorem only asserts a set theoretic equality. Scheme theoretic equality is true, but will be discussed later as it involves some different ideas.

Recall that $H^{0}(T_{C}(p))=0\ \forall p\in C$ when $g\geq 2$. Hence in the long exact sequence of cohomology associated to

$0\longrightarrow T_{C}\longrightarrow T_{C}(p)\longrightarrow T_{C}(p)_{|p}\longrightarrow 0$

We will need this same notion regarding classes in $H^{1}(\mathcal{O}_{C})$ so we note the following:

$0\longrightarrow\mathcal{O}_{C}\longrightarrow\mathcal{O}_{C}(p)\longrightarrow\mathcal{O}_{C}(p)_{|p}\longrightarrow 0$.

We wish to generalize the preceding definitions to reduced divisors. Let $D=\sum_{i=1}^{d}p_{i}$ with the $p_{i}$ distinct.

$\langle\sigma_{i}\rangle\subset H^{1}(\mathcal{O}_{C})$ is the vector space spanned by the $\sigma_{p_{i}}=\sigma_{i}$ i.e. pick local parameters $z_{i}$ around $p_{i}$ and let $\sigma_{i}=\partial\left(\frac{1}{z_{i}}\right)\in H^{0}(\mathcal{O}_{C}(p)_{|p})$, then $\langle\sigma_{i}\rangle=\left\{\sum_{i=1}^{d}a_{i}\sigma_{i}\,|\,a_{i}\in k\right\}$.

Similarly let $\tau_{i}=\partial\left(\frac{1}{z_{i}}\frac{\partial}{\partial z_{i}}\right)\in H^{0}(T_{C}(p)_{|p})$ then

$T(D)=\left\{\sum_{i=1}^{d}a_{i}\tau_{i}\,|\,a_{i}\in k\right\}$. $T(D)$ is the set of Shiffer variations supported on $D$.

Unless the degree of $D$ is large, and in particular if $\deg(D)\leq 2g-3$, then $H^{0}(T_{C}(D))$ $=0$. Hence in this case, $\dim T(D)=\deg D$.

Recall that any $\xi\in H^{1}(T_{C})$ induces $\xi:H^{0}(K_{C})\longrightarrow H^{1}(\mathcal{O}_{C})$. In terms of Shiffer variations we can describe the action of $T(D)$ as follows:

Pick local coordinates $z_{i}$ around $p_{i}$ and let $\tau_{i}=\frac{1}{z_{i}}\frac{\partial}{\partial z_{i}}\in H^{1}(T_{C})$, $\sigma_{i}=\frac{1}{z_{i}}\in H^{1}(\mathcal{O}_{C})$. Suppose $\omega\in H^{0}(K_{C})$ has a local representation $f(z_{i})dz_{i}$ with $f(0)=a_{i}$. Then, for $\tau=\sum_{i=1}^{d}\tau_{i}$

Both $\sigma_{i}$ and $\tau_{i}$ are defined in terms of boundaries (Say $\sigma_{i}=\partial(\widetilde{\sigma_{i}})\ and\ \tau_{i}=\partial(\widetilde{\tau_{i}})$) $\partial:H^{0}(\mathcal{O}_{D}(D))\longrightarrow H^{1}(\mathcal{O}_{C})\ and\ \partial:H^{0}(T_{C}(D)_{|D})\longrightarrow H^{1}(T_{C})$. As such it is clear we have a commutative diagram for any $\omega\in H^{0}(K_{C})$

But $\tau_{i}\cup\omega=\frac{1}{z_{i}}\frac{\partial}{\partial z_{i}}\cup f(z_{i})dz_{i}=\frac{f(z_{i})}{z_{i}}=\frac{a_{i}}{z_{i}}+$ holomorphic fnc. $f(z_{i})/z_{i}=\frac{a_{i}}{z_{i}}$ in $H^{0}(\mathcal{O}_{p_{i}}(p_{i}))$ so denoting by $\widetilde{\tau_{i}}=\frac{1}{z_{i}}\frac{\partial}{\partial z}\in H^{0}(T_{C}(p_{i})_{|p_{i}})$ $\tau_{i}(\omega)=\partial(\widetilde{\tau_{i}}\vee\omega)=\partial\left(\frac{a_{i}}{z_{i}}\right)=a_{i}\sigma_{i}$. Since $T_{C}(D)_{|D}\ and\ \mathcal{O}_{D}(D)$ are skyscraper sheaves it follows that $\tau(\omega)=\sum_{i=1}^{d}\sigma_{i}(\omega)=\sum_{i=1}^{d}a_{i}\sigma_{i}$ as claimed. ∎

