text:
\section{Introduction}

 At present, one of the main challenges
of Theoretical Physics is to devise a quantum theory
which will provide a complete description of gravity.
Due to the complexity of the theories involved,
the Einstein-Hilbert gravity theory for instance,
this task faces imposing technical
difficulties. However, equally important or even more serious
are the conceptual problems that arise. These are not only due
to a variety of unfamiliar features which are peculiar
to  diffeomorphism-invariant theories
but are also due to our present lack of an adequate
formulation and interpretation of Quantum Mechanics. To an large
extent, we ignore what a quantum theory {\it really} is -- i.e., we
ignore what the adjective ``quantum'' really means --  and how we
should interpret these theories.  Because of that, we ignore not only how to
produce a quantum theory of gravity but also what would constitute
a successful completion of this task.

In this context ``toy theories'' should have
a crucial concept-clarifying role to play; no wonder
a variety of them are currently
under study. Prominent among them are
the 2D dilaton models of gravity, which are two-dimensional
general-covariant models whose gravity sector involves,
along with the space-time metric $g_{\mu\nu}$, a scalar field $\phi$,
the dilaton (for a review, see Ref. \cite{[Strominger]}). These models, while
much simpler to handle than are their higher-dimensional cousins, share
with them not only the conceptual problems which are peculiar
to diffeomorphism-invariant theories but also their most
relevant physical features,
such as the formation of black holes and their subsequent evaporation.
Some of these models can be obtained, via dimensional reduction,
from realist, higher-dimensional theories, but this is not a necessary,
nor even convenient, fact to bear in mind when approaching these
models, as they may be regarded as something like two-dimensional Brans-Dicke
theories.

If attention is restricted to the usual category of models -- that is,
those with second-order Euler-Lagrange equations of motion --
a very useful result to take into account is that, excepted isolated
pathologies which may arise, the actions of these models can all
be brought, by means of appropiate redefinitions of the
dilaton field and conformal redefinitions of the metric,
to the generic form \cite{[Banks],[Gegenberg]}

\begin{equation} {S}_{{GDG}}= S_V - S_M\label{GDG}\end{equation}
where
\begin{equation} S_V=\int d^2x\sqrt{-{g}}
\left({R}{\phi} + {V} ({\phi})\right) \label{matterlessGDG} \end{equation}
and $S_M$ is a gravity-matter interaction term which
may involve the dilaton field as well as the metric .

Despite their being much simpler that their higher-dimensional
cousins, few of these models have been solved classically,
let alone quantum mechanically, when matter fields are present.
As is well known, solvability, classical or quantum,
is usually related to the presence of invariances,
which is the reason that classical
solvability usually implies quantum solvability.
Notwithstanding this fact, little attention had been paid to
the symmetries of these models, apart from the conformal ones
(see, however, Ref. \cite{[Mann]}). Recently,
a large variety of symmetries, which are in general
non-conformal, have been uncovered for the
matterless 2D dilaton models and they have
been shown to explain the (classical)
solvability of these theories \cite{[sym]}.

In the present paper, we shall consider the generic
2D dilaton gravity with matter, eq. (\ref{GDG}),
and shall explore how much of what has been done
in the matterless case can be extended to theories with matter.
Nonetheless, to make the present paper as self-contained as possible
and to make it clear how the symmetries actually
underly the solvability of these models,
a brief but systematic view of the matterless theories shall be
presented in Sect. 2. It is worth discussing beforehand
some general features of these models.

The general variation of the Lagrangian in eq. (\ref{GDG}) yields

\begin{eqnarray}
\delta {\cal L} &=& \sqrt{-g}
\left\{\vphantom{{1\over2}}
\left[ R + V'(\phi)-T_\phi\right]\delta \phi\right. \nonumber \\
&&+\left[g_{\mu\nu}\square\phi -\frac12g_{\mu\nu}V(\phi)
- \nabla_\mu\nabla_\nu\phi-T_{\mu\nu}\right]\delta g^{\mu\nu}\label{deltaGDG}\\
&&+\left({E-L}\right)_A\delta f^A\nonumber\\
&&- \nabla_\alpha s^\alpha\nonumber\end{eqnarray}
where $\nabla_\alpha s^\alpha$ includes all the terms that appear due to
the ``integrations by part'' which are required to produce the equations
of motion.

The equations of motion can be brought to the form:

\begin{eqnarray}
R+V'(\phi)&=&T_\phi\nonumber\\
\square\phi&=&V+T\label{eqsofmotGDG}\\
\nabla_\mu\nabla_\nu\phi &=&
\frac12g_{\mu\nu}V+g_{\mu\nu}T-T_{\mu\nu}\nonumber\\
\left({E-L}\right)_A&=&0\nonumber\end{eqnarray}
where $T$ is the trace of $T_{\mu\nu}$.

Although much of our present study is meant to apply to all
kinds of matter, we shall exemplify much of our developments
with a massless scalar field $\xi$, with action

\begin{equation} S_M=\frac12\int d^2x\sqrt{-{g}}\,\Omega(\phi)(\nabla\xi)^2\end{equation}
and unspecified function $\Omega$. [The particular case of
dimensionally reduced spherically symmetric
Einstein-Hilbert gravity minimally coupled to a
massless scalar field is recovered with
$V=2/\sqrt{\phi}$ and
$\Omega=G\phi$, with $G$ the Newton constant].

For this example, we have

\begin{eqnarray} T_\phi&=&\frac{\Omega'(\phi)}2(\nabla\xi)^2\nonumber\\
T_{\mu\nu}&=&\frac{\Omega}2\left\{\nabla_\mu\xi\nabla_\nu\xi
-\frac12g_{\mu\nu}(\nabla\xi)^2\right\}\label{Txi}\end{eqnarray}

Let us now go back to eq. (\ref{eqsofmotGDG}). It is apparent that
the second(-from-above) equation is redundant
as it follows from the third one. However, the same is true for the
first one. To show this let us first indicate that invariance under
diffeomorphsims of $S_M$ implies that, when the equations of motion of the
matter fields are satisfied, the following generalized conservation
law for the energy-momentum tensor $T_{\mu\nu}$ must be fulfilled

\begin{equation}  2\nabla^\nu T_{\mu\nu}+T_\phi\nabla_\mu\phi=0\end{equation}
which implies

\begin{equation} T_\phi=-\frac2{(\nabla\phi)^2}\nabla^\mu\phi\nabla^\nu T_{\mu\nu}\label{Tphi}\end{equation}
On the other hand, the affine connection $\nabla_\mu$ and the
Riemann tensor $R_{\alpha\mu\beta\nu}$ obey

\begin{equation} [\nabla_\nu,\>\nabla_\beta]\zeta_\mu =-
\zeta_\rho {R^\rho}_{\mu\beta\nu} \end{equation}
for any vector field $\zeta_\mu$.
Particularizing this equality to $\zeta_\mu=\nabla_\mu\phi$ and
using the fact that in two-dimensions $R_{\mu\nu}=\frac12g_{\mu\nu}R$ yield,
after a bit of algebra,

\begin{equation} R\nabla_\mu\phi +V'\nabla_\mu\phi=-2\nabla^\nu T_{\mu\nu}\end{equation}
which, together with eq. (\ref{Tphi}), yields the desired equality.

Therefore, all the equations of motion of the theory are
encapsulated in the last and next-to-last equations
in eq. (\ref{eqsofmotGDG}).
As we shall see, this result facilitates
finding the general classical trajectories.

Given that in the models we are dealing with
it is easier to find the conserved currents than the
associated (Noether) symmetries,
let us remind the reader, before proceeding,
of the Noether theorem, suitably taylored for the occasion:
 Let ${\cal L}={\cal L} \left(\Psi^a\right)$  be an arbitrary
Lagrangian with general variation $\delta {\cal L}=
\left(E-L\right)_a\delta\Psi^a-\nabla_{\mu}s^{\mu}$. Let  $j_0^\mu$
be a current which is conserved on shell and  $\delta_0 \Psi^a$
a transformation of the fields.
Then $j_0^\mu$ is the Noether current
associated to $\delta_0 \Psi^a$ iff,
without using the equations of motion,
the following equality holds as an identity:
\begin{equation}
\left(E-L\right)_a\delta_0\Psi^a=\nabla_{\mu}j_0^{\mu}\>.
\label{Noethertheorem}
\end{equation}
In general, due to semi-invariance,
the current $j_0^\mu$ will not equal $s^\mu(\delta_0 \Psi^a)$.

\section{Symmetries and general solution
for the matterless theories}

With the scalars $\phi$ and $\varphi\equiv(\nabla\phi)^2$ and the vector field
$\nabla_\mu\phi$ scalars, we can built vector and tensor fields
and we can check whether or not they are conserved.
Let $J(\phi)$ be a primitive of $V$, $\d J/\d\phi=V$.
It can be shown that the following results hold:

\noindent{\bf Conserved scalars.}
The local energy \cite{[Mann],[Gegenberg]}

\begin{equation} E=\frac12\left((\nabla\phi)^2-J\right), \label{E}\end{equation}
is a conserved scalar: $\nabla_\mu E=0$, and consequently also is
$f(E)$ for any function $f$. Moreover, these are
the only conserved scalars which can be constructed with
$\phi$ and $(\nabla\phi)^2$.

The Noether symmetry associated to $E$ is given by

\begin{equation}
\delta_a \phi=0,\quad
\delta_a g_{\mu\nu}=
g_{\mu\nu}a_\sigma\nabla^\sigma\phi-\frac12
\left( a_\mu \nabla_\nu\phi +
a_\nu \nabla_\mu \phi \right)\label{deltaa}\end{equation}
with arbitrary constant bivector $a^\mu$.

\noindent{\bf Conserved currents.}
The conserved currents of the form
$j^\mu =A\left(\phi,\>\varphi\right)\nabla^\mu\phi$
can all be written as follows:

\begin{equation} j_{f}^{\mu}=f(E){\nabla^{\mu}\phi\over
\left(\nabla\phi\right)^2}\label{jf}\end{equation}
for some function $f$. The associated symmetries are:

\begin{eqnarray} \delta_f\phi&=&0\>,\nonumber\\
\delta_f g_{\mu\nu}&=& -\epsilon f'(E)
\left(g_{\mu\nu}
-{\nabla_{\mu}\phi\nabla_{\nu}\phi\over\left(\nabla\phi\right)^2}
\right) +
\epsilon f(E)\left({g_{\mu\nu}\over\left(\nabla\phi\right)^2}
-2{\nabla_{\mu}\phi\nabla_{\nu}\phi\over\left(\nabla\phi\right)^4}
\right)\label{deltaf}\end{eqnarray}
In particular, for $f=1$, the corresponding current and symmetry
are, respectively,

\begin{equation} j_{1}^{\mu}={\nabla^{\mu}\phi\over\left(\nabla\phi\right)^2}\label{j1}\end{equation}

\begin{equation} \delta_1\phi=0,\quad
\delta_1 g_{\mu\nu}=
\epsilon\left({g_{\mu\nu}\over\left(\nabla\phi\right)^2}
-2{\nabla_{\mu}\phi\nabla_{\nu}\phi\over\left(\nabla\phi\right)^4}\right)
\label{delta1}\end{equation}

Now, let $j^\mu_R$ be defined by $\nabla_\mu j^\mu_R=R$.
It is easy to see that the following current is conserved

\begin{equation} j_2^{\mu}=j_R^{\mu}+V{\nabla^{\mu}
\phi\over\left(\nabla\phi\right)^2}\label{j2}
\end{equation}
with symmetry

\begin{equation} \delta_2\phi = \epsilon, \quad
\delta_2 g_{\mu\nu}=\epsilon
V\left({g_{\mu\nu}\over\left(\nabla\phi
\right)^2}-2{\nabla_{\mu}\phi\nabla_{\nu}\phi
        \over\left(\nabla\phi\right)^4}\right)
\label{delta2}\end{equation}

\noindent{\bf Conserved tensors.}
There exists a great variety of conserved 2-tensors of the form

\[S^{\mu\nu}= A(\phi,\varphi)\nabla^\mu\phi\nabla^\nu\phi +
B(\phi,\varphi)g^{\mu\nu}\]
They are given by the general solution of the equation
($A_J=\d A/d J,\>A_\varphi=\d A/d\varphi$)

\[\frac32 A + A_J\varphi +A_\varphi\phi + B_J + B_J=0\]
which implies that $A$ ($B$) can be written as $A=A(J,\varphi)$
($B=B(J,\varphi)$).
For $A=0$, we recover the conserved scalars described above.
Another useful solution is the following traceless tensor,
which is basically unique,

\begin{equation} S^{\mu\nu}=
\frac{\nabla^\mu\phi\nabla^\nu\phi}{(\nabla\phi)^4} -
\frac12\frac{g^{\mu\nu}}{(\nabla\phi)^2}\label{tracelesstensor}\end{equation}

