Elementary Transformations of Pfaffian Representations of Plane Curves

TL;DR

This paper studies elementary transformations of linear pfaffian representations of smooth plane curves, providing explicit constructions and applications to bundles and theta characteristics, showing all representations are interconnected through these transformations.

Contribution

It introduces explicit methods for elementary transformations of pfaffian representations and demonstrates their ability to connect any two such representations on a smooth curve.

Findings

Any two pfaffian representations of a smooth curve can be connected by elementary transformations.

Explicit constructions of pfaffian representations and transformations are provided.

Applications to Aronhold bundles and theta characteristics for quartic curves are discussed.

Abstract

Let $C$ be a smooth curve in $\PP^2$ given by an equation F=0 of degree $d$. In this paper we consider elementary transformations of linear pfaffian representations of $C$. Elementary transformations can be interpreted as actions on a rank 2 vector bundle on $C$ with canonical determinant and no sections, which corresponds to the cokernel of a pfaffian representation. Every two pfaffian representations of $C$ can be bridged by a finite sequence of elementary transformations. Pfaffian representations and elementary transformations are constructed explicitly. For a smooth quartic, applications to Aronhold bundles and theta characteristics are given.

Figures

Equations

$$A=\left[\begin{array}[]{ccccc}0&L_{1\,2}&L_{1\,3}&\cdots&L_{1\,2d}\\ -L_{1\,2}&0&L_{2\,3}&\cdots&L_{2\,2d}\\ -L_{1\,3}&-L_{2\,3}&0&&\\ \vdots&\vdots&&\ddots&\vdots\\ -L_{1\,2d}&-L_{2\,2d}&&\cdots&0\end{array}\right]$$

$$\mathrm{Pf}A(x_0,x_1,x_2)=cF(x_0,x_1,x_2)\text{mbox}forsomec\in k,c\ne 0.$$

$$A^{\prime }=XA{X}^t.$$

$$0\to \underset{i=1}{\overset{2d}{⨁}}{\mathcal{O}}_{{\mathbb{P}}^2}(-1)\stackrel{A}{⟶}\underset{i=1}{\overset{2d}{⨁}}{\mathcal{O}}_{{\mathbb{P}}^2}\to \mathcal{E}\to 0.$$

$$\det M(x_0,x_1,x_2)=cF(x_0,x_1,x_2)\text{mbox}forsomec\in k,c\ne 0.$$

$$M^{\prime }=XMY.$$

$$\{\text{mbox}linebundle\mathcal{L}:\mathrm{deg}\mathcal{L}=\frac{1}{2}d(d-3),h^0(C,\mathcal{L})=0\}.$$

$$\left[\begin{array}[]{cc}0&M\\ -M^{t}&0\end{array}\right].$$

$$\left[\begin{array}[]{cc}0&XMY\\ -(XMY)^{t}&0\end{array}\right]=\left[\begin{array}[]{cc}X&0\\ 0&Y^{t}\end{array}\right]\left[\begin{array}[]{cc}0&M\\ -M^{t}&0\end{array}\right]\left[\begin{array}[]{cc}X^{t}&0\\ 0&Y\end{array}\right].$$

$$\sum _{i=1}^{2d}{\alpha }_i{e}_i\wedge \sum _{j=1}^{2d}{\beta }_j{e}_j\mapsto \sum _{i,j=1}^{2d}({\alpha }_i{\beta }_j-{\alpha }_j{\beta }_i)(E_{ij}-E_{ji}),$$

$$A=x_{1}\left[\begin{array}[]{cccc}I&0&\cdots&0\\ 0&I&\cdots&0\\ \vdots&&\ddots&\vdots\\ 0&0&\cdots&I\end{array}\right]-x_{2}\left[\begin{array}[]{cccc}D_{1}&0&\cdots&0\\ 0&D_{2}&\cdots&0\\ \vdots&&\ddots&\vdots\\ 0&0&\cdots&D_{d}\end{array}\right]+x_{0}A_{0},$$

$$I=\left[\begin{array}[]{cc}0&1\\ -1&0\end{array}\right]\ \mbox{ and }\ D_{i}=\left[\begin{array}[]{cc}0&p_{i}\\ -p_{i}&0\end{array}\right].$$

$$A_L=x_1{A}_1+x_2{A}_2$$

$$J=\left[\begin{array}[]{cccc}J_{1}&0&\cdots&0\\ 0&J_{2}&\cdots&0\\ \vdots&&\ddots&\vdots\\ 0&0&\cdots&J_{d}\end{array}\right]$$

