text:
\section{Introduction}
Let $\mathcal X_g$ be an irreducible, smooth, projective curve of genus $g \geq 3$, defined over the complex field $\mathbb C$.
We denote by $\mathcal M_g$ the coarse moduli space of smooth curves of genus $g$ and by $\mathcal H_g$ the hyperelliptic locus in
$\mathcal M_g$. It is well known that $\dim\ \mathcal M_g = 3g-3$ and $\mathcal H_g$ is a $(2g-1)$ -- dimensional subvariety of $\mathcal M_g$. A
curve $\mathcal X_g$ is called \emph{bielliptic} if it admits a degree 2 morphism $\pi: \mathcal X_g \to E$ onto an elliptic
curve. This morphism is called a \emph{bielliptic structure} in $\mathcal X_g$; see \cite{BC}. The locus $\mathcal M_g^b$ of
bielliptic curves is a $(2g-2)$ -- dimensional subvariety of $\mathcal M_g$.

From the Castelnouvo-Severi inequality it follows that $\mathcal X_g$ admits precisely one bielliptic structure for $g
\geq 6$, but if $g \leq 5$ then there are curves which admit more than one bielliptic structure. Since every
bielliptic structure corresponds to an involution in the automorphism group $\mathrm{Aut}(\mathcal X_g)$ of the curve, then these
results can be obtained easily if the list of groups that occur as automorphism groups is known for a given genus
$g$. Lately, algorithms have been developed to determine such lists of groups for reasonably small $g$; see
\cite{MS}.

A bielliptic curve of genus $g \geq 4$ can not be hyperelliptic (Castelnouvo-Severi inequality), but for $g=3$
this can be the case. The bielliptic (non-hyperelliptic) curves of genus $3$ were studied in \cite{BC}; see also
\cite{ST}. In this paper we will focus in the hyperelliptic case. Such curves are known in the literature also as
hyperelliptic curves with \emph{extra involutions}. This extends our previous work in \cite{SV1}, \cite{GS},
\cite{SS}.

In the second section we give a brief description of the dihedral invariants and how they are used to describe the
loci of curves with fixed automorphism groups. Such invariants can be helpful in determining the automorphism
group of the curve and determining its field of moduli. For further details on dihedral invariants we refer to
\cite{GS}. The reader can also check \cite{SV1} for the case $g=2$. Further in this section we briefly define the
classical invariants of binary forms.

In section three, we focus on genus 3 hyperelliptic curves. We define the invariants of binary
octavics and compute them explicitly in terms of the coefficients of the curve for curves with
extra involutions. The locus $\mathcal L_3$ of genus $3$ hyperelliptic curves with extra involutions is a
3-dimensional subvariety of $\mathcal H_3$. Using such explicit expressions for the invariants of the
binary forms we find a birational parametrization of the locus $\mathcal L_3$ via dihedral invariants.
Further, we make use of such invariants to study certain subvarieties of the moduli space of
hyperelliptic curves of genus 3. The list of groups that occur as automorphism groups of genus 3
hyperelliptic curves is described. Then, for each group in the list we describe algebraic relations
that define the corresponding locus.

If $\mathcal X \in \mathcal L_3$ then $V_4 \hookrightarrow \mathrm{Aut}(\mathcal X)$. Let $G$ be a group which occurs as an automorphism group
of hyperelliptic curves of genus 3 and such that $V_4 \hookrightarrow G$. We describe each locus of curves
with automorphism group $G$ in terms of the dihedral invariants and prove that the field of moduli
of such curves is a field of definition. If $|G|>4$ then a rational model of the curve is provided
over its field of moduli. As far as we are aware, this is the first time that such rational models
are known for genus 3 curves.

\medskip

\noindent \textbf{Notation:} We denote a hyperelliptic curve of genus 3 by $\mathcal X_3$. $D_n$ denotes
the dihedral group of order $2n$ and $V_4$ the Klein 4-group. Further, $\mathbb Z_n$ denotes the cyclic
group of order $n$.

\section{Dihedral invariants of hyperelliptic curves}
In this section we give a brief review of some of the basic results on hyperelliptic curves with extra involutions
and their dihedral invariants. For details see \cite{GS}.