\subsection{General local solution}

Once the symmetries are at our disposal, it is easy to solve the equations
of motion. It can be shown \cite{[sym]} that the following
``metric'' is invariant under $\delta_2$

\[ \g_{\mu\nu}=\frac1{(\nabla\phi)^2}\left(g_{\mu\nu}-
\frac{\nabla_\mu\phi\nabla_\nu\phi}{(\nabla\phi)^2}\right)\]
which implies

\[g_{\mu\nu}=(\nabla\phi)^2\g_{\mu\nu} +
\frac{\nabla_\mu\phi\nabla_\nu\phi}{(\nabla\phi)^2}\]

As $\nabla^\mu\phi\g_{\mu\nu}=0$, we must have

\[ \g_{\mu\nu}=Ak_\mu k_\nu\]
with $k^\mu$ the vector density
\begin{equation} k^\mu=\frac{\epsilon^{\mu\nu}}
{\sqrt{-g}}\nabla_\nu\phi\label{kmu}\end{equation}

Now, as $\nabla_\mu k_\nu=\nabla_\nu k_\mu$,
$k_\mu$ is, at least locally, a total derivative:
$k_\mu=\nabla_\mu t$ for some function $t$. It is then natural to
choose as local coordinates $r\equiv\phi$ and $t$.
Now, it is easy to see that $k^\mu$ is a Killing
vector \cite{[Gegenberg]}, which implies $A=A(r)$.
The equations of motion (\ref{eqsofmotGDG}) imply $A=-1$.
Therefore, we finally arrive at the general solution

\begin{eqnarray} \phi&=&r\nonumber\\
 g_{\mu\nu} &=& (2M-J(r))\nabla_\mu t\nabla_\nu t -
\frac{\nabla_\mu r\nabla_\nu r}{2M-J(r)}\end{eqnarray}
where $M=-E$ is an arbitrary constant.

Another elegant way of arriving at the general solutions
is by using free-field methods. Here we choose the conformal
gauge

\begin{equation} \d s^2=2\hbox{e}^\rho\d x^+ \d x^-\label{conformalmetric}\end{equation}
for which we have $R= -2\hbox{e}^{-\rho}\partial_+\partial_-\rho$ and
$\square = 2\hbox{e}^{-\rho}\partial_+\partial_-$.

In the conformal gauge, the conservation law for the traceless
tensor in eq. (\ref{tracelesstensor}) takes the form

\[ 0=\partial_+T_{--}=\partial_+\left(\frac{\partial_-\phi\partial_-\phi}{(\nabla \phi)^4}\right),
\qquad 0=\partial_-T_{++}=\partial_-\left(\frac{\partial_+\phi\partial_+\phi}{(\nabla \phi)^4}\right)\]
which imply $\frac{\partial_+\phi}{(\nabla \phi)^2}=\partial_+p,\>
\frac{\partial_-\phi}{(\nabla \phi)^2}=\partial_-m$,
for some $p=p(x^+)$ and $m=m(x^-)$.

A bit of algebra leads us to

\[\hbox{e}^\rho=2(\nabla\phi)^2\partial_+p\partial_-m=2(2E+J)\partial_+p\partial_-m\]
By fixing the residual gauge as $p=\frac12x^+,
\>m=-\frac12x^-$, we finally arrive at:

\begin{eqnarray}\int^\phi {d\tau\over 2M-J\left(\tau\right)}
&=&-\frac12(x^+ -x^-)\nonumber\\
e^\rho&=&\frac12(2M-J)\end{eqnarray}

\section{Symmetries in 2D dilaton gravity with matter}

Let us now introduce matter fields.
Our most general result in this case is that the
symmetry (current) $\delta_2$ $(j^\mu_2)$
above can be generalized for whatever kind of matter is present.
More precisely: if $S_M$ is invariant under diffeomorphisms and
the equations of motion (\ref{eqsofmotGDG})
are obeyed, the following current is conserved

\begin{equation}  j_2^{\mu}=j_R^{\mu}+V\frac{\nabla^\mu \phi}
{\left(\nabla\phi\right)^2} + 2\frac{\nabla_\nu\phi}
{\left(\nabla\phi\right)^2}T^{\mu\nu}\label{j2matter}
\end{equation}

For us to show this, it suffices to make use of the equations of motion
(\ref{eqsofmotGDG}), eq. (\ref{Tphi}) and of the fact that
in two dimension, and for any tensor ${N^\nu}_\mu$, the following
quantity vanishes identically ($N={N^\rho}_\rho$)

\[ ({N^\mu}_\alpha -\frac12{\delta^\mu}_\alpha N)({N^\alpha}_\nu
-\frac12{\delta^\alpha}_\nu N)-
\frac12{\delta^\nu}_\mu ({N^\beta}_\alpha
-\frac12{\delta^\beta}_\alpha N)({N^\alpha}_\beta
-\frac12{\delta^\alpha}_\beta N)\]
or, equivalently,

\begin{equation} {N^\alpha}_\mu {N^\nu}_\alpha=\frac12\delta^\nu_\mu
 {N^\beta}_\alpha {N^\alpha}_\beta+N{N^\nu}_\mu
-\frac12\delta^\nu_\mu N^2 \label{algebraicT}\end{equation}

The associated Noether symmetry is

\begin{eqnarray}
\delta_2\phi &=& \epsilon\\
\delta_2 g_{\mu\nu}&=&\epsilon\left\{
V\left({g_{\mu\nu}\over\left(\nabla\phi
\right)^2}-2{\nabla_{\mu}\phi\nabla_{\nu}\phi
        \over\left(\nabla\phi\right)^4}\right) +
2\frac{T_{\mu\nu}}{(\nabla\phi)^2}-
2T\frac{g_{\mu\nu}}{(\nabla\phi)^2}\right.\label{delta2matter}\\
&&-\left.\frac2{(\nabla\phi)^4}\left(\nabla_\nu\phi
\nabla^\alpha\phi T_{\alpha\mu} +
\nabla_\mu\phi \nabla^\alpha\phi T_{\alpha\nu}\right)
+ 4\frac{g_{\mu\nu}}{(\nabla\phi)^4}\nabla^\alpha\phi\nabla^\beta
\phi T_{\alpha\beta}\right\}\nonumber\end{eqnarray}
The variation $\delta f^A$ of the matter fields
is such that off-shell

\begin{equation} \left({E-L}\right)_A\delta f^A= 2\frac{\nabla_\nu\phi}
{\left(\nabla\phi\right)^2}\nabla_\mu T^{\mu\nu}\end{equation}
For our exemplifying scalar matter field,
this transformation is

\begin{equation} \delta\xi = \epsilon\frac{\nabla^\mu\phi}{(\nabla\phi)^2}\nabla_\mu\xi =
\epsilon\frac{\nabla\phi\nabla \xi}{(\nabla\phi)^2}\end{equation}
It is easy to see that, in spite of what happens in the matterless
case, this symmetry does not correspond to the diffeomorphism
generated on-shell by the vector field $s^\mu =
\frac{\nabla^\mu\phi}{(\nabla\phi)^2}$.

Let us now consider currents of the form

\begin{equation} S^\mu=A(\phi,\>\varphi)\nabla^\mu\phi+
B(\phi,\>\varphi)\nabla_\nu T^{\mu\nu}\end{equation}
Conservation implies

\begin{eqnarray} 0&=&A_\phi (\nabla\phi)^2 +A_\varphi V(\nabla\phi)^2 +AV
-\frac12B(\nabla\phi)^2 T_\phi\nonumber\\
&&+\left[ 2A_\varphi(\nabla\phi)^2  +A +\frac12BV \right] T \nonumber\\
&&+\left[ B+B_\varphi(\nabla\phi)^2\right] \left(T^2-
T_{\alpha\beta}T^{\alpha\beta}\right)\\
&&+\left[-2A_\varphi +B_\phi
+B_\varphi V\right]\nabla_\mu\phi\nabla_\nu\phi T^{\mu\nu}
\nonumber\end{eqnarray}
where eq. (\ref{algebraicT}) has been used.
It follows that

\noindent 1) If $T_\phi=0,\>T=0$ and $V'=\beta V$
(with $\beta=$ constant)  the following current is conserved

\begin{equation} S^\mu_\beta =-\beta \nabla^\mu\phi +
V \frac{\nabla^\mu \phi}{\left(\nabla\phi\right)^2} +
2\frac{\nabla_\nu \phi}{\left(\nabla\phi\right)^2}T^{\mu\nu}\end{equation}
The Noether symmetry associated to $j^\mu_\beta=j^\mu_2-S^\mu_\beta$
is conformal and is given by \cite{[sym]}

\begin{equation} \delta_\beta\phi=\epsilon,\quad
\delta g_{\mu\nu}=-\epsilon\beta g_{\mu\nu},\quad \delta f^A=0
\label{deltabeta}\end{equation}

\noindent 2) If $T_\phi=0,\>T=0$ and
$T_{\alpha\beta}T^{\alpha\beta}=0$ (chiral matter)

\begin{equation} S^\mu_f =f(E)\nabla_\nu\phi T^{\mu\nu}\end{equation}
is a conserved current for any function $f$.
The associated symmetry transformation of the gravity sector is

\begin{equation}\delta\phi=0,\qquad \delta g_{\mu\nu}= T_{\mu\nu} \end{equation}

\noindent{\bf Conserved 2-tensors.}
Consider now (symmetric) tensor fields of the form

\begin{equation} S^{\mu\nu}= A(\phi,\varphi)\nabla^\mu\phi\nabla^\nu\phi +
B(\phi,\varphi)g^{\mu\nu} + C(\phi,\varphi)T^{\mu\nu}\end{equation}
It can be show that

\noindent 1) For $T_\phi=0$ and $T=0$ the tensor fields with

\begin{eqnarray} C&=&C(\phi),\quad \hbox{$C$ an arbitrary function}\nonumber\\
B&=&(C_\phi +\alpha)\varphi +2\alpha J +\frac12\int^\phi VC_\phi\\
A&=&-C_\phi -2\alpha\nonumber\end{eqnarray}
are conserved. If $T\neq0$, these tensors are still conserved if
$\alpha=0$.
The tensor with $C=\phi,\>\alpha=-\frac12$,

\begin{equation}  J^{\mu\nu}=\frac12g^{\mu\nu}\left((\nabla\phi)^2
-J(\phi)\right) +\phi T^{\mu\nu}\label{energymatter}\end{equation}
can be regarded as the generalization of the local energy
of the massless models.
The conserved tensor with $C=0$ is purely kinematical -- it
does not depend on the matter fields:

\begin{equation} S_0^{\mu\nu}=\nabla^\mu\phi\nabla^\nu\phi -
\frac12g^{\mu\nu}\left(\nabla\phi\right)^2 -g^{\mu\nu}J\end{equation}

\subsection{Chiral matter}

We define chiral matter as the one whose energy-momentum tensor obeys

\begin{equation} T_\phi=0,\qquad {T^\mu}_\mu=0
\quad \hbox{and}\quad T_{\mu\nu}T^{\mu\nu}=0\end{equation}
In the conformal gauge we have

\begin{equation} 0=T_{\mu\nu}T^{\mu\nu}=2\hbox{e}^{-2\rho}T_{++}T_{--}  \end{equation}
Therefore the set of all solution splits in two sectors of
left-moving  and right-moving fields. Let us choose $T_{--}=0$.
The conservation law for the tensor in
eq. (\ref{energymatter}) implies:

\begin{eqnarray} \partial_-E &=&0\quad\Longrightarrow E=P(x^+)\nonumber\\
\partial_+ E + \nabla^+\phi T_{++}&=&0\quad\Longrightarrow
\nabla^+\phi=p(x^+)\end{eqnarray}
Thus, we are led to the equations

\begin{eqnarray} e^{-\rho}\partial_-\phi&=&p(x^+)\nonumber\\
\partial_+\phi&=&\frac{P}{p} +\frac1{2p}J\\
\frac{\partial_+P}{p}&=&-T_{++}\nonumber\end{eqnarray}

However, it does not appear that, in the present coordinates,
 these equations can be solved in general for arbitrary
$J$ and $T_{++}$. To go further,
let us make a change of coordinates

\begin{equation} x^+=u, \qquad x^-=x^-(u,r)\end{equation}
so that $\phi\equiv r$. The metric and the energy-momentum tensor
in this coordinates takes the form

\begin{eqnarray} \d\hbox{s}^2&=&e^\lambda\left(2\d\phi\d u+e^\lambda A(\d u)^2\right)\nonumber\\
T&=&T_{uu}(du)^2\end{eqnarray}
with $T_{uu}=T_{uu}(u)$.
Now, by writting down in this gauge the equations of motion
(\ref{eqsofmotGDG}), it is easy to show that $\hbox{e}^\lambda$
is pure gauge, and finally arrive at the following general solution

\begin{equation} \d\hbox{s}^2= 2\d r \d u + (2M(u)-J) (\d u)^2\end{equation}
with

\begin{equation} M(u)=\int^u \d\tilde{u} T_{uu}(\tilde{u})\label{Mu}\end{equation}
This metric generalizes, for arbitrary  potential $V$,
the Vaidya solution of Einstein gravity.

