Vertex Operators of the KP hierarchy and Singular Algebraic Curves

We revisit the vertex operators of the KP-hierarchy [5]. We apply them to quasi-periodic solutions, that is, solutions which are expressed by Riemann’s theta functions of non-singular algebraic curves [22]. The problem we consider here is what kind of solutions we get in this way. To study this problem we use the Sato Grassmannian [36]. We show that solutions obatined here correspond to certain singular algebraic curves whose normalizations are the non-singular curves of the seed quasi-periodic solutions. It implies two things. First is that the action of the vertex operator of the KP-hierarchy has the effect of creating certain singularities on a curve. Second is that the solutions created by vertex operators describe certain limits of quasi-periodic solutions, since singular curves may be considered as limits of non-singular curves. We check, by computer simulations, that the solutions here represent solitons on the quasi-periodic backgrounds, where the soliton matrices can be extracted from the parameters of the vertex operators. It implies that wave patterns of quasi-periodic solutions of the KP equation contain various shapes of soliton solutions [16, 17] as a part. Recently interactions of solitons and quasi-periodic solutions attracted much attention in relation with soliton gases [10, 18, 7]. It is interesting to study whether the results in this paper can have some application to this subject.

Now let us explain the results in more detail along the history of researches. During the last two decades it is revealed that the shapes of soliton solutions of the KP equation form various wave patterns like web diagrams [4, 16, 17]. Those wave patterns are related with combinatorics of non-negative Grassmannians and cluster algebras [20, 21]. Mathematically soliton solutions, in terms of tau function, are those described by linear combinations of exponential functions and are known to be constructed from singular algebraic curves of genus 0 [23, 38]. Quasi-periodic solutions of the KP equation are those written by Riemann’s theta function of non-singular algebraic curves of positive genus [22, 37]. It is expected that quasi-periodic solutions tend to soliton solutions in certain genus zero limits [23, 38, 1]. Therefore it is quite interesting to study the wave patterns of quasi-periodic solutions incorporating the recent development on soliton solutions.

One strategy to study this problem is to take limits of quasi-periodic solutions and make correspondence between quasi-periodic solutions and soliton solutions. However, to carry out this program was not very easy because it is difficult to compute limits of period matrices. We have avoided this difficulty by using the Sato Grassmannian and have calculated limits of quasi-periodic solutions for several examples [31, 32, 33, 34]. Recently there are important progress in computing the limits of quasi-periodic solutions [2, 3, 12, 11]. In [12], in particular, some kind of limits have been computed for any Riemann surface. However it seems difficult to understand how solitonic structure is incorporated in the wave patterns of quasi-periodic solutions from those results.

In this paper we change the direction of study. Instead of studying the limits of quasi-periodic solutions we construct solutions corresponding to degenerate algebraic curves which are more covenient to see the relation with soliton solutions. In course of studying the degeneration of quasi-periodic solutions we found the following formula (Theorem 4.5 of [33] ):

	$\displaystyle\lim\tau_{g,0}(t)=C{\rm e}^{-2\sum_{l=1}^{\infty}\alpha^{l}t_{2l}}$
	$\displaystyle\times\Bigl{(}{\rm e}^{\eta(t,\alpha^{1/2})}\tau_{g-1,0}(t-[\alpha^{-1/2}])+(-1)^{n}{\rm e}^{\eta(t,-\alpha^{1/2})}\tau_{g-1,0}(t-[-\alpha^{-1/2}])\Bigr{)},$

where $C$ is a certain constant, $t=(t_{1},t_{2},t_{3},...)$, $[\kappa]=(\kappa,\kappa^{2}/2,\kappa^{3}/3,...)$, $\eta(t,p)=\sum_{j=1}^{\infty}t_{j}p^{j}$, $\tau_{n,0}$ is a quasi-periodic solution of the KdV hierarchy, expressed in some standard form, corresponding to a hyperelliptic curve of genus $n$, lim means pinching a pair of branching points and $\alpha$ is the parameter correponding to the pinched point. The term inside the bracket of the right hand side has a very special form. It is rewritten using the vertex operator introduced in [5]:

	$\displaystyle X(p,q)=e^{\eta(t,p)-\eta(t,q)}e^{-\eta(\tilde{\partial},p^{-1})+\eta(\tilde{\partial},q^{-1})},$
	$\displaystyle\tilde{\partial}=(\partial_{1},\partial_{2}/2,\partial_{3}/3,...).$

In fact, for any function $\tau(t)$, we have

$\displaystyle e^{\eta(t,q)}e^{aX(p,q)}\tau(t-[q^{-1}])=e^{\eta(t,q)}\tau(t-[q^{-1}])+ae^{\eta(t,p)}\tau(t-[p^{-1}]).$

If we take $(p,q)=(-\alpha^{1/2},\alpha^{1/2})$, $a=(-1)^{n}$ and $\tau(t)=\tau_{g-1,0}$, we recover the above formula. The vertex operator transforms a solution of the KP-hierarchy to another one, that is, if $\tau(t)$ is a solution then $e^{aX(p,q)}\tau(t)$ is again a solution for any constant $a$ [5]. The above fact suggests that the limits of quasi-periodic solutions may be described by the action of vertex operators on quasi-periodic solutions. The aim of this paper is, in a sense, to show that this is actually the case.

Let $\tau_{0}(t)$ be the quasi-periodic solution of the KP-hierarchy constructed from the data $(C,L_{\Delta,e},p_{\infty},z)$, where $C$ is a compact Riemann surface of genus $g>0$, $L_{\Delta,e}$ is a certain holomorphic line bundle on $C$ of degree $g-1$, $p_{\infty}$ a point of $C$ and $z$ is a local coordinate around $p_{\infty}$ [22, 37, 15]. We apply vertex operators with various parameters successively to $\tau_{0}(t)$. More precisely, let $M,N\geq 1$, $p_{i},q_{j}$, $1\leq i\leq N$, $1\leq j\leq M$ distinct complex parameters, $A=(a_{i,j})$ an $M\times N$ matrix. We consider the vertex operator of the form

$\displaystyle G=e^{\sum_{i=1}^{M}\sum_{j=1}^{N}a_{i,j}X(q_{i}^{-1},p_{j}^{-1})}.$

We make a certain shift on $\tau_{0}(t)$, apply $G$ and multiply it by a constant times exponential function and a get new solution $\tau(t)$ (cf. (3.3)). We show that $\tau(t)$ is a solution constructed from the data $(C^{\prime},{\cal W}_{e},p^{\prime}_{\infty},z)$ where $C^{\prime}$ is a certain singular algebraic curve whose normalization is $C$, ${\cal W}_{e}$ is a certain torsion free sheaf of rank one on $C^{\prime}$ , $p^{\prime}_{\infty}$ is a point of $C^{\prime}$ and $z$ a local coordinate around $p^{\prime}_{\infty}$. To prove these properties we use the Sato Grassmannian, which we denote by UGM. It is the set of certain subspaces of the vector space $V=\mathbb{C}((z))$ and parametrizes all formal power solutions of the KP-hierarchy. We recall here that if $\tau(t)$ is a solution of the KP-hierarchy so is $\tau(-t)$. W remark that the descriptions of the points of UGM corresponding to $\tau(t)$ and $\tau(-t)$ are not very symmetric. The point corresponding to $\tau(-t)$ is more suitable to describe the geometry. By this reason we determine the point $W_{e}$ of UGM corresponding to $\tau(-t)$. This is done by examining the properties of the wave function associated with $\tau(t)$ (cf. (2.4)).

The subspace of $V$ corresponding to $\tau_{0}(-t)$ is a module over the affine coordinate ring $R$ of $C\backslash\{p_{\infty}\}$. Since $W_{e}$ is a subspace of this space, we consider the stabilizer $R_{e}$ of $W_{e}$ in $R$. Following Mumford [27] and Mulase[25, 26] we define $C^{\prime}$and ${\cal W}_{e}$ as a scheme and a sheaf on it respectively using $R_{e}$ and $W_{e}$.

Finally it should be mentioned that the relation of the vertex operator and the degeneration of Riemann’s theta function has also been observed by Yuji Kodama from different point of view [18, 19]. It is interesting to investigate the relation of the results of this paper and those of Kodama.