ker $(\tau_{i})=H^{0}(K_{C}(-p_{i}))$ and for $\tau\in T(D)$ ker$(\tau)\supset H^{0}(K_{C}(-D))$.

$\tau_{i}(\omega)=a_{i}\sigma_{i}$ where $\omega=f(z_{i})dz_{i}\ and\ f(0)=a_{i}$. Hence $\tau_{i}(\omega)=0\Leftrightarrow a_{i}=0\Leftrightarrow\omega\in H^{0}(K_{C}(-p_{i}))$. If $\tau=\sum b_{i}\tau_{i}$ then

The rank of $\xi\in H^{1}(T_{C})$ is the rank of $\xi:H^{0}(K_{C})\longrightarrow H^{1}(\mathcal{O}_{C})$. $R^{k}(C)=\{\xi\in H^{1}(T_{C})\,|\,\mbox{rank}(\xi)\leq k\}$.

Suppose $E=kp$ is a divisor. Then we define generalized Shiffer variations as follows: let $\tau_{p}^{j}=\partial(\widetilde{\tau}_{p}^{j})\ 1\leq j\leq k$ : where ${\widetilde{\tau_{p}}}^{j}=\frac{1}{z^{j}}\frac{\partial}{\partial z}\in H^{0}(T_{C}(kp)_{|kp})$. If $D=\sum_{i=1}^{d^{\prime}}k_{i}p_{i}$ then

From the definition it follows that $T(D)$ is the linear space spanned by the divisor D. Thus the following is clear.

The purpose of this thesis is to show that the relationship between $\operatorname{Sec}^{j-1}(C)$ and $\ R^{j}(C)$ is controlled by the Clifford index. This will follow from an analysis of Shiffer variations and classes. To that end we need to understand for any $\tau\in T(D)$ what the possible kernels and images can be. Since $\mbox{ker}(\tau)\supset H^{0}(K_{C}(-D))$ in any event set $W_{D}=W=H^{0}(K_{C})/H^{0}(K_{C}(-D))$ and $S={\langle\sigma_{i}\rangle}_{p_{i}\in D}$ so $\tau:W\longrightarrow S$. Note $\dim(W)=g-(g-d+r)=d-r$ since $D$ defines a $g^{r}_{d}$. We calculate $\dim(S)$:

$\sum a_{i}\sigma_{i}=0\Leftrightarrow\exists f\in H^{0}(\mathcal{O}_{C}(D))$ s.t. in the same local coordinates $z_{i}$ used to define $\sigma_{i}$ one has $f(z_{i})=a_{i}/z_{i}$ + holomorphic function.

since $\partial(H^{0}(\mathcal{O}_{D}(D)))=\langle\sigma_{i}\rangle$ so i) follows from $h^{0}(\mathcal{O}_{D}(D))=d$ and$\dim(\mbox{Im}(\rho))=r$ ($D$ is a $g_{d}^{r}$). ii) This is just an explication of i). Choosing local coordinates $z_{i}$ around $p_{i}$ and identifying $H^{0}(\mathcal{O}_{p_{i}}(p_{i}))$ with $\frac{1}{z_{i}}$. Then if $f=\frac{a_{i}}{z_{i}}$ + holomorphic function near $p_{i}$ then $\rho(f)=\bigoplus_{i\in D}\frac{a_{i}}{z_{i}}$ so $\partial(\rho(f))=0=\,\sum a_{i}\sigma_{i}$. ∎

Remark: If for $D=\sum_{i=1}^{d^{\prime}}k_{i}p_{i}$ with $\sum k_{i}=d$ and we define $\widetilde{\sigma}_{i}^{j}\ \ 1\leq j\leq k_{i}$ to be the class of $\frac{1}{z_{i}^{j}}\in\mathcal{O}_{k_{i}p}(k_{i}p)\ \ \sigma_{i}^{j}=\partial(\widetilde{\sigma}^{j}_{i})$ then the lemma extends in a straight forward manner to non-reduced divisors.