\section{Conclusions}

We have analysed in a rather systematic way the conserved currents
and the symmetries of the 2D dilaton models
of gravity with and without matter.
In particular we have shown that $\delta_2$
can be extended to models coupled to any kind of matter.
We have also shown how analytic solvability is directly related with
the existence of invariances -- and hence of conserved currents.
In fact, almost all the models
which are known to be solvable fall into one of the categories of
symmetric models that we have described in the present paper.
The Jackiw-Teitelboim model, $V=4\lambda^2\phi$, coupled
to conformal matter may be regarded as an exception
to this rule,  as it is solvable \cite{[canonicaltr]} even though,
apart from $\delta_2$, it is not known to have any
additional symmetry. This model is solvable because,
in the conformal gauge, the first equation
in eq. (\ref{eqsofmotGDG}) is a Liouville equation.
This Liouville equation, despite being a limiting case of the free-field
equation of the exponential model which is associated to the symmetry in eq.
(\ref{deltabeta}) (see also Ref. \cite{[sym]}),
does not appear to be related itself to any invariance.
In fact, the whole problem of solving the generic 2D dilaton
models of gravity (with conformal matter)
can be regarded as a generalization of Liouville theory.
It would be interesting to see
if the machinery of integrable systems is useful here.

The massless 2D dilaton models have been shown
to be related to Poisson-$\sigma$-models, and this
seems to explains their highly symmetric nature
(see, for instance, Ref. \cite{[Kummer],[Poisson]}
and references therein).  On the other hand,
the underlying reason why $\delta_2$ is so general
remains a mistery to us as of the writing of this paper. It seems
to indicate that these models have a degree of unity
previously unexpected. This unity may have important
consequences in relation to their solvability.

\section*{Acknowledgements}
The author has profitted from discussions with
J. Cruz and J. Navarro-Salas.
He acknowledges the Spanish MEC, CSIC and IMAFF (Madrid)
for a research contract.

This work was partially supported by the
Comisi\'on Interministerial de Ciencia y Tecnolog\'{\i}a
and DGICYT.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

Given a finite morphism of smooth projective  irreducible curves, one defines the attached Prym variety as the component of the origin of the kernel of the induced map between their corresponding Albanese (Jacobian) varieties. By construction, this is a polarized abelian variety whose geometry reflects many of the properties of the finite map and, in many cases, of the Brill-Noether loci of the source and target curves. This theory is specially rich in the case of unramified double coverings. Let us denote by $\mathcal R_g$ the moduli of isomorphism classes of pairs $(C,\eta)$, where $C\in \mathcal M_g$ and $\eta $ is a non-trivial $2$-torsion point in the Jacobian of $C$. This is equivalent to an irreducible double cover of $C$. After the seminal work of Mumford (\cite{mu}), the Prym map between moduli stacks:
\[
\mathcal P_g:\mathcal R_g \longrightarrow \mathcal A_{g-1},
\]
sending $[(C,\eta)]$ to  its Prym variety $P(C,\eta)$,
has been extensively studied. Although is known to be generically injective and non-injective for $g\ge 7$ very few is known about its behavior when restricted to special subvarieties. It is worthy to  mention that Beauville extended $\mathcal P_g$ to a partial compactification $\overline {\mathcal R}_g$ allowing some coverings of stable curves (see \cite{be_invent}). The main property of this extended map $\overline {\mathcal P}_g$ is its properness.

More recently, double coverings ramified in an even number $r>0$ of points have been considered from the Prym map scope.
Let us denote by $\mathcal R_{g,r}$ the moduli of ramified coverings of curves of genus $g$ ramified in $r$ points. This classifies isomorphism classes of elements $(C,\eta, B)$, where $C\in\mathcal M_g$, $\eta \in Pic^{\frac r2}(C) $ and $B$ is a reduced divisor in $\vert \eta^{\otimes 2}\vert $.  Nagaraj and Ramanan studied the $r=4$ case in \cite{nagaraj_ramanan} and later Marcucci and Pirola considered the injectivity of
\[
\mathcal P_{g,r}:\mathcal R_{g,r} \longrightarrow
\mathcal A_{g-1+\frac r2}^{\delta}
\]
in general, see \cite{mp} ($\delta $ is the type of the polarization, it is of the form $(1,\ldots ,1,2,\ldots,2)$). They proved the generic Torelli Theorem for most of the values of $(g,r)$. Finally, Ortega and the first author proved in \cite{naranjo-ortega2} the global Torelli Theorem for $r\ge 6$. Moreover, the fibers of $\mathcal P_{g,r}$ have been studied in  \cite{fns} when $\dim \mathcal R_{g,r}>\dim \mathcal A_{g-1+\frac r2}^{\delta }$.

The structure of the fibers of $\mathcal P_{g,2}$ and $\mathcal P_{g,4}$ is rather complicated, there exist positive dimensional fibers (e.g. if $C$ is hyperelliptic), the restriction to the tetragonal locus $\mathcal R_{g,r}^{tet}$ has degree $3$, etc. Notice that $\mathcal P_{g,2}$ can be seen as the restriction of the classical (unramified)  extended Prym map $\overline {\mathcal P}_g$ to a divisor of the boundary $\overline {\mathcal R}_g \setminus  \mathcal R_g$.
A first step in the analysis of these maps is to consider the differential $d\mathcal P_{g,r}$. As proved in \cite{mp}, the codifferential is  given by the multiplication map
\begin{equation}\label{codifferenziale}
	d\mathcal P_{g,r}^* (C, \eta, B): Sym^2 H^0(C, \omega_C \otimes \eta) \longrightarrow H^0(C, \omega_C^2 \otimes \mathcal O(B)).
\end{equation}

Using \cite[Remark 2.2]{naranjo-ortega2}, is easy to show that, if the differential $d\mathcal P_{g,r}$ is not injective at $[(C, \eta, B)]$, then:
	\begin{itemize}
		\item[1.]  $r=2$ and $\eta=\mathcal O_C(x+y-z)$ for $x,y,z\in C$ or $r=4$ and $h^0(C,\eta)>0$. Otherwise
		\item[2.] $r=2$ and $C$ is hyperelliptic, trigonal or a quintic plane curve or $r=4$ and $C$ is hyperelliptic.
	\end{itemize}
Observe that in the first case the line bundle $\eta $ is special and in the second case the curve $C$ has Clifford index $\le 1$. This is not a characterization, apart from the hyperelliptic case we have no information on the rest of the possibilities.

Our aim in this paper is to clarify the situation for double coverings of plane quintic curves with $r=2$. We  prove first the injectivity of the differential of $\mathcal P_{6,2}$ at a generic element:

\begin{teo}(Infinitesimal Torelli, Theorem \ref{differenziale iniettivo})
	Let $[(C, \eta, B)]$ be  a  general element in $ \mathcal{RQ}_{6,2}$. Then $d\mathcal P_{g,2}$ is injective at $[(C, \eta, B)]$.
\end{teo}

Moreover, we prove the generic Torelli Theorem for the Prym map restricted to the locus of quintic planes curves:

\begin{teo} (Generic Torelli) \label{thm_gen_tor}
The restriction of $\mathcal P_{6,2}$ to the quintic plane locus $\mathcal {RQ}_{6,2}$ is generically injective.
\end{teo}

Notice that the tetragonal construction (see \cite{do_tetr}) applies in this case: indeed, given $(C,\eta,B)\in \mathcal {RQ}_{6,2}$ we can identify the points of $B$ to get a nodal curve which $3$ natural tetragonal series. Then, there are other coverings in $\overline {\mathcal R}_7$ with the same Prym variety. The previous theorem implies that these other coverings do not belong to  $\mathcal {RQ}_{6,2}$ (as subspace of $\overline {\mathcal R}_7$).

Since the proof of the second theorem is set-theoretical, we can not deduce the first, since $\mathcal {RQ}_{6,2}$ could be contained in some ramification or singular locus in $\mathcal R_{6,2}$.

The structure of the paper is as follows: after some preliminaries on geometric properties of quintic plane curves (section 2), we  devote sections 3 and 4 to prove the two main theorems. In both cases we can convert the statement in some deformation problems: the study of extensions in $Ext^1(\omega_C\otimes \eta,\eta^{-1})$ with coboundary map of ranks $0$ and $1$.

\textbf{Acknowledgments:} We are very grateful to Paola Frediani, Mart\'i Lahoz and Gian Pietro Pirola for stimulating discussions on this subject. We thank Angela Ortega for pointing out an inaccuracy in a first version of the paper.

\section{Preliminaries}

\subsection{Plane quintics and conics}

It is an easy application of Riemann-Roch Theorem that a smooth plane quintic $C$ is neither hyperelliptic nor trigonal. Moreover, any $g^1_4$ on $C$ is obtained  from the $g^2_5$ by subtraction of a point, so the $W^1_4(C)$ is isomorphic to the curve $C$ itself (see \cite{ACGH}, page 225). Here we collect some results describing the triplets $[(C,\eta, B=p_1+p_2)] \in \mathcal RQ_{6,2}$. The adjunction formula gives us: $\omega_C=\mathcal O_{\mathbb{P}^2}(-3+5) \restr{C}=\mathcal O_{C}(2)$. Thus, it turns out that conics, and specially conics tangent to $C$ in several points, play a crucial role in the analysis.

Our first result is the following:

\begin{lemma} \label{eta_two_minus_one}
 Let $[(C,\eta, B=p_1+p_2)] \in \mathcal RQ_{6,2}$. Assume that $p_1+p_2$ are general in $C$. Then $h^0(C,\eta \otimes \mathcal O_C(x))=0$ for all $x\in C$. In other words, $\eta $ is not of the form $\mathcal O(y+z-x)$, $x,y,z\in C$.
\end{lemma}
\begin{proof}
 By contradiction, assume that $\eta \cong \mathcal O(y+z-x)$. Then $2y+2z\sim 2x+p_1+p_2$ gives a $g^1_4$. So there is a point $q\in C$ and there exist lines $r_1$, $r_2$ such that $r_1\cdot C=2y+2z+q$ and $r_2 \cdot C=2x+p_1+p_2+q$. Hence, the line determined by $p_1$ and $p_2$ (that is, $r_2$) intersects $C$ in a point that belongs to a bitangent of $C$, this contradicts the genericity of the points.
\end{proof}
When also the curve $C$ is general, we can go further. Indeed, we can state the following:
\begin{prop}\label{eta_three_minus_two} Let $[C,\eta, B]\in \mathcal{RQ}_{6,2}$ be general. Then $\eta $ is not of the form $\mathcal O_C(x_1+x_2+x_3-y_1-y_2)$.
\end{prop}
\begin{proof}
 By contradiction, assume that $\eta \cong \mathcal O_C(x_1+x_2+x_3-y_1-y_2)$. Then:
 \[
 2 x_1 +2 x_2+ 2x_3 \sim 2 y_1 + 2y_2 +p_1 + p_2.
 \]
 Riemann-Roch formula tells us that $h^0(\Omega_C-(2 x_1 +2 x_2+ 2x_3))\geq1$, namely that there exists $z_1,z_2,z_3,z_4$ such that $2 x_1 +2 x_2+ 2x_3+ z_1+z_2+z_3+z_4$ and that $2 y_1 + 2y_2 +p_1 + p_2+z_1+z_2+z_3+z_4$ belong to $\vert \mathcal{O}_C(2)\vert$. This is equivalent to say that the pencil of conics passing through $z_1,z_2,z_3,z_4$ contains both a 3-tangent conic $Q_1$ and a 2-tangent conic $Q_2$ passing through $p_1,p_2$.

 Let $C$ be a degenerate plane quintic given by 5 lines in general position: $C=\bigsqcup_{i=1}^5l_i$. Take a conic $Q_1$ 3-tangent in $x_1,x_2,x_3$. Let $R_4:=Q_1\cdot C-2x_1-x_2-x_3$. Let $\mathcal{Q}$ be the pencil of conics determined by $R_4$. By construction, the four points of $R_4$ will lie in two lines of $C$, let us assume $l_1,l_2$. If there existed a 2-tangent conic $Q_2\in\mathcal{Q}$ then $Q_2$ would be tangent to two of the lines of $Q$, assume $l_3,l_4$. Thus $Q_2\cdot l_5=p_1+p_2$. That would say that the 2 points $p_1,p_2$ would lie in a line of $Q$. By assumption of generality this is not true.