$$J_{i}=\left[\begin{array}[]{cc}0&x_{1}-p_{i}x_{2}\\ -x_{1}+p_{i}x_{2}&0\end{array}\right].$$

$${\cup }_{i=1}^d\left\{\text{mbox}2--dimensionalbasisof\mathcal{E}(P_i)\right\}.$$

$$M_C(2,K_C)-\{\mathcal{K}:h^0(C,\mathcal{K})>0\}.$$

$$A\mapsto RA{R}^t,R\in \mathrm{GL}(2d,k).$$

$$\left[\begin{array}[]{cccc}R_{1}&0&\cdots&0\\ 0&R_{2}&&0\\ \vdots&&\ddots&\vdots\\ 0&0&\cdots&R_{d}\end{array}\right],$$

$$R\,A_{0}\,R^{t}=\left[\begin{array}[]{ccl}0&&\\ &\ddots&R_{i}\,\left[\begin{array}[]{cc}a^{0}_{2i-1,2j-1}&a^{0}_{2i-1,2j}\\ a^{0}_{2i,2j-1}&a^{0}_{2i,2j}\end{array}\right]\,R_{j}^{t}\\ &&\ddots\\ &&\ \ \ \ \ \ \ \ 0\end{array}\right],\ \ i,j=1,\ldots,d.$$

$$Q=\left[\begin{array}[]{cccccc}1&-1&0&0&\cdots&0\\ 0&0&1&-1&&0\\ \vdots&&&\ddots&&\vdots\\ 0&0&\cdots&&1&-1\\ 0&1&0&0&\cdots&0\\ 0&0&0&1&&0\\ \vdots&&&\ddots&&\vdots\\ 0&0&\cdots&&0&1\end{array}\right]$$

$$\left[\begin{array}[]{cc}0&\operatorname{Id}\\ -\operatorname{Id}&0\end{array}\right],\ \ \left[\begin{array}[]{cc}0&D\\ -D&0\end{array}\right],$$

$$\frac{\partial F(x_0,x_1,x_2)}{\partial x_k}=\sum _{i,j}{a}_{ij}^k(-1{)}^{i+j}{\mathrm{Pf}}^{ij}A(x_0,x_1,x_2).$$

$$\tilde{A}=\left[\begin{array}[]{ccl}0&&\\ &\ddots&(-1)^{i+j}\operatorname{Pf}^{ij}A\\ &&\ddots\\ &&\ \ \ \ \ \ \ \ 0\end{array}\right].$$

$$\tilde{A}A=\mathrm{Pf}A\text{mbox}{Id}_{2d}.$$

$$\mathrm{Coker}A(x)\cong \mathrm{Ker}A(x)$$

$$\sum _{k=0}^2{x}_k\frac{\partial F}{\partial x_k}(\lambda )$$

$$T\,[u_{\lambda},v_{\lambda}]^{t}\,A(x)\,[u_{\mu},v_{\mu}]\,T^{\prime}=\left[\begin{array}[]{cc}u_{\lambda}^{t}A(x)u_{\mu}&u_{\lambda}^{t}A(x)v_{\mu}\\ v_{\lambda}^{t}A(x)u_{\mu}&v_{\lambda}^{t}A(x)v_{\mu}\end{array}\right]$$

$$F(x)\left[\begin{array}[]{cc}0&0\\ 0&0\end{array}\right],\ F(x)\left[\begin{array}[]{cc}0&0\\ 0&1\end{array}\right]\mbox{ or }F(x)\left[\begin{array}[]{cc}0&1\\ 1&0\end{array}\right].$$

$$\mathrm{Pf}(y_1{\sigma }_2-y_2{\sigma }_1+\gamma )=cf(y_1,y_2)c\ne 0,$$

Full text

1. Introduction

Let $k$ be an algebraically closed field and $C$ a nonsingular curve defined by an irreducible polynomial $F(x_{0},x_{1},x_{2})$ of degree $d$ in $\mathbb{P}^{2}$. A linear pfaffian representation of $C$ is a $2d\times 2d$ skew-symmetric matrix

[TABLE]

with linear forms $L_{ij}=a_{ij}^{0}x_{0}+a_{ij}^{1}x_{1}+a_{ij}^{2}x_{2}$ such that

[TABLE]

Its cokernel is a rank 2 vector bundle on $C$. Throughout the paper we equate the notion of vector bundles and locally free sheaves.

Two pfaffian representations $A$ and $A^{\prime}$ are equivalent if there exists $X\in\operatorname{GL}_{2d}(k)$ such that

[TABLE]

There is a one to one correspondence between linear pfaffian representations (up to equivalence) of $C$ and rank 2 vector bundles (up to isomorphism) on $C$ with certain properties. This well known result is summed up in the following theorem of Beauville [3, Corollary 2.4].