Let $\mathcal X_g$ be a genus $g$ hyperelliptic curve defined over $\mathbb C$ and $\mathrm{Aut}(\mathcal X_g)$ its automorphism
group. We say that $\mathcal X_g$ has an \emph{extra involution} when there is a non-hyperelliptic
involution in $\mathrm{Aut}(\mathcal X_g)$. If the fixed field of such an extra involution is an elliptic field
then sometimes these are called \emph{elliptic involutions}. The hyperelliptic involution $\alpha_0 \in
\mathrm{Aut}(\mathcal X_g)$ is in the center of $\mathrm{Aut}(\mathcal X_g)$. We denote $\overline {\mathrm{Aut}}(\mathcal X_g) := \mathrm{Aut}(\mathcal X_g)/\<\alpha_0\>$ and
call it the \emph{reduced automorphism group} of $\mathcal X_g$.

Let $\mathcal X_g$ be a genus $g$ hyperelliptic curve with an extra involution $\alpha_1 \in \mathrm{Aut}(\mathcal X_g)$. Then,
$\mathcal X_g$ is isomorphic to a curve given by an equation
$$Y^2=X^{2g+2} + a_{g} X^{2g} + \cdots + a_1 X^2 +1.$$
Such equation is called the \emph{normal equation} of the curve $\mathcal X_g$. There is a degree 2 map
$$\phi_1 : \mathcal X_g \to C_1$$
where $C_1$ is the hyperelliptic curve with equation
$$Y^2=X^{g+1} + a_{g} X^{g}+ \cdots + a_1 X +1$$
and genus $g_1= \left[ \frac g 2 \right]$. Since every hyperelliptic curve $\mathcal X_g$ has the
hyperelliptic involution $\alpha_0 \in \mathrm{Aut}(\mathcal X_g)$, then the extra involutions come in pairs $(\alpha_1,
\alpha_2=\alpha_0 \alpha_1)$. The extra involution $\alpha_2$ determines another degree 2 covering
$$\phi_2 : \mathcal X_g \to C_2$$
where $C_2$ is the hyperelliptic curve with equation
$$Y^2=X(X^{g+1} + a_{g} X^{g}+ \dots + a_1 X +1)$$
and genus $g_2= \left[ \frac {g+1} 2 \right]$. The curve $\mathcal X_g$ is called \emph{bielliptic} if
$C_1$ is an elliptic curve. This always happens if $g=2$ or 3.

The Jacobian $J_{\mathcal X_g}$ of $\mathcal X_g$ is isogenous to $J_{C_1} \times J_{C_2}$. We say that $J_{\mathcal X_g}$
splits. Our goal is to determine the locus of such curves $\mathcal X_g$ in the variety of moduli. The
locus of genus $g$ hyperelliptic curves with an extra involution is an irreducible $g$-dimensional
subvariety of $\mathcal H_g$ which we denote by $\mathcal L_g$. The following
$$u_i := a_1^{g-i+1} \, a_i \, + \, a_g^{g-i+1} \, a_{g-i+1}, \quad 1\leq i\leq g$$
are called \emph{dihedral invariants} of genus $g$. The next theorem shows that $\mathcal L_g$ is a rational variety; see
\cite{GS}.
\begin{theorem}\label{thm1}
Let $g\geq 2$ and $(u_1,\dots,u_g)$ be the $g$-tuple of dihedral invariants. Then,
$k(\mathcal L_g)=k(u_1,\dots,u_g)$.
\end{theorem}