 Since the contradiction holds for this specific configuration we claim that the same still holds for the general element $[C,\eta, B]\in \mathcal{RQ}_{6,2}.$
\end{proof}

\subsubsection{Pluritangent conics}

The aim of this section is to prove Lemma \ref{one_section}, which is a technical result with important implications in the rest of the paper. The proof of this lemma is a consequence of rather elementary considerations on conics which we discuss now.

\begin{prop}\label{proposizione coniche}
Assume that $C$ is a general plane quintic, and
 let $\mathcal Q_k\subset \mathbb P^5$ be the closure of the set of conics of rank $\ge 2$ tangent to $C$ at $k$ different points. Then the codimension of $\mathcal Q_k$ in $\mathbb P^5$ is $k$ for $1\le k \le 5$.
\end{prop}
\begin{proof}
First, consider in $\mathbb P^5\times C^{(k)} $ the closed subset $I_k$ defined as the closure of the set of pairs  $(Q,p_1+\ldots +p_k)$ such that all the points $p_i$ are different and $Q$ is a conic tangent to $C$ at all $p_i$, that is $T_C(p_i)\subset T_Q(p_i)$.

Since all the fibers of $I_1 \longrightarrow C$ are projective spaces of dimension $3$ we get that $I_1$ is irreducible of dimension $4$. Since the fibers of $I_1 \longrightarrow \mathbb P^5$ are finite, we get that the image of $I_1$, which is $\mathcal Q_1$, is an irreducible hypersurface.  Notice that this hypersurface contains the Veronese surface of double lines $\mathcal V\subset \mathbb P^5$.

On the other hand, the generic fibers of  $I_2 \longrightarrow C^{(2)}$ are lines. As before, projecting on $\mathbb P^5$, we obtain the codimension 2 subvariety, $\mathcal Q_2$, parametrizing bitangent conics. Notice that also this variety contains $\mathcal V$.

The tritangent conics can be studied similarly but there are some differences with respect to the previous cases: the map into $C^{(3)}$ is no longer surjective and the preimage of the Veronese surface $pr_1^{-1}(\mathcal V)$ gives now a component of the locus we want to study. Define $T'\subset C^{(3)}$ to be the image of $I_3$. Notice that $T'$ contains $pr_2(pr_1^{-1}(\mathcal V))$, which is  the surface $\Gamma_3=\{D\in C^{(3)} \mid h^0(C,\mathcal O_C(1)(-D))>0\}$. Given $a,b\in C$, with different tangent lines, then $T'$ intersects the curve $a+b+C$ in a finite number  of points: the conics tangent to $C$ in $a$ and $b$ is a pencil providing a morphism $C\longrightarrow \mathbb P^1$ of degree $6$ with a ramification divisor of degree $22$. Thus, $T'\cdot (a+b+C)=22$ for all $a+b\in C^{(2)}$ and $T'$ has dimension $2$. Hence, we have a decomposition $T'=T\cup \Gamma_3$. This ends the case $k=3$. To go further in the study of $\mathcal Q_3$ we need the following Lemma, which is obvious.

\begin{lemma}
 Let $Q$ be a conic tangent to $C$ in  three different points $p_1, p_2, p_3$. Then $Q$ is degenerate if and only if at least one of these cases occurs:
 \begin{enumerate}
     \item [a)] the three points are in a line $r$, and $Q$ is the double line $r^2$,
     \item [b)] two of the points, say $p_1, p_2$, have the same tangent line $b$ (a bitangent) and $Q$ is formed by $b$ and the line tangent to $C$ in $p_3$,
     \item [c)] two of the points, say $p_1, p_2$, have the same tangent line $b$ (a bitangent) and $p_3 \in b$; in this case $Q$ is formed by $b$ and any line through $p_3$.
 \end{enumerate}
\end{lemma}

 We say we are in the ``bitangent case'' in the case c) of the lemma (this is the only case where the conic is not completely determined by the divisor $D$ of degree $3$). Moreover, observe that the case a) corresponds to the map $pr_1^{-1}(V)\rightarrow \Gamma_3$ which  has degree $1$.
 In general, the fibers of   $I_3 \rightarrow T'$ are  points, except to in a finite number of points, the bitangent cases, where the fiber is isomorphic to $\mathbb P^1$. Let $\widetilde T $ be the blow-up of $T$ at the bitangent points.

Now we define a  map on the open set $U\subset T$ of the non-bitangent points. Denote by $Q_D$ the only conic such that $Q_D\cdot C \ge 2D$ for a $D\in U$. Then we consider the other points of intersection of the $Q_D$ with $C$:
\[
 \varphi: U\rightarrow C^{(4)}, \quad D\mapsto Q_D\cdot C-2D.
\]
Notice that this map could also be defined in the other component
$\Gamma_3$. In this case the image is simply the small diagonal $\Delta_{2,2}=\{2x+2y\mid x,y\in C\}$. Moreover, observe  that the map on $T$ extends to $\widetilde T$: an element $(Q,x_1+x_2+x_3)\in \widetilde T$ is in the exceptional divisor if $Q$ is a couple of lines $r\cdot s$ such that $r$ is a bitangent with $r \cdot C=2x_1+2x_2+x_3$ and $s$ satisfies $s\cdot C=x_3+a_1+a_2+a_3+a_4$. Then $\varphi (Q,x_1+x_2+x_3)=a_1+a_2+a_3+a_4$.

Let us denote by $\Delta_4\subset C^{(4)}$ the big diagonal, that  is, the image of the addition $\Delta \times C^{(2)}\longrightarrow C^{(4)}$, where $\Delta \subset C^{(2)}$ is the usual diagonal. Observe that $\Delta_{2,2} \subset \Delta_4 $.

 Since the image of the exceptional divisors are clearly not contained in $\Delta_4$ (but with non-empty intersection with $\Delta_4$) we deduce that $T_4:= \varphi (\widetilde T)\cap \Delta_4$ has dimension $1$. This proves the case $k=4$.

 Finally, the case $k=5$  reduces to notice that there is a bijection between $\mathcal{Q}_5$ and the non-trivial 2-torsion points $\alpha \in JC$ with
$h^0(C,\mathcal{O}_C(1)\otimes\alpha) > 0$. Hence finiteness is given by the theory of theta characteristics. \\ \end{proof}

The following holds.

\begin{lemma}\label{one_section} For a generic element $[(C,\eta, B)]\in \mathcal RQ_{6,2}$, we have that:
 \[
 h^0(C,\mathcal O_C(1)\otimes\eta)=1
 \]
\end{lemma}
\begin{proof} By Riemann-Roch Theorem, it is equivalent to show that $h^0(C,\mathcal O_C(1)\otimes\eta^{-1})=0$. Assume, by contradiction, that $h^0(C,\mathcal O_C(1)\otimes\eta^{-1})>0$, namely, that $\forall \eta$ there exist $x_1,x_2,x_3,x_4\in C$ such that $\eta (x_1+x_2+x_3+x_4)\cong  \mathcal O_C(1)$. This is equivalent to require that $\forall p_1,p_2\in C$ there exist $x_1,x_2,x_3,x_4\in C$ such that $ p_1+p_2 +2x_1+2x_2+2x_3+2x_4$ is the intersection of $C$ with a conic $Q$ which will be $4$-tangent to the quintic C. If the rank of $Q$ were $1$ then $p_1=p_2$ which is not possible. Hence, we have a $2$-dimensional family of conics of rank $\ge 2$ which are tangent at $4$ points of $C$. This contradicts Proposition \ref{proposizione coniche}.
\end{proof}

\subsection{Rank 2 vector bundles on curves}

The following result will be very useful along the paper:

\begin{lemma}{Beauville \cite[\S X.7]{beauBook}}\label{lemma beauville}
	Let $C$ be a curve and $E$ a rank 2 vector bundle on it. If \[2h^0(C, E)-3>h^0(C, \det E)\]
	then the kernel of the map $\alpha$ has a decomposable element defining (up to a saturation of the base locus) a line bundle $L\subset E$ such that $h^0(C, L)\geq 2$ and the quotient $E/L$ is a line bundle.
\end{lemma}

\section{Infinitesimal Torelli for plane quintics}

The aim of this section is to prove the injectivity of  the differential of the Prym map in a general $[(C,\eta,B)]\in  \mathcal RQ_{6,2}$.

\begin{teo}\label{differenziale iniettivo}
	Let $[(C, \eta, B)]$ be  a  general element in $ \mathcal{RQ}_{6,2}$. Then $d\mathcal P_{g,2}$ is injective at $[(C, \eta, B)]$.
\end{teo}
	\begin{proof}
		By contradiction, assume there exists $\xi\in H^1(T_C(-p_1-p_2))$ such that
		\[
		\begin{aligned}
		 d\mathcal P_{g,2}: H^1(T_C(-p_1-p_2))&\rightarrow S^2H^1(\eta^{-1})=Hom^s(H^1(C,\eta ^{-1})^*, H^1(C,\eta ^{-1})) \\
		 \xi&\mapsto0
		\end{aligned}
		\]
		This means that $\xi\in Ext^1(\mathcal O_C, T_C(-p_1-p_2))$ corresponds to the exact sequence \[\xi: \;\; 0\rightarrow T_C(-p_1-p_2)\rightarrow E\rightarrow \mathcal O_C\rightarrow 0  \] and that
		\[0\rightarrow\eta^{-1}\rightarrow E\otimes \omega_C\otimes\eta\rightarrow \omega_C\otimes\eta\rightarrow 0 \]
		has zero coboundary map. Thus, the study of the injectivity of the differential of the Prym map corresponds to the study of extensions $Ext^1(\omega_C\otimes\eta, \eta^{-1})$ of rank zero. Putting $F:=E\otimes \omega_C\otimes\eta$, we get \[H^0(C,F)=H^0(C,\omega_C\otimes\eta)=6. \]
		Furthermore, $\det F=\omega_C. $ Hence the map \[\Lambda^2H^0(C,F)\rightarrow H^0(C,\Lambda^2 F)=H^0(C,\omega_C)\]
		has kernel of dimension 9. By Lemma \ref{lemma beauville}, the kernel contains at least a decomposable element defining a line bundle $L$ such that $h^0(C,L)\geq 2$ and $F/L$ is a line bundle. Line bundles $L\hookrightarrow F$ of such type are parametrized by points in $Grass(2,6)$ intersected with the Kernel mentioned above. They fit in the following diagram:
		\begin{equation}\label{diagramma croce}
		\begin{tikzcd}
&	&0	\arrow[d] & &\\
&	&L \arrow{dr}{\tau}\arrow[d] & &\\
0\arrow[r]&\eta^{-1}\arrow[r]&F\arrow[r]\arrow[d]&\omega_C\otimes\eta\arrow[r]&0\\
& &M	\arrow[d] & &\\
& &0 & &
		\end{tikzcd}
		\end{equation}
Notice that	the map $\tau$ is necessarily non-zero, therefore we always have that:
\begin{equation}\label{condizione tau }
	h^0(C,\omega_C\otimes \eta\otimes L^{-1})>0.
\end{equation}

The vertical exact sequence of diagram \eqref{diagramma croce} gives us \begin{align}\label{condizioni}
			h^0(C,L)+h^0(C,M)&\geq 6\\
			h^0(C,L)-h^1(C,L)&=\deg L -5.\notag
		\end{align}
Since $M=\omega_C\otimes L^{-1}$, the sum of the two lines of \eqref{condizioni} gives us $2h^0(C,L)\geq \deg L +1.$
Moreover, since $C$ is non-hyperelliptic, Clifford's Theorem gives $\deg (L)>2(h^0(C,L)-1)$. Hence $\deg(L)+1\ge 2 h^0(C,L)$.  All together gives $5\le \deg(L)+1=2h^0(C,L)$. Since $h^0(C,L)\le 6$, the only possible values for $(\deg(L),h^0(C,L))$ are  $(5,3)$, $(7,4)$, $(9,5)$ and $(11,6)$.