The main theorem of this section is Thereom 2.1 which characterizes the rank filtration in terms of the Clifford index. I recall the statement.

$\mbox{Sec}^{j}(C)=R^{j}(C)$ for $j<\mbox{Cliff}(C)$

$\mbox{Sec}^{j}(C)\subsetneq R^{j}(C)$ for $j\geq\mbox{Cliff}(C)$.

This theorem is a geometric characterization of the Clifford index. Griffiths has commented upon the importance of understanding the rank filtration in terms of doing Hodge theory, but perhaps there is more to be said algebraically.

The main tool used is a theorem that characterizes the possible ranks of a generalized Shiffer variation supported on $D$ in terms of data about $D$, in particular the Clifford index $d-2r$ of $D$. A calculation of the dimension of $\mbox{Sec}^{j}(C)$ which allows one to bound the number of Shiffer variations needed to express any element of $H^{1}(T_{C})$ finishes the proof. The relationship between the rank filtration and the Clifford index is governed by:

Let $\tau=\sum_{p_{i}\in D}a_{i}\tau_{i}\,\in\,T(D)\ a_{i}\neq 0$ be a generalized Shiffer variation. Then:

The upper bound is always achieved and the lower bound is achieved if $D\ and\linebreak\ K_{C}(-D)$ are base point free.

Remarks: Notice that if $D$ computes the Clifford index of $C$ then $D$ satisfies ii). Also note the choice of local coordinates $z_{i}$ is irrelevant as different choices of $z_{i}$ just scale $\sigma_{i}$ and $\tau_{i}$ differently.

Since $\tau:W\longrightarrow S$, $\operatorname{rk}(\tau)\leq\dim(S)=d-r$. Let $p_{1},\ldots,p_{d-r}\in D$ be such that $h^{0}(\mathcal{O}_{C}(p_{1},\ldots,p_{d-r}))=1$. That is to say that the points $p_{1},\ldots,p_{d-r}$ are linearly independent. One can check easily by induction that such points exist. If we take $\tau=\sum_{i=1}^{d-r}\tau_{i}$ then $\mbox{rank}\,\tau=d-r$ because if $\omega\in H^{0}\left({K_{C}}\right)\,$ then $\tau(\omega)=\sum_{i=1}^{d-r}b_{i}\sigma_{i}$ where $b_{i}\in k$ is the value of $\omega$ at $z_{i}$, that is to say that locally $\omega=f_{i}(z_{i})dz_{i}$ where $f_{i}(0)=b_{i}$. Thus $\tau(\omega)=0$ if and only if $b_{i}=0$ for all $i$, since the $\sigma_{i}$ were constructed to be linearly independent. Hence $\tau=\sum_{i=1}^{d-r}\tau_{i}$ achieves the upper bound.

Next we show the lower bound $d-2r\leq\mbox{rank}(\tau)$. Recall that the action of a Shiffer variation $\tau$ on one forms can be calculated by considering the action of $\widetilde{\tau}\in H^{0}(T_{C}(D)_{|D})$ representing $\tau$. In fact we have the following commutative diagram.