Theorem 1.1**.**

Let $C$ be a smooth plane curve defined by a polynomial $F$ of degree $d$ and let ${\mathcal{E}}$ be a rank 2 bundle on $C$ with determinant ${\mathcal{O}}_{C}(d-1)$ and $H^{0}(C,{\mathcal{E}}(-1))=0$. Then there exists a $2d\times 2d$ skew-symmetric linear matrix $A$ with $\operatorname{Pf}A=F$ and an exact sequence

[TABLE]

Conversely, let $A$ be a linear skew-symmetric $2d\times 2d$ matrix with $\operatorname{Pf}A=F$. Then its cokernel is a a rank 2 bundle with $\det{\mathcal{E}}\cong{\mathcal{O}}_{C}(d-1)$ and $H^{0}(C,{\mathcal{E}}(-1))=0$.

Using this, in [5] all linear pfaffian representations of $C$ (up to equivalence) were found and related to the moduli space $M_{C}(2,K_{C})$ of semistable rank $2$ vector bundles on $C$ with canonical determinant. In particular, pfaffian representations of $C$ can be parametrised by the open set $M_{C}(2,K_{C})-\{\mathcal{K}\,:\,h^{0}(C,\mathcal{K})>0\}$. The properties of the moduli space were extensively studied in [18], [22] and  [21]; for example it is an irreducible, normal projective variety and for $C$ of genus $g\geq 2$ it has dimension $3(g-1)$.

Study of pfaffian representations is strongly related to and motivated by determinantal representations. A linear determinantal representation of $C$ is a $d\times d$ matrix $M(x_{0},x_{1},x_{2})$ of linear forms such that

[TABLE]

Determinantal representations $M$ and $M^{\prime}$ are equivalent if there exists $X,Y\in\operatorname{GL}_{d}(k)$ such that

[TABLE]

By [25] all linear determinantal representations of $C$ (up to equivalence) can be parametrised by the open set in the Jacobian variety

[TABLE]

Determinantal representations can be seen as a special case of pfaffian representations. Indeed, every determinantal representation $M$ induces a * decomposable pfaffian representation*

[TABLE]

The corresponding cokernel equals $\operatorname{Coker}M\oplus(\operatorname{Coker}^{-1}M\otimes{\mathcal{O}}(d-1))$ as described in [5] and [7]. In the moduli space, decomposable pfaffian representations correspond to an open subset of the singular locus of $M_{C}(2,K_{C})$. Note that the equivalence relation is also well defined since

[TABLE]

Elementary transformations of determinantal representations were introduced by M. S. Livsic and Kravitsky in operator theory. Vinnikov et al generalised these ideas in [23], [2] using notions of vessels and Cauchy kernels. An explicit and complete description of elementary transformations of determinantal representations can be found in [24]. Elementary transformations of vector bundles are due to Maruyama [16]. For a modern exposition and proofs we refer to Abe [1]. The most traditional example of elementary transformations is that of ruled surfaces (i.e., of rank 2 bundles). A good reference on ruled surfaces is [11, Chapter V. 2].

Throughout the paper we will be using the following properties of the wedge product. It is well known that $\bigwedge^{2}k^{2d}$ can be identified with $2d\times 2d$ skew-symmetric matrices. Let $\Omega_{n}$ denote the set of vectors $\sum_{i=1}^{n}w_{i}\wedge w^{\prime}_{i}$ in $\bigwedge^{2}k^{2d}$, where

$\dim\{w_{1},\ldots,w_{n},w^{\prime}_{1},\ldots,w^{\prime}_{n}\}=2n$. Elements of $\Omega_{n}$ are said to have irreducible length $n$ since they can be written as a sum of $n$ and not less than $n$ pure nonzero products. In [26] it is shown that $\Omega_{n}$ is isomorphic to the set of all rank $2n$ skew-symmetric matrices. The isomorphism equals

[TABLE]

where $\{e_{1},\ldots,e_{2d}\}$ is the standard basis for $k^{2d}$ and $\{E_{ij}\}$ is the standard basis for $2d\times 2d$ matrices. Note that these are the Plücker coordinates in Gr$(2,2d)$.