The paper is organized as follows. In section 2 the relation of the KP-hierarchy and the Sato Grassmannian is reviewed. The main results are formulated and stated in section 3. The formula for a quasi-periodic solution, the definition and the properties of the vertex operator and the result of the action of the vertex operator on a quasi-periodic solution are given here. In section 4 the singular algebraic curve and the sheaf on it associated with the solution considered in the main theorem are defined and their properties are studied. The example of genus one is studied in detal in section 5. Figures of computer simulation by Mathematica are presented here. The proof of the main theorem is given in section 4 and 5. Regarding the readability of the paper proofs of assertions in section 4 are given in appendix A to E.

The KP-hierarchy is the equation for a function $\tau(t)$, $t=(t_{1},t_{2},t_{3},...)$ of the form

$\displaystyle\oint\tau(t-s-[\lambda^{-1}])(t+s+[\lambda^{-1}])e^{-2\eta(s,\lambda)}\frac{d\lambda}{2\pi i}=0,$

(2.1)

where $s=(s_{1},s_{2},s_{3},...)$,

$\displaystyle[\mu]=(\mu,\frac{\mu^{2}}{2},\frac{\mu^{3}}{3},...),\hskip 14.22636pt\eta(t,\lambda)=\sum_{n=1}^{\infty}t_{n}\lambda^{n},$

and $\oint\cdot\frac{d\lambda}{2\pi i}$ means taking the residue at $\lambda=\infty$. By expanding in $\{s_{j}\}$ the KP-hierarchy is equivalent to the infinite set of differential equations which include the KP equation in the bilinear form:

$\displaystyle(D_{1}^{4}-3D_{2}^{2}-4D_{1}D_{3})\tau\cdot\tau=0,$

(2.2)

where the Hirota derivatives $D_{i}$ is defined, in general, for a function $f(t)$, as the Taylor coefficients of the expansion of $f(t+s)f(t-s)$ in $s$:

$\displaystyle f(t+s)f(t-s)=\sum\frac{D^{\alpha}f\cdot f}{\alpha!}s^{\alpha},$

where $\alpha=(\alpha_{1},\alpha_{2},...)$, $D^{\alpha}=D_{1}^{\alpha_{1}}D_{2}^{\alpha_{2}}\cdots$, $\alpha!=\alpha_{1}!\alpha_{2}!\cdots$, $s^{\alpha}=s_{1}^{\alpha_{1}}s_{2}^{\alpha_{2}}\cdots$. If $x=t_{1}$, $y=t_{2}$, $t=t_{3}$ and $u=2\partial_{x}^{2}\log\tau(t)$, (2.2) implies the KP equation

$\displaystyle 3u_{yy}+(-4u_{t}+6uu_{x}+u_{xxx})_{x}=0.$

(2.3)

The Sato Grassmannian, which we denote by UGM after Sato, parametrizes all formal power series solutions of the KP-hierarchy. It is defined as follows.

Let $V={\mathbb{C}}((z))$ be the space of formal Laurent series in one variable $z$, $V_{\phi}={\mathbb{C}}[z^{-1}]$ the subspace of polynomials in $z^{-1}$ and $V_{0}=z{\mathbb{C}}[[z]]$ the subspace of formal power series vanishing at $z=0$. Then $V=V_{\phi}\oplus V_{0}$. The Sato Grassmannian is the set of subspaces of $V$ with the same size as $V_{\phi}$. More precisely let $\pi:V\rightarrow V_{\phi}$ be the projection. Then UGM is the set of a subspace $U$ such that ${\rm Ker}\pi|_{U}$ and ${\rm Coker}\pi|_{U}$ is both finite dimensional and their dimensions are same.

Here we shall give a criterion for a subspace of $V$ to be a point of UGM. To this end, for $f(z)\in V$, define the order of $f$ by,

$\displaystyle\operatorname{ord}f=-N\text{ if $f(z)=cz^{N}+O(z^{N+1})$ with $c\neq 0$.}$

It describes the order of a pole at $z=0$ if $N$ is negative. We set

$\displaystyle V(n)=\{f\in V\,|\,\operatorname{ord}f\leq n\},$

and, for a subspace $U$ of $V$, set $U(n)=U\cap V(n)$. Then

Next we explain the correspondence between solutions of the KP-hierarchy and points of UGM. For a point of UGM the corresponding solution of the KP-hierarchy is constructed as a series using Schur functions and Plücker coordinates. For this see [32].