Here $r$ is the restriction map and $\widetilde{\tau}$ is such that $\partial(\widetilde{\tau})=\tau$. The key point is if $D=\sum_{i=1}^{j}k_{i}p_{i}$ and $\widetilde{\tau}=\sum_{i=1}^{j}\sum_{l=1}^{k_{i}}b_{il}\tau_{i}^{l}$ then $\widetilde{\tau}$ is an isomorphism if and only if $b_{ik_{i}}\neq 0\ \forall i$ ; i.e. the highest pole order terms are nonzero. This means that all coefficients are non-zero if $D$ is reduced. To see this its clearly enough to check at any point $p_{i}$ that the map $H^{0}({K_{C}}_{|k_{i}p})\stackrel{{\scriptstyle\widetilde{\tau}_{i}}}{{\longrightarrow}}H^{0}(\mathcal{O}_{k_{i}p}(k_{i}p))$ is an isomorphism $\Leftrightarrow b_{ik_{i}}\neq 0$. Working in local coordinates since $z_{i}^{j}\tau_{i}^{k}=\tau_{i}^{k-j}$ one sees that $\mbox{rank}\,\tau_{i}^{j}=j\ and\ \mbox{Im}\,\tau_{i}^{j}=z_{i}\mbox{Im}(\tau_{i}^{j+1})$ so $\sum_{j=1}^{k_{i}}b_{ij}\tau_{i}^{j}$ has $\mbox{rank}\,m\Leftrightarrow b_{i_{m}}\neq 0$ and $b_{i_{j}}=0\ j>m$ and hence $\mbox{rank}\,\widetilde{\tau_{i}}=k_{i}\Leftrightarrow b_{ik_{i}}\neq 0$.

Now returning to our diagram, the map $\widetilde{\tau}$ is an isomorphism, and $r$ is injective, so $\mbox{ker}(\tau)=\mbox{ker}(\partial\circ\widetilde{\tau}\circ r)=\mbox{ker}(\partial_{|\rm{\scriptstyle{Im}}(\widetilde{\tau}\circ r)})$. Since $\dim(\mbox{ker}\,\partial)=r\geq\dim(\mbox{ker}(\partial_{|\rm{\scriptstyle{Im}}(\widetilde{\tau}\circ r)}))$ so $\dim\,\mbox{ker}(\tau)\leq r$ and hence $\mbox{rank}\,\tau=\dim(W)-\dim(\mbox{ker}(\tau))\geq(d-r)-r=d-2r$.

To finish we must show that if $K_{C}(-D)$ is base point free there exists a $\tau$ with $\mbox{rank}(\tau)=d-2r$. Because $D$ moves and is base point free, ($h^{0}(\mathcal{O}(D))=1$ is the case $r=0$) we may find $D^{\prime}$ linearly equivalent to $D$ and reduced, $D^{\prime}=\sum_{i=1}^{d}p_{i}$ where $p_{i}\neq p_{j}$ for $i\neq j$. We may set $D=D^{\prime}$ since we are only interested in existence. Now since $K_{C}(-D)$ is base point free there exists $\omega\in H^{o}(K_{C}(-D))$ such that $\omega$ has simple zeroes at $p_{i}$ for all $i$. Pick local coordinates s.t. $z_{i}$ on $U_{i}\ni p_{i}^{\prime}$ with $\omega(z_{i})=z_{i}dz_{i}$ and set $\sigma_{i}=\partial\left(\frac{1}{z_{i}}\right)\in H^{1}(\mathcal{O}_{C})$ and $\tau_{i}=\partial\left(\frac{1}{z_{i}}\frac{\partial}{\partial z_{i}}\right)\in H^{1}(T_{C})$. Let $f_{1}\ldots f_{r}$ be a basis for $H^{0}(\mathcal{O}_{C}(D))/H^{0}(\mathcal{O}_{C})$. From our choices we have $f_{i}\omega=\eta_{i}\in H^{0}(K_{C})\ and\ \tau_{i}(\eta_{j})=a_{i}^{(j)}$ where $f_{j|U_{i}}=a_{i}^{(j)}/z_{i}$+hol. function. Note that $\eta_{i}$ are linearly independent elements of $H^{0}(K_{C})$ because if $\sum_{j=1}^{r}R_{j}\eta_{j}=0$ for $R_{j}\in k$ then $\left(\sum_{j=1}^{r}R_{j}f_{j}\right)\omega=0$. But $\omega$ is a holomorphic one form so this can only happen if $\sum_{j=1}^{n}R_{j}f_{j}=0$ which contradicts the linear independence of $f_{j}$. Now let $\tau=\sum_{i=1}^{d}\tau_{i}$. $\tau(\eta_{j})=\sum_{i=1}^{d}\tau_{i}(\eta_{j})=\sum_{i=1}^{d}a_{i}^{(j)}\sigma_{i}=0$ (because $\tau_{i}(\eta_{j})=a_{i}^{(j)}$ by Lemma 2.2). Hence we have exhibited $r$ linearly independent elements of $\mbox{ker}\,\tau$ i.e. $\mbox{rank}(\tau)\leq d-2r$ hence $\mbox{rank}(\tau)=d-2r.$ ∎