Let us analyze all these possibilities:
		\begin{enumerate}
		\item[1.] $ h^0(C,L)=3 $ and $\deg L= 5$. In this case, necessarily $L=\mathcal O_C(1)$.
		\item[2.] $ h^0(C,L)=4 $ and $\deg L=7$. This is not allowed since  $\omega_C\otimes L^{-1}$ would be a $g^1_3$.
		\item[3.] $ h^0(C,L)=5 $ and $\deg L=9$. This implies that $L=\mathcal O_C(K_C-x)$, for some $x$ in $C$. As observed in \eqref{condizione tau }, diagram \eqref{diagramma croce} would require $0<h^0(C, \omega_C(\eta)\otimes L^{-1})=h^0(\eta +x)$. Therefore, there would exist $y,z\in C$ such that $\eta +x\sim y+z$. This contradicts Lemma \ref{eta_two_minus_one}.
		\item[4.] $ h^0(C,L)=6$ and $\deg L=11$. This is not possible, since diagram \eqref{diagramma croce} requires $h^0(C,\omega_C(\eta)\otimes L^{-1})>0$.
		\end{enumerate}

Hence, we can assume from now on that $L=\mathcal O_C(1)$ and therefore also $M=\mathcal O_C(1)$. By assumption, the coboundary map of the vertical exact sequence $\delta : H^0(C,\mathcal O_C(1))\rightarrow H^1(C,\mathcal O_C(1)) $ is the zero map. The map $\delta$ is given by the cup product with $\varsigma\in Ext^1(\mathcal O_C(1),\mathcal O_C(1))\cong H^1(C,\mathcal O_C)\cong H^0(C,\omega_C)^*. $ We consider
the cup-product map:
\[
H^0(C,\mathcal O_C(1))\otimes H^1(C,\mathcal O_C ) \rightarrow H^1(C,\mathcal O_C(1))
\]
that induces
\[
H^0(C,\mathcal O_C(1)) \otimes H^0(C,\mathcal O_C(1))\otimes H^1(C,\mathcal O_C )\rightarrow \mathbb{C}.
\]
Hence we obtain that $\varsigma\cdot H^0(C,\omega_C)=0$, namely $\varsigma=0$.
This shows that the vertical exact sequence splits and thus the horizontal exact sequence becomes
\[
0\rightarrow \eta^{-1}\rightarrow \mathcal O_C(1)\oplus \mathcal O_C(1)\rightarrow \omega_C\otimes 	\eta\rightarrow 0.
\]

The non-zero section of $\mathcal O_C(1)\otimes\eta$ determines a skyscraper subsheaf (roughly speaking, the quotient $\mathcal O_C(1)/\eta^{-1}$) of $\omega_C\otimes 	\eta$. Since this is impossible we conclude.  \end{proof}

\section{Generic Torelli in the locus quintic plane quintics}

In view of the results of the previous section it makes sense to ask whether the  restriction of $\mathcal P_{g,2}$ to $\mathcal RQ_{6,2}$ is generically injective. All this section is devoted to the proof of the generic Torelli Theorem \ref{thm_gen_tor}.

\begin{defin}
The \textit{semicanonical curve} $C_\eta$ is the image of $C$ through the projective map associated with the line bundle $\omega_C\otimes\eta$.

\end{defin}
We denote with $I_2(C_\eta)$ the space of the homogeneous quadratic  polynomials vanishing on $C_\eta$, namely the kernel of the multiplication map \eqref{codifferenziale} (\cite{green_laz}). Our goal is to recover the curve $C_\eta$ in the intersection of such quadrics. Then $C_\eta$ identifies, up to isomorphism, the pair $(C,\eta)$. Since $C$ is non-hyperelliptic, then $h^0(C,\eta^{\otimes 2})=1$, and the divisor $B$ is completely determined by $C$ and $\eta$.

We recall that there is a natural bijection between the points of $\mathbb{P}H^0(C,\omega_C\otimes \eta)^*$ lying in the intersection of the quadrics of $I_2(C_\eta)$ and the extensions (up to isomorphism and multiplication by a scalar)
\begin{equation}\label{estensioni}
	0\rightarrow\eta^{-1}\rightarrow E\rightarrow \omega_C\otimes\eta\rightarrow 0
\end{equation}
	with coboundary map of rank 1. Moreover, given $p$ in the intersection of the quadrics, the image of the corresponding $\delta: H^0(C,\omega_C\otimes\eta)\rightarrow H^1(C,\eta^{-1})$ identifies a 1-dimensional space in $ H^0(C,\omega_C\otimes\eta)^*$ which corresponds exactly to $p\in \mathbb{P}H^0(C,\omega_C\otimes \eta)^* $. This is Lemma 1.2 of \cite{lange_sernesi}.

So let us classify extensions \eqref{estensioni} with coboundary map of rank 1. This is equivalent to require $h^0(C,E)=5$, hence we have
\[
2h^0(C,E)-3=7 > h^0(C,\det E)=h^0(C,\omega_C)=6.
\]
Thus, we can apply Lemma \ref{lemma beauville}: $E$ has a sub-line bundle $L$ such that $h^0(C,L)\geq 2$ and $E/L$ is a line bundle. The situation is summarized in a diagram as the one in \eqref{diagramma croce}. In the rest of the paper we consider two different subsets of rank 1 extensions in $Ext^1(\omega_C\otimes\eta, \eta^{-1})$:
\[Ext^1_{ns}(\omega_C\otimes\eta, \eta^{-1})\quad\text{and}\quad Ext^1_{s}(\omega_C\otimes\eta, \eta^{-1}). \]
The first consists of rank 1 extensions $E\in Ext^1(\omega_C\otimes\eta, \eta^{-1})$  admitting non-special sub-line bundles $L\subset E$, while the second deals with the special cases.

\subsection{L non-special.} In this case, since $h^0(C,M)=h^0(C,\omega_C \otimes L^{-1})=h^1(C,L)$, we get that $h^0(C,L)=h^0(C,E)=5=h^0(C,\omega_C
\otimes \eta)-1 $. Notice that the map $L\rightarrow \omega_C\otimes \eta$ is non-zero, therefore  $L= \omega_C\otimes \eta(-p) $ and so, in cohomology, we obtain the following diagram:
	\begin{equation}
	\begin{tikzcd}[column sep=1.2em]
	&	&0	\arrow[d] &0	\arrow[d] &\\
	&	&H^0(C,\omega_C \otimes \eta(-p))\arrow[d]\arrow[equal]{r} &H^0(C,\omega_C\otimes \eta(-p) )\arrow[d] & \\
	0\arrow[r]&H^0(C,\eta^{-1})\arrow[r]&H^0(C,E)\arrow[r]\arrow[d]&H^0(C,\omega_C\otimes\eta)\arrow[r]\arrow[d]&H^0(C,\omega_C\otimes\eta)^*\\
	& &H^0(C,\eta^{-1}(p))\arrow[r]	\arrow[d] &\mathbb{C}_p\arrow[ur] &\\
	& &0. & &
	\end{tikzcd}
	\end{equation}
	The fact that $ H^0(C,\omega_C\otimes \eta(-p))\rightarrow H^0(C,\omega_C\otimes\eta)^*  $ is the zero map, allows us to consider the map $\mathbb{C}_p\rightarrow H^0(C,\omega_C\otimes\eta)^* $, therefore with obtain the image of $p$ in $C_{\eta}$.
		\begin{remark}
		Notice that the  vertical extension splits for every $p$. Indeed we have \[
	\begin{aligned}
	&Ext^1(\eta^{-1}(p),\omega_C\otimes \eta(-p)\cong Ext^1(\mathcal O_C,\omega_C\otimes \eta ^{\otimes 2}(-2p))\\ & \cong H^1(C,\omega_C\otimes \eta ^{\otimes 2}(-2p))\cong H^0(C,\mathcal O_C(2p-p_1-p_2))^*=0.
	\end{aligned}
	\]
	\end{remark}
	This remark suggests that, vice versa, the extension
	\[
	0 \rightarrow \eta ^{-1} \rightarrow \eta^{-1}(p) \oplus \omega_C \otimes \eta (-p) \rightarrow \omega_C \otimes \eta \rightarrow 0.
	\]
	corresponds exactly to the point $p\in\mathbb{P}H^0(\omega_C\otimes\eta)^*$. Thus, we have the following:
	\begin{prop}
	  The extensions in $Ext^1_{ns}(\omega_C\otimes\eta, \eta^{-1})$ trace out the semicanonical curve $C_\eta$ in $\mathbb{P}H^0(\omega_C\otimes\eta)$.
	\end{prop}

\subsection{L special.} In this case, using diagram \eqref{diagramma croce} and the Riemann-Roch formula, we get conditions similar to \eqref{condizioni} and we obtain that $2h^0(C,L)-2 \deg(L) \ge 0$. Therefore
\[
1\leq Cliff(L)=\deg(L)- 2 h^0(C,L)+2 \leq 2.
\]
Summarizing, the line bundle $L$ satisfies the following three conditions:
\begin{enumerate}
    \item [a)] $Cliff(L)=1,2$,
    \item [b)] $2\le h^0(C,L) \le h^0(C,E)=5$,
    \item [c)] $h^0(C,\omega_C\otimes \eta \otimes L^{-1})>0$.
\end{enumerate}

We need some notation to state the next lemma. By Lemma \ref{one_section} we have that $ h^0(C,\mathcal O_C(1)\otimes \eta)=1 $. Let $t$ be the generator and let $D$ be the unique effective divisor in the linear series $\vert \mathcal O_C(1)\otimes \eta \vert $. Put $D=a_1+\ldots +a_6$. We have the following:

\begin{lemma} Let $C$ be a general plane quintic and let $L$ be a line bundle satisfying these three conditions. We have the following possibilities:
\begin{enumerate}\label{casi possibili}
    \item [a)] If $Cliff(L)=1$, then $L= \mathcal O_C(1)$.
    \item [b)] If $Cliff(L)=2$, then either there exists $p\in C$ such that $L\cong \mathcal O_C(1)(-p)=:L_p$, or $L=\mathcal O_C(1)(a_i)=:M_{a_i}$, $i=1,\ldots ,6$.
 \end{enumerate}
\end{lemma}
\begin{proof}
Assume first that $Cliff(L)=1$, in particular $\deg(L)=2h^0(C,L)-1$. Notice that $h^0(C,L)=2$ would give a trigonal series, and that $h^0(C,L)=3$ implies $L\cong \mathcal O_C(1)$.
 On the other hand, $h^0(C,L)=4$, $\deg L=7$ implies that $\omega_c\otimes L^{-1}$ is a $g^1_3$. Finally, if $h^0(C,L)=5$ and $\deg L=9$, we have that $L=\omega_C(-p)$, for some $p\in C.$ Then, by the condition c),  $\eta \otimes \mathcal O_C(p)\cong \mathcal O_C( x_1+x_2)$, for some $x_1,x_2\in C$. This contradicts Lemma \ref{eta_two_minus_one}.

 Assume now that $Cliff(L)=2 $ and thus $ 2 h^0(C,L)=\deg L$. If $h^0(C,L)=2$, then $L$ is a $g^1_4$ and then is of the form $L_p$ as in the statement. Let us consider the cases $(h^0(C,L),\deg (L))=(3,6), (4,8),(5,10)$.

If $\deg L=6$ and $ h^0(C,L)=3$, then $\omega _C\otimes L^{-1}$ gives a $g^1_4$, hence $L=\mathcal O_C(1)(p)=:M_p$ for some $p\in C$. The condition $h^0(C,\omega_C(\eta)\otimes L^{-1})=h^0(C,\mathcal O_C(1)\otimes \eta (-p))>0$ forces $p=a_i$.

In the case $h^0(C,L)=4$ and $\deg(L)=8$, we obtain that $\eta$ is of the form $\mathcal O_C(x_1+x_2+x_3-y_1-y_2)$ and this contradicts Proposition \ref{eta_three_minus_two}.