We need, in this paper, the converse construction of the point of UGM from a solution of the KP-hierarchy. Let $\tau(t)$ be a solution of the KP-hierarchy. Define the wave function $\Psi(t;z)$ and the adjoint wave function $\Psi^{\ast}(t;z)$ by

$\displaystyle\Psi(t;z)=\frac{\tau(t-[z])}{\tau(t)}e^{\eta(t,z^{-1})},\hskip 28.45274pt\Psi^{\ast}(t;z)=\frac{\tau(t+[z])}{\tau(t)}e^{-\eta(t,z^{-1})}.$

(2.4)

It is easy to verify that if $\tau(t)$ is a solution of the KP-hierarchy so is $\tau(-t)$.

Let $C$ be a compact Riemann surface of genus $g>0$, $\{\alpha_{i},\beta_{i}\}_{i=1}^{g}$ a canonical basis of $H^{1}(C,{\mathbb{Z}})$, $\{dv_{i}\}_{i=1}^{g}$ the normalized basis of holomorphic one forms, $\Omega=(\Omega_{i,j})_{1\leq i,j\leq g}$ with $\Omega_{i,j}=\int_{\beta_{j}}dv_{i}$ the period matrix, $J(C)={\mathbb{C}}^{g}/L_{\Omega}$ with $L_{\Omega}={\mathbb{Z}}^{g}+\Omega{\mathbb{Z}}^{g}$ the Jacobian variety of $C$, $p_{\infty}$ a point of $C$, $I(p)=\int_{p_{\infty}}^{p}dv$ with $dv={}^{t}(dv_{1},...,dv_{g})$ the Abel map, $K$ Riemann’s constant, $\theta(z|\Omega)$ Riemann’s theta function

$\displaystyle\theta(z|\Omega)=\sum_{n\in{\mathbb{Z}}^{g}}\exp(\pi i{}^{t}n\Omega n+2\pi i{}^{t}nz),\quad z={}^{t}(z_{1},...,z_{g}),$

where $n\in{\mathbb{Z}}^{g}$ is considered as a column vector.

We extend the the definition of the Abel map to divisors of any degree by, for $\displaystyle D=\sum_{j=1}^{m}p_{j}-\sum_{j=1}^{n}q_{j}$,

$\displaystyle I(D)=\sum_{j=1}^{m}I(p_{j})-\sum_{j=1}^{n}I(q_{j}).$

We denote by $\Delta$ the Riemann divisor. It is the dvisor of degree $g-1$ which satisfies $2\Delta\equiv\Omega_{C}^{1}$ and $I(\Delta)=K$, where $\equiv$ signifies the linear equivalence of divisors and $\Omega^{1}_{C}$ is the linear equivalence class of divisors of holomorphic one forms. The Riemann divisor is uniquely determined from the canonical homology basis by the condition [28]

$\displaystyle\{I(p_{1}+\cdots+p_{g-1}-\Delta)\,|\,p_{1},...,p_{g-1}\in C\}=\{z\in J(C)\,|\,\theta(z|\Omega)=0\}.$

Notice that the left hand side does not depend on the choice of the base point $p_{\infty}$ of the Abel map.

Let $E(p_{1},p_{2})$ be the prime form [8, 29] (see also [30]):

$\displaystyle E(p_{1},p_{2})=\frac{\theta[\delta](\int_{p_{1}}^{p_{2}}dv)}{h_{\delta}(p_{1})h_{\delta}(p_{2})},$

where $\delta=\binom{\delta^{\prime}}{\delta^{\prime\prime}}$, $\delta^{\prime},\delta^{\prime\prime}\in\frac{1}{2}{\mathbb{Z}}^{g}$ is a non-singular odd half characteristic and $h_{\delta}(p)$ is the half differential satisfying

$\displaystyle h_{\delta}^{2}(p)=\sum_{j=1}^{g}\frac{\partial\theta[\delta]}{\partial z_{j}}(0)dv_{j}(p).$