Remark: The construction of the second part is related to classical ideas about constructing quadrics containing the canonical curve. In fact general $\omega\in H^{0}(K_{C}(-D))$ have simple zeroes on $D$ and if ${\omega}^{\prime}\ and\ \eta_{i}^{\prime}$ are another such choice then $\eta_{i}{\omega}^{\prime}-\eta_{i}^{\prime}\omega\in H^{0}(K_{C})^{\otimes 2}$ is a quadric containing $C$ i.e. $\eta_{i}{\omega}^{\prime}=f_{i}\omega\cdot{\omega}^{\prime}=f_{i}{\omega}^{\prime}\cdot\omega=\eta_{i}^{\prime}\omega$ so $\eta_{i}{\omega}^{\prime}-\eta_{i}^{\prime}\omega\in\mbox{ker}(H^{1}(K_{C})^{\otimes 2}\longrightarrow H^{0}(K_{C}^{\otimes 2}))$.

$\dim(\mbox{Sec}^{j}(C))=\min(2j+1,3g-4)$

To show $\pi_{2}$ is quasifinite it is enough to show: $(\ast)\ \exists U\subset\mbox{Sym}^{j}(C)$ open and non-trivial s.t. for $D\in U\ \exists$ an open subset $V_{D}\subset\mbox{Sym}^{j}(C)$ such that $D\in V_{D}$ and $\forall E\in V_{D}\ \ H^{0}(T_{C}(D+E))=0.$

Statement $(\ast)$ may be translated as saying for general $D\in\mbox{Sym}^{j}(C)$ and any nearby $E\subset\mbox{Sym}^{j}(C)\langle D\rangle\cap\langle E\rangle=\emptyset$. But by $(\ast)$ if $D_{1}$ and $D_{2}\in U$ are distinct divisors then for $p_{i}\in\pi_{i}^{-1}(D_{i})$ for $i=1,2$ then $\pi_{2}(p_{1})\neq\pi_{2}(p_{2})$. Hence $\pi_{2}$ is quasifinite on the open set $\pi_{1}^{-1}(U)$. However $(\ast)$ is clear. Since $j\leq\frac{3g-5}{2}$, $\deg(T_{C}(E+D))\leq g-3$ for $D,E\in\operatorname{Sym}^{j}(C)$ and the general divisor of degree $g-3$ is not effective. By semi-continuity for general $D,E$ $h^{0}\left({\mathcal{O}_{C}(D+E)}\right)=0$ once it is true for one special $D,E$. ∎

Finally we will show how Theorem 2.6 implies Theorem 2.1.

Geometric Riemann-Roch and the Definition of the Clifford Index of a General Line Bundle

In the previous section we have proved that the Clifford index of a curve can be characterized in terms of the geometry of the curve, specifically the geometry of the bicanonical embedding. The purpose of this section is to generalize these ideas to a much wider class of line bundles. The Clifford index can be given a more geometric interpetation which allows one to make sense of the notion of a Clifford index for any very ample line bundle. In fact the definition makes sense for any line bundle, but it doesn’t seem to be useful unless the line bundle is very ample.

The main result we are aiming to prove is one that compares secant varieties of curves to certain rank loci for non special line bundles of large degree. While secant varieties have nice geometric properties, rank loci have the important property that their equations are (by definition!) determinants of a given degree. For example to say that a curve in some given embedding is a rank one locus is to say that the curve is defined by the vanishing of two by two minors of some matrix. This brings to the forefront the issue of whether the two scheme structures defined by the secant structure and the rank loci structure coincide. We show that this is true in a large range of circumstances. Finally we relate this to earlier work of Eisenbud, Koh, and Stillman on determinantal presentations of curves and their secant varieites.

The key idea is as follows. First use Geometric Riemann-Roch to define a general Clifford index. For a very ample line bundle L interpet $L^{\otimes 2}$ as giving an embedding of C in a space of matrices. Finally use the Clifford Index to bound from below the rank of a matrix. This allows one to say that up to a given integer $d$, $\mbox{Sec}^{d}(C)$ is set-theoretically a determinantal locus. Scheme theoretic equality then comes from a different argument.