A brief outline of the paper is the following. In section 2 we present pfaffian representations in canonical forms. This enables us to write an algorithm which computes all the pfaffian representations of $C$ and thus gives a description of $M_{C}(2,K_{C})$. For suitable choices of vectors in the cokernel bundles discussed in Section 3, we define elementary transformations of pfaffian representations in Section 4. In Section 6 elementary transformations of pfaffian representations are related to elementary transformations of vector bundles. The main Theorem 7.3 in Section 7 proves that any two pfaffian representations can be bridged by a finite sequence of elementary transformations of Type I and II. In other words, we can explicitly construct all pfaffian representations of $C$ from a given one (for example, from a decomposable representation with symmetric blocks induced by one of the $2^{g-1}(2^{g}+1)$ even theta characteristics). Section 8 is an exposition on plane quartics. Concrete examples and algorithms for computing Aronhold bundles and theta characteristics are given.

2. The Canonical Form

Canonical forms play an important role in explicit descriptions of the moduli space $M_{C}(2,K_{C})$. In the sequel we outline an algorithm for such computation.

Let $A=x_{0}A_{0}+x_{2}A_{2}+x_{2}A_{2}$ be a representation of $C$. We can always assume that after a projective change of coordinates $C$ intersects the line $L:x_{0}=0$ in distinct points $P_{1}=(p_{1},1,0),\ldots,P_{d}=(p_{d},1,0)$. We will prove that $A$ is equivalent to a representation in the canonical form.

Proposition 2.1**.**

For every pfaffian representation $A$ of $C$ there exists a basis of $k^{2d}$ in which $A$ has the canonical form

[TABLE]

where

[TABLE]

Proof.

As above, let $P_{i}=(p_{i},1,0)$ be $d$ distinct points in the intersection of $F(x_{0},x_{1},x_{2})=0$ with the line $L:x_{0}=0$. By restricting to $L$, we obtain the pencil of skew-symmetric matrices

[TABLE]

with $\operatorname{Pf}A_{L}=F|_{L}=F(0,x_{1},x_{2})=\prod_{i=1}^{d}(x_{1}-p_{i}x_{2})$. Since all the matrices in the representation are skew-symmetric, $\operatorname{Coker}A$ and $\operatorname{Ker}A$ define the same rank 2 vector bundle ${\mathcal{E}}$. Note that ${\mathcal{E}}(P_{i})$ is the kernel of $p_{i}A_{1}+A_{2}$. Thus ${\mathcal{E}}(P_{i}),\ i=1,\ldots,d$ are $2$-dimensional subspaces in $k^{2d}$. In the proof of [5, Proposition 3.4] we showed that $h^{0}(C,{\mathcal{E}}(-1))=0$ implies the following important fact: the union of bases of the vector spaces ${\mathcal{E}}(P_{i})$ span the whole space $k^{2d}$. In this basis $A_{L}$ is equivalent to

[TABLE]

where

[TABLE]

No $2\times 2$ block $J_{i}$ is identically equal to [math], otherwise the rank of $A$ at $P_{i}$ would be at most $2d-4$. ∎

Let $A$ be a representation of $C$ and ${\mathcal{E}}$ its cokernel. Choose the basis of $k^{2d}$

[TABLE]

and denote by $P\in\operatorname{GL}(2d,k)$ the change of basis matrix. Then $P$ acts on the representation by $A\mapsto P\,A\,P^{t}$. Thus $A$ is equivalent to a representation in the canonical form by Proposition 2.1.

In [5] we established a one to one correspondence between the pfaffian representations (up to equivalence) of $C$ and the open set

[TABLE]

It thus suffices to find all pfaffian representations of $C$ in the canonical form, which will yield a set of equations describing the open subset (5) in the moduli space.

There are $d(2d-1)$ parameters in the representation (4), namely the entries of the $2d\times 2d$ skew-symmetric matrix $A_{0}=[a^{0}_{ij}]$. Since $\operatorname{Pf}A$ equals $F$, we get $\frac{1}{2}d(d+1)$ relations among $a^{0}_{ij}$. Indeed, every monomial $x_{0}^{\alpha}x_{1}^{\beta}x_{2}^{\gamma},\ \alpha\neq 0,\,\alpha+\beta+\gamma=d$ in $F$ gives one equation. By the implicit function theorem we are left with $d(2d-1)-\frac{1}{2}d(d+1)=\frac{3}{2}d(d-1)$ parameters $a^{0}_{ij}$. Recall that pfaffian representations are equivalent under the action

[TABLE]

By a suitable $R$ we can further reduce the number of parameters in $A_{0}$. Of course we only consider $R$ whose action preserves the canonical form. This way we will reduce the number of equivalent representations in each equivalence class to 1. The following lemma is an elementary exercise in linear algebra.

Lemma 2.2**.**

The action $A\ \mapsto\ R\,A\,R^{t}$ preserves the canonical form of the first two matrices in the representation if and only if $R$ equals

[TABLE]

where every $R_{i}$ is an invertible $2\times 2$ matrix with determinant 1.