Take a local coordinate $z$ around $p_{\infty}$ and write

	$\displaystyle E(P_{1},P_{2})=\frac{E(z_{1},z_{2})}{\sqrt{dz_{1}}\sqrt{dz_{2}}},\hskip 14.22636ptP_{i}\in C,\quad z_{i}=z(P_{i}),$
	$\displaystyle d_{z_{1}}d_{z_{2}}\log E(z_{1},z_{2})=\left(\frac{1}{(z_{1}-z_{2})^{2}}+\sum_{i,j=1}^{\infty}q_{i,j}z_{1}^{i-1}z_{2}^{j-1}\right)dz_{1}dz_{2},$		(3.1)
	$\displaystyle dv_{i}=\sum_{j=1}^{\infty}v_{i,j}z^{j-1}dz.$

We set

$\displaystyle{\cal V}=(v_{i,j})_{1\leq i\leq g,1\leq j},\hskip 28.45274ptq(t)=\sum_{i,j=1}^{\infty}q_{i,j}t_{i}t_{j}.$

Then

$\displaystyle\tau_{0}(t)=e^{\frac{1}{2}q(t)}\theta({\cal V}t+e|\Omega)$

is a solution of the KP-hierarchy for arbitrary $e\in{\mathbb{C}}^{g}$ [37](see also [35, 15, 30]).

Let

	$\displaystyle X(p,q)=e^{\eta(t,p)-\eta(t,q)}e^{-\eta(\tilde{\partial},p^{-1})+\eta(\tilde{\partial},q^{-1})},$
	$\displaystyle\eta(t,p)=\sum_{j=1}^{\infty}t_{j}p^{j},\hskip 28.45274pt\tilde{\partial}=(\partial_{1},\partial_{2}/2,\partial_{3}/3,...),$

be the vertex operator.

The following theorem is known.

Vertex operators satisfy

$\displaystyle X(p_{1},q_{1})X(p_{2},q_{2})=\frac{(p_{1}-p_{2})(q_{1}-q_{2})}{(p_{1}-q_{2})(q_{1}-p_{2})}:X(p_{1},q_{1})X(p_{2},q_{2}):$

(3.2)

where $:\quad:$ denotes the normal ordering taking all differential operators to the right of all multiplication operators, that is,

$\displaystyle:X(p_{1},q_{1})X(p_{2},q_{2}):=e^{\sum_{j=1}^{2}(\eta(t,p_{j})-\eta(t,q_{j}))}e^{\sum_{j=1}^{2}(-\eta(\tilde{\partial},p_{j}^{-1})+\eta(\tilde{\partial},q_{j}^{-1}))}.$

The following properties follow from this.

	$\displaystyle X(p_{1},q_{1})X(p_{2},q_{2})$	$\displaystyle=$	$\displaystyle X(p_{2},q_{2})X(p_{1},q_{1})\quad\text{if $p_{1}\neq q_{2}$ and $q_{1}\neq p_{2}$,}$
	$\displaystyle X(p_{1},q_{1})X(p_{2},q_{2})$	$\displaystyle=$	$\displaystyle 0\quad\text{if $p_{1}=p_{2}$ or $q_{1}=q_{2}$}.$

Let $M,N$ be positive integers, $q_{i}$, $1\leq i\leq M$, $p_{j}$, $1\leq j\leq N$ non-zero complex numbers and $(a_{i,j})$ an $M\times N$ complex matrix. We set $p_{N+j}=q_{j}$ and use both notation $p_{N+j}$ and $q_{j}$.

Set

$\displaystyle G=e^{\sum_{i=1}^{M}\sum_{j=1}^{N}a_{i,j}X(q_{i}^{-1},p_{j}^{-1})}.$

Define

$\displaystyle\tau(t)=\Delta(p_{1}^{-1},...,p_{N}^{-1})e^{\sum_{j=1}^{N}\eta(t,p_{j}^{-1})}G\,\tau_{0}(t-\sum_{j=1}^{N}[p_{j}]),$

(3.3)

where $\Delta(p_{1},...,p_{n})=\prod_{1\leq i<j\leq n}(p_{j}-p_{i})$. It can be computed explicitly as follows.