Denote by $\overline{D}$ the span of $D$ in $P$. If $D$ has no multiple points this is clear. In general if $V\subset H^{0}\left({C,L}\right)$ is of codimension $m$ the $H^{0}\left({C,L}\right)\longrightarrow H^{0}\left({C,L}\right)/V$ determines a $m-1$ dimensional subspace. $\overline{D}$ is the space corresponding to $V=H^{0}\left({L(-D)}\right)$.

If $s\in H^{0}\left({\mathcal{O}_{C}(D)}\right)$ is a section which vanishes on $D$ (i.e. $"1"\in H^{0}\left({\mathcal{O}_{C}}\right)\hookrightarrow H^{0}\left({\mathcal{O}_{C}(D)}\right)$) then under the natural multiplication map $\mathcal{O}_{C}(D)\otimes L(-D)\longrightarrow L$ we can identify $I(\overline{D})=\left\{{\xi\in H^{0}\left({L}\right)\,|\,\xi|_{\overline{D}}=0}\right\}$ with $s\otimes H^{0}\left({L(-D)}\right)$.

This is the definition since $h^{0}\left({L(-D)}\right)=h^{0}\left({L}\right)-d+r$ means that codimension of $H^{0}\left({L(-D)}\right)$ in $H^{0}\left({L}\right)$ is $d-r$.

Tensoring this with $L(-D)$ and taking global sections give us (ii).

Thus we may rephrase the definition of the Clifford index as

$\operatorname{Cliff}(L,C)=\min\{\operatorname{Cliff}(L,D)\,|\,r_{L}(D)>0\text{ and the span of }D\\ \text{is of codimension two or greater}\}$

Remark It is a standard notation that $\operatorname{Cliff}(D)=\operatorname{Cliff}(K_{C},D)$. This definition is specific for curves. As far as I can tell, the important point is that $D$ consists of points not divisors. We now record the basic properties of $r_{L}(D)$ and $\operatorname{Cliff}(L,D)$.

$r_{L}(D)=\operatorname{codim}\left(\operatorname{im}\left(H^{0}\left({L}\right)\longrightarrow H^{0}\left({L,D}\right)\right)\right)$.

$r_{L}(D)=h^{0}\left({K_{C}\otimes L^{-1}(D)}\right)-h^{0}\left({K_{C}\otimes L^{-1}}\right)$.

$\operatorname{Cliff}(L,D)=\operatorname{Cliff}(L(-D))-\operatorname{Cliff}(L)$.

$\operatorname{Cliff}(L,D)=\operatorname{Cliff}(L,L^{\otimes 2}\otimes K_{C}^{-1}(-D))$.

$r_{L}(D)$ is defined by $h^{0}\left({L(-D)}\right)=h^{0}\left({L}\right)-d+r_{L}(D)$. Using

(since $H^{1}\left({L|_{D}}\right)=0$) and $h^{0}\left({L|_{D}}\right)=d$, we see that $\dim\left(\operatorname{im}\left(H^{0}\left({L}\right)\longrightarrow H^{0}\left({L|_{D}}\right)\right)\right)\linebreak=d-r_{L}(D)$ which is (i).

From (i) we get $r_{L}(D)=h^{1}\left({L(-D)}\right)-h^{1}\left({L}\right)$. Applying Serre duality the result follows.

$\operatorname{Cliff}(L,D)=d-2r_{L}(D)=\left(d-r_{L}(D)\right)-r_{L}(D)$, which by (i) and (ii) $=\left(h^{0}\left({L}\right)-h^{0}\left({L(-D)}\right))-(h^{1}\left({L(-D)}\right)-h^{0}\left({L}\right)\right)=\left(h^{0}\left({L}\right)+h^{1}\left({L}\right)\right)-\linebreak\left(h^{0}\left({L(-D)}\right)+h^{1}\left({L(-D)}\right)\right)$. (iii) now follows from the well known

$h^{0}\left({L}\right)+h^{1}\left({L}\right)=g+1-\operatorname{Cliff}(L)$.