Finally, if $\deg L=10$ and $ h^0(C,L)=5$, then $h^0(C,L)-h^1(C,L)=5-h^1(C,L)=\deg(L)+1-6=5$. So $L$ is non-special.
\end{proof}

Let us study separately the cases in the Lemma \ref{casi possibili}. We will denote the three possibilities as $Ext^1_{s,g^2_5}(\omega_C\otimes\eta,\eta^{-1})$, $Ext^1_{s,g^1_4}(\omega_C\otimes\eta,\eta^{-1})$ and $Ext^1_{s,g^2_6}(\omega_C\otimes\eta,\eta^{-1})$

\vskip 3mm
\noindent	\textbf{
Case $L=\mathcal O_C(1)$.}
\vskip 3mm

By Lemma \ref{one_section} we have $H^0(C,\mathcal O_C(1)\otimes \eta)=\langle t\rangle$. Tensoring the vertical sequence of \eqref{diagramma croce} with $\eta$ we get $H^0(C,\mathcal O_C(1)\otimes \eta)\hookrightarrow H^0(C,E\otimes\eta)$. Thus, using the section $t$, we can construct the following diagram:
 	\begin{equation}\label{diagramma croce g^2_5}
 \begin{tikzcd}
 &	&0	\arrow[d] &0\arrow[d] &\\
 &	&\mathcal O_C(1) \arrow[d]\arrow[equal]{r} &\mathcal O_C(1) \arrow[d]&\\
 0\arrow[r]&\eta^{-1}\arrow[r]\arrow[equal]{d}&E\arrow[r]\arrow[d]&\omega_C\otimes\eta\arrow[r]\arrow[d]&0\\
0 \arrow[r] &  \eta^{-1} \arrow[r] & \mathcal O_C(1)	\arrow[d]	\arrow[r] &\mathcal{O}_D \arrow[r] \arrow[d] & 0\\
 & &0 & 0&
 \end{tikzcd}
 \end{equation}

Therefore, in cohomology we have:
	\begin{equation}\label{diagramma croce g^2_5 in coomologia}
 \begin{tikzcd}
 	&0	\arrow[d] &0	\arrow[d] &\\
 	&H^0(C,\mathcal O_C(1)) \arrow[d]\arrow[equal]{r} &H^0(C,\mathcal O_C(1)) \arrow[d]&\\
 0\arrow[r]&H^0(C,E)\arrow[r]\arrow[d]&H^0(C,\omega_C\otimes\eta)\arrow{r}{\delta}\arrow[d]&H^0(C,\omega_C\otimes\eta)^* \arrow[equal]{d}\\
 0\arrow[r] &H^0(C,\mathcal O_C(1))		\arrow[r] &H^0(C,\mathcal{O}_D)\arrow{r}{f} &   H^0(C,\omega_C\otimes\eta)^* \\
  & & &
 \end{tikzcd}
 \end{equation}

Notice that Im$(\delta)\subset $Im $(f)$.  Since $f$ is simply the evaluation at the points in the support of $D$, we have that $\mathbb P(\text {Im} f)$ is the linear variety generated by the image of the points by the semicanonical map
$\phi : C\longrightarrow \mathbb P H^0(C, \omega_C \otimes \eta )^*$, that is $\langle \phi(a_1),\ldots ,\phi(a_6) \rangle =:\langle \phi(D) \rangle $. Since $H^0(C,\omega _C\otimes \eta (-D))=H^0(C,\mathcal O_C(1))\cong \mathbb C^3$, we have that
$\langle \phi(D) \rangle$ is a projective plane in $\mathbb P H^0(C,\omega_C\otimes\eta)^*$. Then the point corresponding to the rank $1$ horizontal extension lands in this plane, which does not depend on the extension. Thus we have shown the following:
\begin{prop}\label{immagini estensioni g^5_2}
  The rank 1 special extensions in $Ext^1_{s, g^2_5}(\omega_C\otimes\eta, \eta^{-1})$ correspond to points in the plane $\langle\phi(D)\rangle\subset\mathbb{P}H^0(C,\omega_C\otimes\eta)^*$.
\end{prop}

\begin{remark}
We stress that extensions in $Ext^1_{s, g^2_5}(\omega_C\otimes\eta, \eta^{-1})$ do exist. Indeed they come from non-splitting  extensions \[
 0\rightarrow \mathcal O_C(1)\rightarrow E\rightarrow \mathcal O_C(1)\rightarrow 0
\]
with coboundary map $\delta: H^0(C,\mathcal O_C(1))\rightarrow H^1(C,\mathcal O_C(1))$ of rank 1. We know that these ones do exist since they are described by the elements or rank 1 in the image of map
\[
Ext^1(\mathcal O_C(1),\mathcal O_C(1))\xrightarrow{m^*}Hom(H^0(C,\mathcal O_C(1)), H^1(C,\mathcal O_C(1))
\]
i.e., dualizing to
\[
S^2H^0(C,\mathcal O_C(1))\xrightarrow{m}H^0(C,\omega_C)=H^0(C,\mathcal O_C(2)),
\]
by rank $1$ conics. This shows that they correspond to points of the Veronese surface $\mathcal{V}\subset \mathbb{P}^5$.
\end{remark}

\noindent \textbf{Case $L_p=\mathcal O_C(1)(-p)$, $p\in C$.}
\vskip 3mm

We have that $L_p$ is a $g^1_4$. Let us denote with $M_p=\mathcal O_C(1)(p)$, $p\in C$. We consider extensions  \begin{equation}\label{estensione g^1_4}
\quad	0\rightarrow L_p\rightarrow E\rightarrow M_p\rightarrow 0
\end{equation}
with $h^0(C,E)=5$.
Observe that $h^0(C,L_p)=2$ and $h^0(C,M_p)=3$. Therefore the coboundary map
\[
\delta: H^0(C,M_p)\rightarrow H^1(C,L_p)\cong H^0(C,M_p)^*
\]
is the zero map.  Let us denote by $Ext^1_0(M_p,L_p)\subset Ext^1(M_p,L_p)$ the set of the rank 0 extensions as in \eqref{estensione g^1_4}.

\begin{lemma}\label{estensioni verticali rango 0}
Points in $Ext^1_0(M_p,L_p)$ are in bijection with the co-kernel of the multiplication map:
\[
m: H^0(C,M_p)\otimes H^0(C,M_p)\rightarrow H^0(C,\omega_C(2p)).
\]
\end{lemma}
\begin{proof} Consider an extension as in \eqref{estensione g^1_4}. Then the rank is $0$ if
taking $\omega \in H^0(C,M_p)$, we have $\xi\omega(\omega')=\xi(\omega\omega')=0$ for every $\omega'\in H^0(C,M_p)$.
This determines (and is determined by) an element in the co-kernel of the map:
\[
m: H^0(C,M_p)\otimes H^0(C,M_p)\rightarrow H^0(C,\omega_C(2p))\cong H^1(C,\mathcal O_C(-2p))^*\cong Ext^1(M_p,L_p)^*.
\]
\end{proof}

Notice that in this case the co-kernel is always non-empty. Indeed,  $H^0(C,M_p)=H^0(C,\mathcal O_C(1))$ for every $p$, hence the image of the multiplication map has co-dimension $1$ in the target, that is the co-dimension of $H^0(C,\omega_C)$ in $H^0(C,\omega_C(2p))$. Thus, up to scalars, the non-trivial rank 0 vertical extension in $Ext^1(M_p,L_p)$ is unique and it only depends on the point $p$. This says that the vector bundle $E$ is determined by $L_p\subset E$ for every point $p\in C$. For this reason, from now on we denote it by $E_p$. Furthermore, we can interpret $\mathcal{E}:=Ext^1_0(M_p,L_p)$ as a line bundle over the curve $C$.

In order to present it more precisely, we discuss Lemma \ref{estensioni verticali rango 0} in families. Let $C\times C\xrightarrow{\pi_i}C$ be the projection on the $i$-th factor, $i=1,2$, and let $\Delta\subset C\times C$ be the diagonal. The sheaves \[\mathcal{F}:= \pi_2^*(\mathcal{O}_C(1))\otimes \mathcal{O}_{C\times C}(\Delta) \quad\text{and}\quad\mathcal{G}:= \pi_2^*(\omega_C)\otimes \mathcal{O}_{C\times C}(-2\Delta)\]
and the co-kernel of the map \[{\operatorname{Sym}}^2 R^0\pi_{1*}(\mathcal{F})\longrightarrow R^0\pi_{1*}(\mathcal{G}) \]
yields the definition of $\mathcal{E}$. Notice that, by construction, we have
\[\mathbb{P}(\mathcal{E})\cong C.\]

Let us fix a point $p\in C$ and let us consider the exact sequence \eqref{estensione g^1_4}. Tensoring with $\eta$, we deduce that
\[
H^0(C,E_p\otimes\eta)\cong H^0(C,M_p\otimes \eta)=2.
\]

Therefore, for every $E_p$, we have a 2-dimensional family of diagrams parametrized by sections $s\in H^0(C,M_p\otimes \eta)$
as follows (here we put $(s)_0=D_s$):
\begin{equation}\label{diagramma croce g^1_4}
 \begin{tikzcd}
 &	&0	\arrow[d] & &\\
 &	&L_p \arrow[d]\arrow[equal]{r} &L_p \arrow[d]&\\
 0\arrow[r]&\eta^{-1}\arrow{r}{\cdot s}\arrow[equal]{d}& E_p\arrow[r]\arrow[d]&\omega_C\otimes\eta\arrow[r]\arrow[d]&0\\
 0\arrow[r]&\eta^{-1}\arrow{r}{\cdot s} &M_p	\arrow[d]\arrow[r] &\mathcal O_{D_s}\arrow[r] &0\\
 & &0 & &
 \end{tikzcd}
\end{equation}
 In other words, we have a map:
\begin{align}\label{fibrato rango 2}
    Ext^1_{s,g^1_4}(\omega_C\otimes\eta, \eta^{-1})&\rightarrow\mathcal{E}\\
    (s,p)&\mapsto <E_p>\notag
\end{align}
   with fibers given by $H^0(M_p\otimes \eta)$.

Notice that $\mathbb{C}\cong H^0(C,\mathcal O_C(1)\otimes \eta)\subset H^0(M_p\otimes \eta) $. In particular, the divisor $D+p$ belongs to $ \vert M_p\otimes\eta\vert$. Let $s=t\mu_p$, where $\mu_p(p)=0$, be the section such that $(s)_0=D+p$. We have the following:

   \begin{prop}\label{curva C nella superficie}
     The image of extension  $(t\mu_p,p)\in Ext^1_{s,g^1_4}(\omega_C\otimes\eta, \eta^{-1})$ is the point $p$ in $C_\eta\subset \mathbb{P}H^0(C,\omega_C\otimes \eta)^*$ for every $p\in C$.
   \end{prop}
   \begin{proof}
        Let us consider the element $tl\in H^0(\omega_C\otimes\eta)$: $l$ is a line in $ H^0(C,\mathcal{O}_C(1))$ not passing through $p$. Taking diagram \eqref{diagramma croce g^1_4} in cohomology, one easily checks that the vertical sequence
\[
0\rightarrow H^0(C,L_p)\rightarrow H^0(C,\omega_C\otimes\eta) \xrightarrow{v}H^0(C,\mathcal O_{D_s})\rightarrow H^1(C,L_p)\rightarrow 0,
\]
where $v$ is the evaluation map $\alpha\mapsto (\alpha(a_1), \dots, \alpha(a_6),\alpha(p)), $ dualizes to the horizontal sequence
\[
\begin{aligned}
0\rightarrow H^0(C,M_p)\rightarrow H^0(C,\mathcal O_{D_{s}})&\xrightarrow{v^*} H^1(C,\eta^{-1})\cong \\
& \cong H^0(C,\omega_C\otimes\eta)^*\rightarrow   H^1(C,M_p)=H^0(C,L_p)^*\rightarrow 0.
\end{aligned}
 \]
Since we know that the coboundary map $\delta : H^0(C, \omega_C\otimes \eta)\longrightarrow H^0(C, \omega_C \otimes \eta)^*$ has rank 1, once we show that $\delta(tl) \neq 0$, we are almost done. This follows immediately from the fact that
\[
\delta(tl)=v^*v(tl)=v^*(0,\dots, 0,tl(p)).
\]
Up to a scalar, this yields the evaluation map in $p$, i.e. the point $p\in C_\eta$. \\
   \end{proof}
   \begin{remark}
   Notice that the previous proposition also clarifies how rank 1 extensions in $Ext^1_{s,g^1_4}(C,\omega_C\otimes\eta, \eta^{-1})$, with $s\neq t\mu_p$, contribute to the intersection of the quadrics in $\mathbb{P}I_2(C_\eta)$. Indeed, it shows that changing the section $t\mu_p$ to a section $s\in H^0(C,\mathcal{O}_C(1)(p)\otimes \eta)$ in a neighborhood of it, by continuity, $\delta(tl)$ is again different from zero. This says that to study the image of the map
\[\mathbb{P}H^0(C,\omega_C\otimes\eta)\otimes \mathbb{P}Ext^1_{s,g^1_4}(\omega_C\otimes\eta, \eta^{-1})\rightarrow \mathbb{P}H^0((C,\omega_C\otimes\eta)^*)\]
(which gives us all the points in $\mathbb{P}H^0(C,\omega_C\otimes\eta)^*$ coming from $\mathbb{P}Ext^1_{s,g^1_4}(\omega_C\otimes\eta, \eta^{-1})$)
it is sufficient to look at the image of
\begin{equation}\label{mappa moltiplicazione finale}
   \phi: \{tl\}\otimes \mathbb{P}Ext^1_{s,g^1_4}(\omega_C\otimes\eta, \eta^{-1})\rightarrow \mathbb{P}H^0((C,\omega_C\otimes\eta)^*).
\end{equation}
  By \eqref{fibrato rango 2}, $S_1:=\mathbb{P}Ext^1_{s,g^1_4}(\omega_C\otimes\eta, \eta^{-1})$ is a $\mathbb{P}^1$-bundle over $C$ with fibres $\mathbb{P}H^0(C,\mathcal{O}_C(1)(p)\otimes \eta)$. Thus, the problem finally turns upon the study of the surface $S$ which is  $\phi (S_1)$.
 \end{remark}