Matrix $R$ in Lemma 2.2 acts on the third matrix in the representation by

[TABLE]

In other words, view $A_{0}$ as $d\times d$ matrix of $2\times 2$ blocks. Then $R$ acts on $i,j-$th block by $\ast\mapsto R_{i}\,\ast\,R_{j}^{t}$. In every $R_{i}$ we have a choice of 3 independent parameters since its determinant is 1. Thus each $R_{i}$ reduces the number of $a^{0}_{ij}$’s by 3.

For a general $F(x_{0},x_{1},x_{2})$ this sequence of reductions can be performed explicitly. We are left with

1. $3(g-1)=\frac{3}{2}d(d-3)=$
2. $d(2d-1)$ [number of parameters $a^{0}_{ij}$ in $A$ in the canonical form] $-$
3. $\frac{1}{2}d(d+1)$ [relations since $\operatorname{Pf}A=F$]$-$
4. $3d$ [parameters reduced by the action of equivalence relation $R\cdot A\cdot R^{t}$]

independent variables $a^{0}_{ij}$. As expected, this represents the open subset (5) of the moduli space in $\mathbb{P}^{d(2d-1)-1}$.

Remark 2.3**.**

In the above considerations we could take any other canonical form of the matrix pair $A_{1},A_{2}$. For example, the equivalence relation action $Q\,A\,Q^{t}$ of

[TABLE]

brings the first two matrices in (4) into

[TABLE]

where $D$ is the diagonal matrix $\{p_{1},\ldots,p_{d}\}$. This canonical form is particularly useful since it naturally includes all the decomposable representations (2). The same canonical form was obtained in [12] , where canonical forms for matrix pairs were classified purely by the methods of linear algebra.

3. Tangents

In this section we explain how tangents or lines through two points $\lambda,\mu$ on $C$ can be read from pfaffian representations. Moreover, for any pfaffian representation $A$ we relate vectors in $\operatorname{Coker}A(\lambda)$ and $\operatorname{Coker}A(\mu)$.

Since our aim is to do concrete calculations, we describe the correspondence in Theorem 1.1 explicitly: Denote by $\operatorname{Pf}^{ij}A$ the pfaffian of the $(2d-2)\times(2d-2)$ skew-symmetric matrix obtained by removing the $i$th and $j$th rows and columns from $A$. It is easy to prove the following analogue of the Jacobi’s formula for the derivative of determinants:

[TABLE]

If for some $x=(x_{0},x_{1},x_{2})\in C$ all $2d-2$ pfaffian minors vanish, then $x$ must be a singular point of $F$. By our assumption $F$ is smooth, thus $\operatorname{rank}A(x)\geq 2d-2$ for all $x\in C$. Rank of skew-symmetric matrices is even and $\det A=F^{2}=0,$ thus equality holds. Therefore $\operatorname{Coker}A(x)$ defines a $\operatorname{rank}2$ bundle over $C$. Define the pfaffian adjoint of $A$ to be the skew-symmetric matrix

[TABLE]

Again, by analogy with determinants, the following holds

[TABLE]

More properties and linear algebra of pfaffians can be found in [10, Appendix D].

Since $\operatorname{Pf}\!A=c\,F$, the cokernel can be obtained from $\tilde{A}$ by using (8). Indeed, for any point $x\in C$ every column (or row) of $\tilde{A}(x)$ is in $\operatorname{Coker}A(x)$. Since $A(x)$ is a skew-symmetric matrix with linear entries, we have

[TABLE]

which we denote by ${\mathcal{E}}(x)$. We will often view ${\mathcal{E}}$ as a $\mathbb{P}^{1}$–bundle, or equivalently as a ruled surface $\mathbb{P}{\mathcal{E}}$.

Lemma 3.1**.**

Every representation of $C$ yields tangents and lines through $\lambda,\mu\in C$ in the following way:

• •

Let $v_{\lambda}\in{\mathcal{E}}(\lambda),\,u_{\mu}\in{\mathcal{E}}(\mu)$ be independent vectors. Then $u_{\mu}^{t}A(x)v_{\lambda}$ is either identically [math] or defines a line through ${\lambda}$ and $\mu$.

• •

Let $\mathcal{L}in\{u_{\lambda},v_{\lambda}\}={\mathcal{E}}(\lambda)$. Then $u_{\lambda}^{t}A(x)v_{\lambda}=0$ is an equation of the tangent line at ${\lambda}\in C$.