By (iii), $\operatorname{Cliff}(L,D)=\operatorname{Cliff}(L(-D))-\operatorname{Cliff}(L)=\operatorname{Cliff}(K_{c}\otimes L^{-1}(D))-\operatorname{Cliff}(L)$, since $\operatorname{Cliff}(L)=\operatorname{Cliff}(K_{C}\otimes L^{-1})$. Now $K_{C}\otimes L^{-1}(D)=L\otimes\left(L^{\otimes 2}\otimes K_{C}(-D)\right)^{-1}$ and the result follows.

We first consider the case when $\deg(L)=2g-2$. Depending on how “far away” $L$ is from $K_{C}$, its behavior becomes more “very ample”. “Far away” means more general in a sense to be explained below. I am sure that most, if not all, of this material is well known to the experts. I could not find references to these specific statements, so I am including the proofs.

Suppose $deg(L)$ = $2g-2$ and $L\neq K_{C}$

$L$ is base-point free unless $L=K_{C}(p-q)$, $p,q\in C$. $K_{C}(p-q)$ has a unique base point unless $C$ is hyperelliptic.

Suppose $L$ is base-point free, then $L$ is very ample unless $L=K_{C}(D-E)$, $D,E\in\operatorname{Sym}^{2}(C)$.

Suppose $L$ is very ample, then $C$ is not defined by quadrics if $L=K_{C}(D_{1}-D_{2})$, where $\deg(D_{i})=3$ and $D_{i}$ is general. That is, $L$ has a 3-secant line.

Suppose $p\in C$ is a base point. Then $H^{0}\left({C,L}\right)=H^{0}\left({C,L(-p)}\right)$ and hence $h^{1}\left({L(-p)}\right)=1$. As mentioned above, this means $L(-p)=K_{C}(-E)$ (with $E$ effective). It follows that $\deg(E)=1$, so $L(-p)=K_{C}(-q)$ and $L=K_{C}(p-q)$.

For any $p,q\in C$, $K_{C}(p-q)$ has a base point at $p$. Furthermore, if $p_{1}\neq p$ is another base point, then $K_{C}(p-q-p_{1})=K_{C}(-q_{1})$, so $\mathcal{O}_{C}\cong\mathcal{O}_{C}(p-q+q_{1}-p_{1})$ for some choice of $p_{1}\in C$ which means that there exists a function on $C$ with two poles, i.e. $C$ is hyperellipic.

If $L$ is not very ample, there is a divisor $D$ of degree 2, such that $h^{0}\left({L(-D)}\right)>h^{0}\left({L}\right)-2$. This is just the usual criterion that a divisor is very ample if and only if it separates any two points , including infinitely near points, on $C$ (see p.152) . Since $L$ is base-point free if $D=p_{1}+p_{2}$,

hence $h^{1}\left({L(-D)}\right)=1$ and hence $L(-D)=K_{C}(-E)$ with $\deg(E)=2$ and $E$ effective, so $L=K_{C}(D-E)$.

The condition that $L=K_{C}(D_{1}-D_{2})$ with $D_{i}$ general is equivalent to a line which intersects $C$ in 3 points. This is because $h^{0}\left({L}\right)=g-1$ and

so $D_{1}$ is a line. Finally, any quadric containing $C$ contains $\overline{D}_{1}$, since if $Q$ is any such quadric $\#(\overline{D}_{1}\cap Q)\geq 3$.

The map $C^{d}\times C^{d}\longrightarrow Pic^{2d}(C)$ given by $(E,D)\mapsto E-D$ has $2d$ dimensional image in $P_{2d}(C)$ (). So for $g\geq 3$, a general line bundle is base-point free, and for $g\geq 5$, a general bundle of degree $2g-2$ is very ample. This is because the above theorem shows that the space of line bundles with a base point is two dimensional and the space of non ample line bundles is of dimension 4.

Before moving on to the the Clifford index of line bundles of degree $\geq 2g-2$, I would like to give one more application of this idea.