Set $L=M+N$ and define the $L\times N$ matrix $B=(b_{i,j})$ by

	$\displaystyle b_{i,j}$	$\displaystyle=$	$\displaystyle\delta_{i,j}\quad\text{for $1\leq i,j\leq N$},$
	$\displaystyle b_{N+i,j}$	$\displaystyle=$	$\displaystyle a_{i,j}\prod_{m\neq j}^{N}\frac{p_{j}^{-1}-p_{m}^{-1}}{q_{i}^{-1}-p_{m}^{-1}}\quad\text{for $1\leq i\leq M$, $1\leq j\leq N$},$		(3.4)

that is,

$\displaystyle B=\left(\begin{array}[]{ccc}1&&\\ &\ddots&\\ &&1\\ b_{N+1,1}&\cdots&b_{N+1,N}\\ \vdots&&\vdots\\ b_{N+M,1}&\cdots&b_{N+M,N}\\ \end{array}\right).$

(3.11)

We set $[L]=\{1,...,L\}$ and denote by $\binom{[L]}{N}$ the set of $(i_{1},...,i_{N})$, $1\leq i_{1}<\cdots<i_{N}\leq L$ [16, 17]. For $I=(i_{1},...,i_{N})\in\binom{[L]}{N}$ set

	$\displaystyle\Delta^{-}_{I}=\Delta(p_{i_{1}}^{-1},...,p_{i_{N}}^{-1}),\hskip 14.22636pt\eta_{I}=\sum_{i\in I}\eta(t,p_{i}^{-1}),\hskip 14.22636pt[p_{I}]=\sum_{i\in I}[p_{i}],$
	$\displaystyle B_{I}=\det(b_{i_{r},s})_{1\leq r,s\leq N}.$

By a direct calculation using the commutation relation (3.2) we have

The part $\tau_{0}(t-[p_{I}])$ can further be expressed by theta function. To write it let $d\tilde{r}_{k}$, $k\geq 1$, be the normalized differential of the second kind with a pole only at $p_{\infty}$ of order $k+1$, that is, it satisfies

$\displaystyle\int_{\alpha_{j}}d\tilde{r}_{k}=0\text{ for any $j$},\hskip 14.22636ptd\tilde{r}_{k}=d\left(z^{-k}-O(z)\right)\text{ near $p_{\infty}$}.$

The expansion of $d\tilde{r}_{k}$ near $p_{\infty}$ can be written more explicitly using $\{q_{i,j}\}$. In integral form it is given by

$\displaystyle\int^{p}d\tilde{r}_{k}=z^{-k}-\sum_{j=1}^{\infty}q_{k,j}\frac{z^{j}}{j},\hskip 14.22636ptp\in C,z=z(p).$

(3.13)

Then

This proposition is proved by a direct calculation using the following lemma which can be derived from (3.1) .

For $e={}^{t}(e_{1},...,e_{g})\in{\mathbb{C}}^{g}$ $L_{e}$ denotes the holomorphic line bundle of degree $0$ on $C$ whose characteristic homomorphism is specified by

$\displaystyle\chi(\alpha_{j})=1,\hskip 28.45274pt\chi(\beta_{j})=e^{2\pi ie_{j}}.$

If $c\in{\mathbb{C}}^{g}$ is taken such that $\theta(c)\theta(e+c)\neq 0$ then $\theta(I(p)+e+c)/\theta(I(p)+c)$ is a meromorphic section of $L_{-e}$.

We denote $L_{\Delta}$ the holomorphic line bundle of degree $g-1$ corresponding to $\Delta$. Then we consider $L_{\Delta,-e}:=L_{\Delta}\otimes L_{-e}$ which has $\theta(I(p)+e)/E(p,p_{\infty})$ as a meromorphic section if $\theta(e)\neq 0$.