• •

Analogously, for $\mathcal{L}in\{u_{\mu},v_{\mu}\}={\mathcal{E}}(\mu)$ the equation $u_{\mu}^{t}A(x)v_{\mu}=0$ defines the tangent at ${\mu}\in C$.

Proof.

The first assertion is obvious as $u_{\mu}^{t}A(x)v_{\lambda}$ is linear and equals [math] at the points $\lambda=(\lambda_{0},\lambda_{1},\lambda_{2})$ and $\mu=(\mu_{0},\mu_{1},\mu_{2})$.

By (8), $\tilde{A}$ is a rank 2 skew-symmetric matrix at the points of $C$. Using the Plücker coordinates in (3), $u_{\lambda}\wedge v_{\lambda}$ equals to a multiple of $\tilde{A}(\lambda)$. Then by (7)

[TABLE]

which equals to a nonzero multiple of $\operatorname{Trace}\left[A(x)\cdot(u_{\lambda}\wedge v_{\lambda})\right]=2\ u_{\lambda}^{t}A(x)v_{\lambda}.$ In particular we proved that $u_{\lambda}^{t}A(x)v_{\lambda}$ is not identically [math].

The same way we obtain the tangent to $C$ at $\mu$. ∎

${\mathcal{E}}(x)$ can be viewed as pairs of independent vectors $[u_{x},v_{x}]$ modulo right–hand–side multiplications by invertible $2\times 2$ matrices. Then $T,T^{\prime}\in\operatorname{GL}(2,k)$ can be chosen so that

[TABLE]

equals either

[TABLE]

This enables us to relate points on $C$. A distinct pair of points $\lambda,\mu$ on $C$ with respect to representation $A(x)$ is either:

• •

inadmissible if $u_{\mu}^{t}A(x)v_{\lambda}\equiv 0$ on $\mathbb{P}^{2}$ for all $v_{\lambda}\in\mathbb{P}{\mathcal{E}}(\lambda),\,u_{\mu}\in\mathbb{P}{\mathcal{E}}(\mu)$;

• •

semiadmissible if there exists a unique pair of vectors $u_{\lambda}\in\mathbb{P}{\mathcal{E}}(\lambda),\,u_{\mu}\in\mathbb{P}{\mathcal{E}}(\mu)$ such that $w_{\mu}^{t}A(x)u_{\lambda}\equiv 0$ and $u_{\mu}^{t}A(x)w_{\lambda}\equiv 0$ for all $w_{\lambda}\in\mathbb{P}{\mathcal{E}}(\lambda),\,w_{\mu}\in\mathbb{P}{\mathcal{E}}(\mu)$;

• •

admissible if for every $v_{\lambda}\in\mathbb{P}{\mathcal{E}}(\lambda)$ there exists exactly one $v_{\mu}\in\mathbb{P}{\mathcal{E}}(\mu)$ for which $v_{\mu}^{t}A(x)v_{\lambda}\equiv 0$ on the whole $\mathbb{P}^{2}$.

For admissible $\lambda,\mu$ we thus obtain a one to one correspondence between vectors in $\mathbb{P}{\mathcal{E}}(\lambda)$ and $\mathbb{P}{\mathcal{E}}(\mu)$. Namely, $v_{\lambda}$ corresponds to $v_{\mu}$ iff $v_{\mu}^{t}A(x)v_{\lambda}\equiv 0$. Consider next a semiadmissible pair $\lambda,\mu\in C$ and $v_{\lambda}\in\mathbb{P}{\mathcal{E}}(\lambda),\,v_{\mu}\in\mathbb{P}{\mathcal{E}}(\mu)$. We claim that $v_{\mu}^{t}A(x)v_{\lambda}\equiv 0$ if and only if either $v_{\lambda}=u_{\lambda}$ or $v_{\mu}=u_{\mu}$. Indeed, assume that $v_{\lambda}\neq u_{\lambda}$, then by linearity $v_{\mu}^{t}A(x)w_{\lambda}\equiv 0$ for any $w_{\lambda}\in\mathcal{L}in\{v_{\lambda},u_{\lambda}\}={\mathcal{E}}(\lambda)$. Thus by uniqueness in the definition $v_{\mu}=u_{\mu}$.

In the proof of Lemma 3.1 we showed

Corollary 3.2**.**

A bitangent line through $\lambda,\mu\in C$ is defined by either $u_{\mu}^{t}A(x)v_{\lambda}$, $u_{\lambda}^{t}A(x)v_{\lambda}$ or $u_{\mu}^{t}A(x)v_{\mu}$.

Corollary 3.3**.**

Let $u_{\lambda},v_{\lambda}\in\mathbb{P}{\mathcal{E}}(\lambda)$. Then $u_{\lambda}^{t}A(x)v_{\lambda}\equiv 0$ if and only if $u_{\lambda}=v_{\lambda}$.