One says a divisor $D$ of degree $d$ defines a $d$-pointed $j$ secant if $D$ is of degree $d$ and spans a projective space of dimension $j$. By Geometric Riemann-Roch, $D$ spans a $d-1-r$ plane with $r\geq 0$. A general set of points has $r=0$. One can ask what is the smallest $d$ such that there exists a $d$-pointed $d-1-r$ plane. gives the formula that there exists a $d$-pointed $d-1-r$ plane if: $d\geq r(h^{0}\left({L}\right)-d+r)$ This holds for any very ample $L$ such that $h^{1}\left({L}\right)=0$. The techniques used in are sophisticated.

For $r=1$ and $\deg(L)=2g-2$, we can prove this very simply. In this case, the result is

If $d\geq h^{0}\left({L}\right)-d+1$, then there exists a $d$-pointed $d-2$ plane.

Rearranging terms, we need to show that if $2d\geq h^{0}\left({L}\right)+1$, then there exists a divisor $D$ of degree $d$, such that $h^{1}\left({L(-D)}\right)>0$. Using the same argument of as before, if $g$ is even, the map $\operatorname{Sym}^{\frac{g}{2}}(C)\times\operatorname{Sym}^{\frac{g}{2}}(C)\longrightarrow Pic^{g}(C)$, or if $g$ is odd, the map $\operatorname{Sym}^{\frac{g+1}{2}}(C)\times\operatorname{Sym}^{\frac{g+1}{2}}(C)\longrightarrow Pic^{g}(C)$ ($D,E$) $\mapsto D-E-p_{0}$ for a fixed $p_{0}$ are surjective. In either case we get that for any divisor $L_{0}$ of degree 0, we may write $L_{0}=\mathcal{O}_{C}(D-E)$, with $\deg(D)\leq\frac{g+1}{2}$. Since $L=K_{C}\otimes L_{0}$ for some $L_{0}$, $L(-D)=K_{C}(-E)$, and hence for the embedding given by $L$, $D$ is a $d$-secant $d-2$ plane.

We now consider the case where $\deg(L)\geq 2g+1$. We distinguish between the case when L contains $K_{C}$ as a subsheaf and when L doesn’t. Of course if deg(L) $\geq 3g-2$, then the canonical bundle will always be a subsheaf. These results are used in calculating the bounds for which one can say that curves of high degree, and certain of their secant varieties, are determinantally defined.

If $L=K_{C}(D)$ with $D$ effective of degree $=d\geq 2$, then $\operatorname{Cliff}(L,C)=d-2$. Unless $C$ is hyperelliptic, $D$ uniquely achieves this bound.

$h^{0}\left({L}\right)=g-1+d$ and $h^{0}\left({L(-D)}\right)=h^{0}\left({K_{C}}\right)=g$, so $D$ spans a $d$ secant $d-2$ plane, that is, $r_{L}(D)=1$ and $\operatorname{Cliff}(L,D)=d-2$. If $E$ satisfies $r_{L}(E)>0$ then $h^{1}\left({L(-E)}\right)>0$, i.e. $L(-E)=K_{C}(-E_{1})$. Let $e=\deg(E)$, $e_{1}=\deg(E_{1})$, and set $h^{0}\left({\mathcal{O}_{C}(E_{1})}\right)=r_{1}+1$. Then $h^{1}\left({L(-E)}\right)=h^{1}\left({K_{C}(-E_{1})}\right)=g-e_{1}+r_{1}=g+d-e+r_{1}=g+d-1-e+(r_{1}+1)$, and hence $r_{L}(E)=r_{1}+1$. Hence we get $\operatorname{Cliff}(L,E)=\deg(E)-2r_{L}(E)$. $d+e_{1}-2(r_{1}+1)=d-2+e_{1}-2r_{1}=d-2+\operatorname{Cliff}(E_{1})$. By Clifford’s theorem: $\operatorname{Cliff}(E_{1})\geq 0$, with equality $\Leftrightarrow E_{1}=0$, $K_{C}$, or $C$ is hyperelliptic and $E_{1}=ng_{2}^{1}$ a multiple of the $g_{2}^{1}$. $E_{1}=0$ means $E=D$; $E_{1}=K_{C}$ means $E=L$. Clearly $E=D+ng_{2}^{1}$ will give a divisor with $\operatorname{Cliff}(L,E)=d-2$.