Let $H^{0}(C,L_{\Delta,-e}(\ast p_{\infty}))$ be the vector space of meromorphic sections of $L_{\Delta,-e}$ which are holomorphic on $C\backslash\{p_{\infty}\}$. Using the local coordinate $z$ we embed this space into $V={\mathbb{C}}((z))$ as follows. We consider a section of $L_{\Delta,-e}$ as $E(p,p_{\infty})^{-1}$ times a multi-valued meromorphic function on $C$ whose transformation rule is the same as that of $\theta(I(p)+e)$, that is,

$\displaystyle f(p+\alpha_{j})=f(p),\hskip 14.22636ptf(p+\beta_{j})=e^{-\pi i\Omega_{j,j}-2\pi i(\int_{p_{\infty}}^{p}dv_{j}+e_{j})}f(p).$

We realize a section of $L_{\Delta,-e}$ using a function on $C$ in this way. Then we expand elements of $H^{0}(C,L_{\Delta,-e}(\ast p_{\infty}))$ around $p_{\infty}$ in $z$ as $\sum a_{n}z^{n}\sqrt{dz}$ and get elements $\sum a_{n}z^{n}$ of $V$. In the following we always consider $H^{0}(C,L_{\Delta,-e}(\ast p_{\infty}))$ as a subspace of $V$ in this way.

For the solution $\tau(t)$ of (3.3) the descriptions of the points of UGM correponding to $\tau(t)$ and $\tau(-t)$ are not very symmetric as opposed to $\tau_{0}(t)$. We consider $\tau(-t)$ here, since it is more conveniently related with the geometry of $C$ as in the case of soliton solutions [38, 23].

Set

	$\displaystyle b^{\prime}_{N+j,i}=b_{N+j,i}(p_{i}^{-1}q_{j}),$
	$\displaystyle W_{e}=\{f\in H^{0}(C,L_{\Delta,e}(\ast p_{\infty}))\,|\,f(p_{i})=-\sum_{j=1}^{M}b^{\prime}_{N+j,i}f(q_{j}),\quad 1\leq i\leq N\},$
	$\displaystyle U_{e}=z^{N+1}W_{e}.$		(3.15)

Our main theorem is

In this section we study the geometry of $U_{e}$.

By Theorem 3.6 the point of UGM corresponding to the solution $\tau_{0}(-t)$ is associated with $(C,L_{\Delta,e},p_{\infty},z)$, where, as in the previous section, $C$ is a non-singular algebraic curve of genus $g>0$, $L_{\Delta,e}$ is the holomorphic line bundle on $C$, $p_{\infty}$ a point of $C$ and $z$ a local coordinate around $p_{\infty}$ [37, 15].

We show that the point $U_{e}$ of UGM corresponding to the solution $\tau(-t)$ is associated with $(C^{\prime},{\cal W}_{e},p_{\infty}^{\prime},z)$, where $C^{\prime}$ is a singular algebraic curve whose normalization is $C$, ${\cal W}_{e}$ is a rank one torsion free sheaf on $C^{\prime}$, $p_{\infty}^{\prime}$ is a point of $C^{\prime}$ and $z$ a local coordinate at $p_{\infty}^{\prime}$. It suggests that, geometrically, the action of the vertex operator has an effect of creating some kind of singularities on a curve. Moreover singular curves may be considered as degenerate limits of non-singular curves. Therefore $\tau(-t)$ and consequently $\tau(t)$ can be considered as a certain limit of a quasi-periodic solution.

In order to define the curve $C^{\prime}$ from $U_{e}$ the most appropriate way in the present case is the abstract algebraic method of Mumford and Mulase [27, 25, 26]. Namely we define $C^{\prime}$ as a complete integral scheme and ${\cal W}_{e}$ as a sheaf on it.

We referred to [9, 13, 14, 24] as references on algebraic geometry and commutative algebras.

Let $R:=H^{0}(C,{\cal O}(\ast p_{\infty}))$ be the vector space of meromorphic functions on $C$ which have a pole only at $p_{\infty}$. By expanding functions in the local coordinate $z$ around $p_{\infty}$ we consider $R$ as a subspace of $V={\mathbb{C}}((z))$. It is the affine coordinate ring of $C\backslash\{p_{\infty}\}$. The vector space $H^{0}(C,L_{\Delta,e}(\ast p_{\infty}))$ is an $R$-module and $W_{e}$ is a vector subspace of it. Let

$\displaystyle R_{e}=\{f\in R\,|\,fW_{e}\subset W_{e}\}$

be the stabilizer of $W_{e}$ in $R$. Then