This means that $\lambda,\lambda$ can be always considered as an admissible pair of points on $C$.

4. Elementary Transformations

In this section we define elementary transformations of pfaffian representations of the curve $C:\,F(x_{0},x_{1},x_{2})=0$ in $\mathbb{P}^{2}$, which is not necessarily smooth or irreducible. The standard notation of vessels [2] will be used.

Choose the coordinates of $F$ so that the line $\{x_{0}=0\}$ and the curve intersect in $d$ distinct smooth points. For the sake of clearer notation we move to the affine plane. Consider a linear pfaffian representation

[TABLE]

where $y_{1}=\frac{x_{1}}{x_{0}},\,y_{1}=\frac{x_{1}}{x_{0}}$ are the affine coordinates $f(y_{1},y_{2})=F(1,y_{1},y_{2})$ and $\sigma_{1},\sigma_{2},\gamma$ are $2d\times 2d$ skew-symmetric matrices. Denote

[TABLE]

by ${\mathcal{E}}(y_{1},y_{2})$.

Pick distinct regular affine points $\lambda=(\lambda_{1},\lambda_{2})$ and $\mu=(\mu_{1},\mu_{2})$ on $C$. Then each ${\mathcal{E}}(\lambda),\,{\mathcal{E}}(\mu)$ is a 2 dimensional vector space in $k^{2d}$. For all $v_{\lambda}\in{\mathcal{E}}(\lambda),\,u_{\mu}\in{\mathcal{E}}(\mu)$

[TABLE]

and

[TABLE]

hold. Thus

[TABLE]

which implies

[TABLE]

In other words, for any pair of complex parameters $t_{1},t_{2}$,

[TABLE]

is constant whenever the denominator is nonzero. Denote this constant by $K_{v_{\lambda}u_{\mu}}$.

The pair of vectors $v_{\lambda}\in{\mathcal{E}}(\lambda),\,u_{\mu}\in{\mathcal{E}}(\mu)$ is called admissible if $K_{v_{\lambda}u_{\mu}}$ is not 0. Define

[TABLE]

and

[TABLE]

where $u_{\mu}v_{\lambda}^{t},\ v_{\lambda}u_{\mu}^{t}$ and $v_{\lambda}v_{\lambda}^{t}$ are rank 1 matrices and $\rho$ is an arbitrary constant. It is obvious that $\tilde{\gamma}=-\tilde{\gamma}^{t}$ and $\bar{\gamma}=-\bar{\gamma}^{t}$ are skew-symmetric.

In the sequel the following properties of the wedge product will be useful: for any skew-symmetric matrix $\sigma$ and vectors $w,v$ holds $(\sigma w)^{t}=-w^{t}\sigma,\ w^{t}\sigma v=-v^{t}\sigma w$ and $w\wedge v=2(wv^{t}-vw^{t})$.

Using the above, $\tilde{\gamma}$ rewrites into

[TABLE]

and $\bar{\gamma}$ rewrites into

[TABLE]

Theorem 4.1**.**

Let $y_{1}\sigma_{2}-y_{2}\sigma_{1}+\gamma$ be a representation of $C$ and ${\mathcal{E}}(y_{1},y_{2})$ its cokernel. Choose $v_{\lambda}\in{\mathcal{E}}(\lambda),\,u_{\mu}\in{\mathcal{E}}(\mu)$ such that $K_{v_{\lambda}u_{\mu}}\neq 0$. Then $y_{1}\sigma_{2}-y_{2}\sigma_{1}+\tilde{\gamma}$ and $y_{1}\sigma_{2}-y_{2}\sigma_{1}+\bar{\gamma}$ are pfaffian representations of $C$ since

[TABLE]

Proof.

For any $T$ and skew-symmetric $S$ holds

[TABLE]

by [10, Appendix D]. Recall also that for any two square matrices $T^{\prime},T^{\prime\prime}$

[TABLE]

Therefore proving the equality

[TABLE]

will imply the first statement. (15) equals

[TABLE]

From (12) we obtain that

[TABLE]

Analogously

[TABLE]

Together with (10) this implies

[TABLE]

and

[TABLE]

In other words, $u_{\mu}\in\tilde{{\mathcal{E}}}(\lambda)$ and $v_{\lambda}\in\tilde{{\mathcal{E}}}(\mu)$, where $\tilde{{\mathcal{E}}}(y_{1},y_{2})=\operatorname{Coker}(y_{1}\sigma_{2}-y_{2}\sigma_{1}+\tilde{\gamma})$.