The following general fact holds:
\begin{prop}\label{two_surfaces}
Let $f:S_1\rightarrow C$ be a $\mathbb{P}^1$-bundle over a base curve $C$ with section $\sigma:C \longrightarrow S_1$, we put $C_1:=\sigma(C)$. Let $\tau: S_1\rightarrow \mathbb{P}^N$ be a morphism and call $S_2:=\tau(S_1)$ the image. Assume that $\dim S_2=2$, $\tau(\mathbb{P}^1)$ is a line and $\tau_{\vert C_1}$ is an embedding. Then $\tau$ is birational and  there is at most one point $q$ such that $S_1\setminus \tau ^{-1} (\{ q\})$ is isomorphic to $S_2\setminus \{q\} $.
\end{prop}
\begin{proof}
Let $r_1$ be a line in $S_1$. First notice that $\tau^{-1}(\tau(r_1)$ cannot contain more than one vertical line (namely $r_1$), neither $\lambda r_1$, $\lambda\in \mathbb{C}$. This is because in these cases, the assumption on $\tau_{\vert C_1}$ would be violated. Therefore, assume now, by contradiction, that $\tau^{-1}(\tau(r_1))$ contains $r_1$ plus a curve $\Gamma$ not containing more lines and such that the map to $C$ has degree $\ge 2$. This is impossible since the points in the intersection $l\cap\alpha$, with $l$ any other vertical line on $S_1$, must be mapped to $\tau(r_1)$. Thus, the image $\tau(l)$ can no longer be a line. Therefore, $\tau^{-1}(\tau(r_1))$ at worst is given by $r_1$ plus a section $\beta$. If this section is contracted to a point in $\tau(r_1)$ then we would obtain a cone and out of the vertex we are fine. Since lines transform into lines  there is at worst one point where this can happen, therefore we are done. On the other hand, if the section is not contracted to a point, then $\tau(r_1)$ would be birational to $\tau(C_1)$ and this is impossible.
\end{proof}
\begin{cor}
The morphism $\phi$ satisfies the hypothesis of Proposition \ref{two_surfaces}.
\end{cor}
\begin{proof}
 We already know that $\mathbb{P}Ext^1_{s,g^1_4}(\omega_C\otimes\eta, \eta^{-1})$ is a $\mathbb{P}^1$-bundle over $C$. Moreover, let us observe that the bundle $Ext^1_{s,g^1_4}(\omega_C\otimes\eta, \eta^{-1})$ naturally carries a section which is isomorphic to the curve $C$. It is obtained taking the point $\{t\mu_p\}\in \mathbb{P}H^0(\mathcal{O}_C(1)(p)\otimes \eta)$ for every $p\in C$. By Proposition \ref{curva C nella superficie}, this section is clearly sent by $\phi$ to the curve $C_\eta\subset S$. Finally, since $\phi$ is a multiplication map, the vertical line of the bundle are sent to line in $S$ and furthermore $\dim \text{Im}(\phi)=2$.
\end{proof}
Thus, we can conclude with the following:
\begin{prop}
The rank 1 special extensions in $Ext^1_{s,g^1_4}(\omega_C\otimes\eta, \eta^{-1})$ correspond to a surface $S\subset \mathbb{P}H^0(\omega_C\otimes \eta)^*$ birational to a ruled surface with base curve $C$.
\end{prop}

\vskip 3mm
\noindent \textbf{Case $L=\mathcal{O}_C(1)(a_i)=:M_{a_i}, i=1,\dots,6.$}
\vskip 3mm

We consider extensions \[0\rightarrow M_{a_i}\rightarrow E\rightarrow \omega_C \otimes M_{a_i}^{-1}\rightarrow 0\]
with $h^0(E)=5$ and null coboundary map. An easy reformulation of Lemma \ref{estensioni verticali rango 0} shows that such extensions are in bijection with the co-kernel of the multiplication map
\[m: H^0(C, \omega_C\otimes M_{a_i}^{-1})\otimes H^0(C, \omega_C\otimes M_{a_i}^{-1})\rightarrow H^0(C, \omega_C(-2a_i)). \]
Since the co-kernel has dimension 1, it turns out that the rank 0 vertical extension is unique for every $a_i$, up to scalars. Thus, we have to study just six extensions, let us denote them $E_{a_i}.$

In order to produce a diagram like \eqref{diagramma croce g^1_4}, we observe that $h^0(C,E\otimes \eta)=3$: 2 generators come from $H^0(C,\mathcal{O}_C(1)\otimes \eta(a_i))$ and one from $H^0(C,\mathcal{O}_C(1)\otimes \eta(-a_i))$. The first two are not admissible since they would determine a skyscraper subsheaf of $\omega_C\otimes\eta$. Thus, it only remains to consider the section of $H^0(C,\mathcal{O}_C(1)\otimes \eta(-a_i))$. Notice that we already know the divisor in this linear series: it is $\hat{D_i}:=a_1+\dots+\hat{a}_i+\dots+a_6$. In short, we have to consider just 6 diagrams as the following:
\begin{equation}\label{diagramma croce g^2_6}
 \begin{tikzcd}
 &	&0	\arrow[d] & &\\
 &	&M_{a_i} \arrow[d]\arrow[equal]{r} & M_{a_i}\arrow[d]&\\
 0\arrow[r]&\eta^{-1}\arrow{r}\arrow[equal]{d}& E_{a_i}\arrow[r]\arrow[d]&\omega_C\otimes\eta\arrow[r]\arrow[d]&0\\
 0\arrow[r]&\eta^{-1}\arrow{r} & \omega_C\otimes M_{a_i}^{-1}	\arrow[d]\arrow[r] &\mathcal O_{\hat{D_i}}\arrow[r] &0\\
 & &0 & &
 \end{tikzcd}
\end{equation}
Passing in cohomology, notice that the situation is very close to the one of diagram \eqref{diagramma croce g^2_5 in coomologia}. The point corresponding to the rank 1 horizontal extension in \eqref{diagramma croce g^2_6} lands in the plane $\langle\phi(\hat{D_i})\rangle$, which is exactly the same plane of Proposition \ref{immagini estensioni g^5_2}. Indeed, $H^0(C, \omega_C\otimes\eta(-\hat{D}))= H^0(C,\mathcal{O}_C(1)(a_i))=H^0(C,\mathcal{O}_C(1))\cong \mathbb{C}^3$. Since this is independent of $i$, we have the following:
\begin{prop}
The rank 1 special extensions in $Ext^1_{s,g^2_6}(\omega_C\otimes\eta,\eta^{-1})$ correspond to 6 points in the plane $\langle\phi(D)\rangle\subset\mathbb{P}H^0(\omega_C\otimes \eta)^*$.
\end{prop}

To resume, our analysis shows what follows.
\begin{teo}
Let $(C,\eta, B)\in \mathcal{RQ}_{6,2}$ be a general element and let $\mathbb{P}I_2(C_\eta)$ be the space of the quadrics vanishing on the semicanonical model $C_\eta$ of $C$. Then the intersection of the quadrics consists of points landing on a projective plane and of a surface $S$. Moreover, there is a birational map from  a ruled surface with base curve $C_\eta$ to $S$ which is an isomorphism out of the preimages of a finite number of points in $S$.
\end{teo}

Using this result now we can finish the proof of our main theorem \ref{thm_gen_tor}. Indeed, by using the differential of the Prym map at a generic point $(C,\eta, B)$ we obtain a family of quadrics whose intersection is, up to a finite number of points and a linear variety of dimension $2$, a surface $S$ as in the Theorem above. The curve $C_{\eta}$ is contained in this surface. Assume that there is another element in the fibre $(C',\eta',B')$. Then the curve $C'_{\eta '}$ is also contained in $S$. The preimage by $\phi $ of this curve has only one component  $C''$ birational to $C'_{\eta '}$. Then the map $C''\longrightarrow C$ must be birational and therefore $C\cong C'$.
Moreover, the surface $S$ is either isomorphic to the ruled surface $S_1=\mathbb{P}Ext^1_{s,g^1_4}(\omega_C\otimes\eta, \eta^{-1})$ or it is a cone with base $C_{\eta}$. In both cases the curve $C'_{\eta'}$ projects to $C_{\eta}$ (following the rulling in the first case or projecting from the vertex in the second). Hence this isomomorphism send $\eta '$ to $\eta$.


text:
\section{INTRODUCTION}

In recent years, BCI has been one of the most interesting biomedical engineering research fields. For patients suffering from stroke, it is very meaningful to provide a communication method to deliver brain messages and commands to the external world except from the normal nerve-muscle output pathway \cite{wolpaw2002brain}.
Due to the natural and non-intrusive characteristics, most BCI systems select the EEG signal as input.
Event-related desyn-chronization/synchronization (ERD/ERS) is the Rhythmic activity of EEG. The changes of ERD/ERS induced by Motor Imagery (MI) has been widely used in many BCI systems.
The very aim of EEG signal processing in BCI system is translating raw EEG signals into the class of these signals. In addition, EEG-based BCI can be considered as a complex pattern recognition system which includes several critical steps, signal acquisition, preprocess, feature extraction, classification, control external devices and feedback.
If the EEG signal is decoded correctly, it can be translated into commands to control external equipments such as rehabilitation devices \cite{lew2006non} or other devices  \cite{doud2011continuous}.

In order to improve the accuracy of EEG classification, a lot of related work  has been carried out. The performance of a pattern recognition like system depends on both the features selected and the classification algorithms employed.
A great variety of features have been proposed to used in BCI such as Band Powers (BP) \cite{kaiser2011first}, Power Spectral Density (PSD) values \cite{waldert2009review}, iterative two-dimensional nearest feature line (INFL) and so on. In recent years, the common spatial pattern (CSP) \cite{ramoser2000optimal} has been proved to be an expressive feature of EEG signals. A lot of related work has been proposed such as CSSP \cite{lemm2005spatio}, CSSSP \cite{dornhege2006combined}, FBCSP \cite{ang2008filter}, WCSP \cite{mousavi2011wavelet} and SCSSP \cite{aghaei2016separable}.
Moreover, many machine learning techniques such as naive Bayes, K-NN, SVM, LDA, ANN, and Deep Learning \cite{jirayucharoensak2014eeg} have been applied as BCI classifiers.
Although the number of research studies on EEG-based BCI has been great increasing in past decades, it is very difficult to decode the useful information from EEG signal in a reliable and efficient way.
Most of traditional methods focus on a particular feature only which limited the efficiency of BCI.

The rest of this paper is organized as follows. Our algorithm will be introduced in Section 2. In Section 3, experimental results and performance analysis are provided. Finally, the conclusions will be given in Section 4.

\section{Method}

\begin{figure}[thpb]
    \centering
    \includegraphics[width=3.25in]{algorithmprofile.pdf}
    \caption{Skeleton of our algorithm}
    \label{our algorithm profile}
\end{figure}

In this paper, we divide the algorithm into three main stages. The skeleton of our algorithm (stage1-stage3) is schematically illustrated in Fig. \ref{our algorithm profile}. Due to the mass noise in the EEG signal, the preprocess stage is very significant for the following stages. The second stage called feature extraction and classification. In this stage, we extract the spatial features and spectral features, and then use a appropriate classifier called linear discriminant analysis (LDA) to decode the information from these signals. The last stage is fusion processing that obtain the final classification result from the above stages.

\subsection{Preprocess}
The signals from the different channels correspond to the different functional regions of the human brain. In order to have a positive effect in the subsequent classification algorithm, select some special channels in carefully to obtain a better signal-noise ratio \cite{xu2013channel}.
In addition, the begin and end portions of the signal segment have been cut off to further improve the signal-to-noise ratio.

Wavelets denoising has been proven to be a powerful tool for removing noise from digital signals. We utilize the wavelet transform with symlets wavelet for improving the signal-noise ratio, and use variant of the first option as threshold selection rule compare to the universal threshold $\sigma \sqrt{2ln(\cdot )}$ that uses in many other BCI systems. The experiments results show that heursure threshold selection rule can yield better results following the steps discussed in the following subsections.

Independent component analysis (ICA) is an effective algorithm for finding fundamental factors or components from a composite signal. FastICA \cite{delorme2004eeglab} is a widely used ICA-like algorithm, which can be more efficient than traditional conventional gradient descent methods. Therefore, we apply the fastICA algorithm to remove noises from other parts of the body and reduce dimension of these signals.
At the end of the preprocessing, the butter-worth filter has been used as a bandpass filter to remove high-order noise in the signal.