Now it is easy to verify that

[TABLE]

which together with (10) and (17) finishes the proof of (15).

The second statement will be proved if we show

[TABLE]

Note that

[TABLE]

implies

[TABLE]

This means that $v_{\lambda}\in\bar{{\mathcal{E}}}(\lambda)$, where $\bar{{\mathcal{E}}}(y_{1},y_{2})$ denotes the cokernel of $y_{1}\sigma_{2}-y_{2}\sigma_{1}+\bar{\gamma}$.

The above and the obvious equality

[TABLE]

finish the proof of (18). ∎

By performing elementary transformations of $y_{1}\sigma_{2}-y_{2}\sigma_{1}+\gamma$, we constructed new pfaffian representations of $C$:

Definition 4.2**.**

The Type I elementary transformation $y_{1}\sigma_{2}-y_{2}\sigma_{1}+\tilde{\gamma}$ based on the admissible vectors $v_{\lambda}\in{\mathcal{E}}(\lambda),\,u_{\mu}\in{\mathcal{E}}(\mu)$,

The Type II elementary transformation $y_{1}\sigma_{2}-y_{2}\sigma_{1}+\bar{\gamma}$ based on $v_{\lambda}\in{\mathcal{E}}(\lambda)$ and the constant $\rho\neq 0$.

The fact that $v_{\lambda}\in\tilde{{\mathcal{E}}}(\mu),\ u_{\mu}\in\tilde{{\mathcal{E}}}(\lambda)$ and $v_{\lambda}\in\bar{{\mathcal{E}}}(\lambda)$ implies the following

Corollary 4.3**.**

The Type I elementary transformation of $y_{1}\sigma_{2}-y_{2}\sigma_{1}+\tilde{\gamma}$ based on $u_{\mu}\in\tilde{{\mathcal{E}}}(\lambda),\ v_{\lambda}\in\tilde{{\mathcal{E}}}(\mu)$ brings us back to $y_{1}\sigma_{2}-y_{2}\sigma_{1}+\gamma$. The same way the Type II elementary transformation of $y_{1}\sigma_{2}-y_{2}\sigma_{1}+\bar{\gamma}$ based on $v_{\lambda}\in\bar{{\mathcal{E}}}(\lambda)$ and $-\rho$ brings us back to $y_{1}\sigma_{2}-y_{2}\sigma_{1}+\gamma$.

It can be easily seen that the Type I and II elementary transformations are special rank 2 cases of ”the concrete interpolation problem for meromorphic bundle maps” studied in [2]: For a given array of $m$ constants $\rho_{i}\in k$, $m$ distinct points $\lambda^{i}\in C$ and vectors $w_{i}\in{\mathcal{E}}(\lambda^{i})$ define the Type CONINT elementary transformation of $y_{1}\sigma_{2}-y_{2}\sigma_{1}+{\gamma}$ by

[TABLE]

where $w=[w_{1},\ldots,w_{m}]$ and $-\Gamma$ equals the $m\times m$ symmetric matrix with $\rho_{i}$’s along the diagonal and $K_{w_{i}\,w_{j}}$ at the $(i<j)$–th position. Then $y_{1}\sigma_{2}-y_{2}\sigma_{1}+\check{\gamma}$ is indeed a pfaffian representation of $C$ since $y_{1}\sigma_{2}-y_{2}\sigma_{1}+{\gamma}=Z^{t}(y)\,(y_{1}\sigma_{2}-y_{2}\sigma_{1}+\check{\gamma})\,Z(y)$ for

[TABLE]

An easy exercise in linear algebra shows that

[TABLE]

and $\det Z(y)=\det Z^{-1}(y)=1$. Observe that the only condition for $\check{\gamma}$ to exist is that $\Gamma$ is invertible.

In Section 7 we will show that any two pfaffian representations of $C$ can be bridged by a finite sequence of Type I and Type II elementary transformations. Thus it is enough to restrict our study to these two types.

5. Admissible arrays

We briefly relate the definitions of admissible pairs of points and admissible pairs of vectors. Let

[TABLE]

be a representation of $C$ with the cokernel ${\mathcal{E}}(x)$. As before, let $v_{\lambda}\in{\mathcal{E}}(\lambda),\ u_{\mu}\in{\mathcal{E}}(\mu)$ for distinct $\lambda,\mu\in C$. We claim that

[TABLE]

Indeed, $K_{v_{\lambda}u_{\mu}}=0$ implies $v_{\lambda}^{t}\,\sigma_{1}\,u_{\mu}=v_{\lambda}^{t}\,\sigma_{2}\,u_{\mu}=0$