\subsection{Feature extraction}

To improve the accuracy of classification, we use the common spatial pattern of spatial information \cite{ramoser2000optimal} and statistical values of wavelet transform coefficients as the spectral information.

For extracting spectral information, discrete wavelet transform (DWT) has been applied to each fragment in each channel \cite{robinson2013eeg}. In our work, the symlet wavelets have been used as the mother wavelet to generate the wavelet coefficients.
We can simply use the mean and variance of each fragment as an approximation of the spectral feature to obtain the statistical features of spectral information.

\subsection{Classification}
The classification stage in the BCI system is intended to label the feature vector. We use LDA to classify spatial information and spectral information \cite{lotte2007review} \cite{liu2015uncorrelated}. The purpose of LDA is to separate the training feature vectors representing different classes using the hyperplane to obtain the smallest intraclass dispersion and the maximum interclass dispersion.
\begin{eqnarray}
\begin{aligned}
\hat{\varphi} & = \mathrm{argmax} ~J_{fisher}(\varphi) \\
& = \mathrm{argmax} ~\frac{{\varphi}^{T} S_b \varphi}{{\varphi}^{T} S_w \varphi}
\end{aligned}
\end{eqnarray}
where $S_b$ is the discrete degree between classes, $S_w$ is the discrete degree within the class and $\hat{\varphi}$ is the optimal projection matrix. Naturally, LDA can achieve good result on unseen data, and also good results can be obtained in practical applications.

\subsection{Fusion}

To overcome the limitations and shortcomings of conventional EEG classification algorithms, we use a fusion algorithm based on Bayesian-decision \cite{daunizeau2007symmetrical} to fuse the results of multiple classifiers in different aspects. In a normalized BCI system, there is no favoritism of any class, so we choose the 0-1 lost function to minimize the total error risk. Then Bayesian rule with the smallest error rate using the 0-1 lost function can be formulated as:
\begin{eqnarray}
\begin{aligned}
\hat{j} & = \underset{j \in C} {\mathrm{argmax}} ~P(\omega_i|x) \\
& = \underset{j \in C} {\mathrm{argmax}} ~P(x|\omega_j)P(\omega_j)
\end{aligned}
\end{eqnarray}
where $\hat{j}$ is the class with the largest possible and $P(\omega_i|x)$ is the probability of classified as $i$ which determined by one classifier. Assuming that there are $R$ EEG classifiers which independent of each other and suppose there is no favoritism of any class, the fusion rule will be rewritten as follow:
\begin{eqnarray}
\begin{aligned}
\label{multiply_classifiers_reduct}
\hat{j} & = \underset{j \in C} {\mathrm{argmax}} ~P(\omega_j|x_1,x_2,...,x_R)\\
& =  \underset{j \in C} {\mathrm{argmax}} ~{P(x_1,x_2,...,x_R|\omega_j)P(\omega_j)} \\
& =  \underset{j \in C} {\mathrm{argmax}} ~{P(\omega_j) \prod_{i=1}^{R}{P(x_i|\omega_j)}} \\
& =  \underset{j \in C} {\mathrm{argmax}} ~{P(\omega_j) \prod_{i=1}^{R}{\frac{P(\omega_j|x_i) * p(x_i)}{P(\omega_j)}}} \\
& =  \underset{j \in C} {\mathrm{argmax}} ~{P^{-(R-1)}(\omega_j) \prod_{i=1}^{R}{P(\omega_j|x_i)}}
\end{aligned}
\end{eqnarray}
Suppose that there is only tiny difference between $P(\omega_k)$ and $P(\omega_k|x_j)$, which means:
\begin{equation}
\label{tiny_different_between}
P(\omega_k|x_j) = P(\omega_k)(1+\delta_{kj}), ~~\delta_{kj} << 1
\end{equation}
Ignore items with order greater than 2, then Eq. \eqref{multiply_classifiers_reduct} can be revised as:
\begin{eqnarray}
\begin{aligned}
\hat{j} & =  \underset{j \in C} {\mathrm{argmax}} ~{P^{-(R-1)}(\omega_j) \prod_{i=1}^{R}{P(\omega_j|x_i)}}\\
& =  \underset{j \in C} {\mathrm{argmax}} ~{P(\omega_j) \prod_{i=1}^{R}{ (1+\delta_{kj}) }} \\
& \approx  \underset{j \in C} {\mathrm{argmax}} ~{P(\omega_j) (1+\sum_{k=1}^{R}{\delta_{kj}}) }
\end{aligned}
\end{eqnarray}
We use the accuracy obtained on the validation set as the confidence level of the classifier and set the weight of the classifier to a transformation of the confidence level by a $tan$ like function. Then introduce a scaling factor $s$ to enhance this validity, and the weight $W_i$ will be determined to:
\begin{eqnarray}
\begin{aligned}
W_i^{'} &= ( sign(C_i-\frac{1}{2}) ) \frac{(\frac{1}{2} - \left | C_i-\frac{1}{2} \right |) (s-1)} {s} \\
W_i &= \tan{
    \left( \frac{\pi}{2}
        \left(
            C_i + W_i^{'}
        \right)
    \right)
}
\end{aligned}
\end{eqnarray}
where $s \geq 1$ is the scaling factor and $C_i$ is the confidence of the classifier calculated from the validation set.
As we have derived in equation \ref{tiny_different_between}, assume that each class has the same prior probability in our work. The following equation is then used to determine the final fusion result.
\begin{eqnarray}
\begin{aligned}
\hat{j} & =  \underset{j \in C} {\mathrm{argmax}} ~P(\omega_j|x_1,x_2,...,x_R)\\
& = \underset{j \in C} {\mathrm{argmax}} ~P(\omega_j) \sum_{i=1}^{R}{ W_i P(\omega_j|x_i) }
\end{aligned}
\end{eqnarray}
where $W_i$ is the weight of the classifier estimated in the above steps, and $R$ is the number of classifiers.

\section{EXPERIMENT}

We performed our experiments at the data set IVa that provided by BCI competitions \uppercase\expandafter{\romannumeral3}.
This data set contains 118 channels of EEG sampled signals which divided into five subjects (with named aa, al, av, aw, ay). Each subject contains 280 segments of data.  The label of each piece of data shows that the subject is doing motor image by performing one of the following three movements: (L) left hand, (R) right hand, (F) right foot.

First of all, we choose the 21 channels (CP6, CP4, CP2, C6, C4,
C2, FC6, FC4, FC2, CPZ, CZ, FCZ, CP1, CP3, CP5, C1, C3, C5, FC1, FC3, FC5) from the data set. Second, we carry out the above-mentioned preprocess operation. The influence of preprocessing step on frequency spectrum is shown in Fig. \ref{fig:experiment_preprocess}. This figure clearly shows that our preprocessing operation is useful for feature extraction and classification. Moreover, it can be seen that the useful part of signal after preprocess is concentrated at 8-30 Hz.

\begin{figure}
    \centering
    \subfigure[before]{
        \label{fig:experiment_preprocess:before}
        \includegraphics[width=2.4in]{before_filter-eps-converted-to.pdf}}
    \hspace{1in}
    \subfigure[after]{
        \label{fig:experiment_preprocess:after}
        \includegraphics[width=2.4in]{after_filter-eps-converted-to.pdf}}
    \caption{Frequency power variation in preprocess}
    \label{fig:experiment_preprocess}
\end{figure}

The accuracy of classification algorithm can be calculated simply by:

\begin{equation}
accuracy = \frac{N_{right}}{N_{testset}}
\end{equation}

The experimental results show that our work achieved better result than other algorithms such as PSD, INFL, CSP, CSSP. Experimental results are shown in Table \ref{table: experiment result of different algorithms compare}, and these results demonstrate that our algorithm can achieve a better accuracy in most cases and obtain the best mean value and acceptable variance value.

\begin{table}[h!]
\caption{The experimental results (\%) obtained from each subject using different algorithms}
\label{table: experiment result of different algorithms compare}
\begin{center}
	\begin{tabular}{||c | c cl c c||}
		\hline
		~     & PSD   & INFL  & CSP   & CSSP & Our Work  \\ [0.5ex]
		\hline\hline
		   aa & 64.29 & 75.89 & 71.43 & 77.68 & \textbf{85.27}  \\
		\hline
		   al & 82.14 & 83.93 & 94.64 & 96.43 & \textbf{100}  \\
		\hline
		   av & 55.10 & 58.16 & 61.22 & 63.27 & \textbf{69.89} \\
		\hline
		   aw & 65.18 & 82.14 & 89.28 & \textbf{90.63} & 85.71 \\
		\hline
           ay & 75.00 & 84.92 & 73.02 & 79.37 & \textbf{82.34} \\
        \hline
         mean & 68.34 & 77.00 & 77.92 & 81.48 & \textbf{84.64}   \\
         \hline
         var  & \textbf{87.3}& 98.67& 150.73& 131.54& 92.06\\
 [1ex]
		\hline
	\end{tabular}
\end{center}
\end{table}

\begin{figure}
    \centering
    \subfigure[segment size of signal]{
        \includegraphics[width=1.56in]{parameters_itertor_1-eps-converted-to.pdf}
        \label{figure:parameter_iterator:a}
    }
    \vspace{0in} \hspace{0in}
    \subfigure[dimension of ICA]{
        \includegraphics[width=1.56in]{parameters_itertor_2-eps-converted-to.pdf}
        \label{figure:parameter_iterator:b}
    }

    \subfigure[size of CSP]{
        \includegraphics[width=1.56in]{parameters_itertor_3-eps-converted-to.pdf}
        \label{figure:parameter_iterator:c}
    }
    \vspace{0in} \hspace{0in}
    \subfigure[order of butter-worth filter]{
        \includegraphics[width=1.56in]{parameters_itertor_4-eps-converted-to.pdf}
        \label{figure:parameter_iterator:d}
    }
    \caption{Accuracy curves obtained using different parameter values}
    \label{figure:parameter_iterator}
\end{figure}

In order to optimize the parameters of the algorithm in the experiments, reasonable range of each parameter in our algorithm has been pointed. The classification accuracy results with different parameter values are shown in Fig. \ref{figure:parameter_iterator}, and these figures can indicate the following items:

\begin{itemize}
	\item EEG segment size of the reasonable range is 140-240, which means that we should choose the middle of the motor image signal with the length of 1.4s-2.4s.
	\item The best value for the ICA dimension is 16.
	\item The reasonable size of CSP seems not easy to determined, but the appropriate value in most situations is between 2 and 3.
	\item The reasonable order of butter-worth filter can be determined as 5-6.
\end{itemize}

\begin{table}[h!]
    \caption{Comparison of experiment results (\%) between our fusion classifier and basic spatial classifier}
    \label{table: compare use fusion}
    \begin{center}
        \begin{tabular}{||c | c c||}
            \hline
            ~  & Basic spatial classifier & Our fusion classifier      \\ [0.5ex]
            \hline\hline
            aa & 83.48        & \textbf{85.27}         \\
            \hline
            al & 97.32        & \textbf{100}           \\
            \hline
            av & 60.20        & \textbf{69.89}         \\
            \hline
            aw & 80.13        & \textbf{85.71}         \\
            \hline
            ay & 78.17        & \textbf{82.34}         \\
            \hline
            mean & 79.86        & \textbf{84.64}         \\
            \hline
            var & 141.48       & \textbf{92.06}        \\ [1ex]
            \hline
        \end{tabular}
    \end{center}
\end{table}

For the fusion stage, we compare the accuracy of our fusion classifier with the basic CSP classifier, and the results are shown in Table \ref{table: compare use fusion}. Then, try to optimize the scale factor in the fusion algorithm and find that the reasonable range for this parameter can be determined as 1.4-2.4. These results show that our fusion algorithm plays a positive role in classification.

\section{CONCLUSIONS}
Realizing the information exchange between the brain and the computer based on EEG signal is still a challenge in signal processing and biomedical engineering.
In this paper, our algorithm has been proposed to overcome the limitations and shortcomings of conventional EEG classification algorithms with multiple features.
The signal is preprocessed to improve quality, represented by multiple features in different aspects, and fed to LDA classifiers.
The results of these classifiers is to determine the final classification result of the signal by our fusion algorithm.
Experimental results have shown that the proposed approach in this paper is an effective way to classify EEG signal in a BCI system.

\section{ACKNOWLEDGE}

This work was supported by the National Natural Fund: 91420302 and 91520201. Thanks to the organizers of the BCI competition \uppercase\expandafter{\romannumeral3} to publish their datasets.

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\bibliographystyle{unsrt}