A generic point $\mathfrak p \in \mathcal L_g$ has automorphism group $\mathrm{Aut}(\mathfrak p) {\, \cong\, } V_4$. Singular points of $\mathcal L_g$
have more than one tuple of dihedral invariants and therefore more than one conjugacy class of
involutions in $\overline {\mathrm{Aut}}(\mathfrak p)$. For curves with automorphism group isomorphic to $V_4$ we have the
following:

\begin{corollary}
Let $\mathcal X_g$ and $\mathcal X_g^\prime$ be genus $g$ hyperelliptic curves with automorphism groups isomorphic
to $V_4$, and $(u_1, \dots ,u_g)$, $(u_1^\prime, \dots , u_g^\prime)$ their respective dihedral
invariants. Then,
$$\mathcal X_g {\, \cong\, } \mathcal X_g^\prime \ \iff\ (u_1, \dots , u_g)=(u_1^\prime, \dots, u_g^\prime).$$
\end{corollary}

\begin{theorem} \label{thm_u}
Let $\mathcal X_g $ be a genus $g$ hyperelliptic curve with an extra involution and $(u_1,\dots,u_g)$ its
corresponding dihedral invariants.

i) If $V_4 \hookrightarrow \overline {\mathrm{Aut}}(\mathcal X_g)$ then $2^{g-1}\,u_1^2 = u_g^{g+1}$.

ii) Moreover, if $g$ is odd then $V_4 \hookrightarrow \overline {\mathrm{Aut}}(\mathcal X_g)$ implies that
$$\left(2^r\,u_1 - u_g^{r+1}\right) \, \left(2^r\,u_1 + u_g^{r+1}\right)=0$$
where $r=\frac {g-1} 2$. The first factor corresponds to the case when involutions of $V_4\hookrightarrow
\overline {\mathrm{Aut}}(\mathcal X_g)$ lift to involutions in $\mathrm{Aut}(\mathcal X_g)$. The second factor corresponds to the case when
one of the two involutions lifts to an element of order 4 in $\mathrm{Aut}(\mathcal X_g)$.
\end{theorem}
For the proofs of these statements see \cite{GS}.

From a computational point of view, determining the normal equation of a given curve with an extra
automorphism can be done simply by solving a system of equations; this is quite efficient both
theoretically and in practice. If an extra involution $\alpha=X \to \frac{aX+b}{cX+d}$ is known
explicitly, one can easily find $\sigma$ such that $\alpha^\sigma=X \to -X$, then the equation after applying
$\sigma^{-1}$ has the form $Y^2=F(X^2)$. One more substitution $X \to \lambda\,X$ for certain $\lambda$ will
provide the normal equation.

\subsection{Genus 2 case}
The case $g=2$ has been studied in \cite{SV1}. Every point in $\mathcal M_2$ is a triple $(i_1, i_2, i_3)$ of absolute
invariants, see \cite{SV1} for details. The curve of genus 2 with extra involutions has equation
$$Y^2 = X^6+ a_2 X^4 + a_1 X^2+1.$$
We denote its dihedral invariants by
\begin{equation}
\mathfrak u  := a_1^3 + a_2^3, \, \, \textit{   and    } \, \,  \mathfrak v  := 2 a_1 a_2.
\end{equation}
The following lemma is proved in [\cite[pg.710]{SV1}.

\begin{lemma}
Let $\mathcal X_2$ be a genus 2 curve such that $G:=\mathrm{Aut}(\mathcal X_2)$ has an extra involution and $(\mathfrak u,\mathfrak v)$ its
dihedral invariants. Then,\\

a) $G{\, \cong\, } \mathbb Z_3 \rtimes D_8$ if and only if $(\mathfrak u,\mathfrak v)=(0,0)$ or $(\mathfrak u,\mathfrak v)=(6750,450)$.

b) $G{\, \cong\, } GL_2(3)$ if and only if $(\mathfrak u,\mathfrak v) = (-250,50)$.

c) $G{\, \cong\, } D_{12}$ if and only if $\ \mathfrak v^2 - 220 \mathfrak v -16 \mathfrak u +4500=0$ for $\mathfrak v \neq 18,\,
140+60\sqrt{5},\, 50$.

d) $G{\, \cong\, } D_8$ if and only if $\ 2\mathfrak u^2-\mathfrak v^3=0$, for $\mathfrak v \neq 2, 18, 0, 50, 450$. Cases $\mathfrak v =
0,\,450$ and $\mathfrak u=50$ are reduced to cases a) and b) respectively.
\end{lemma}

Notice that the parameters $u= \displaystyle\frac {\mathfrak v} 2$ and $v=\mathfrak u$ instead of $\mathfrak u, \mathfrak v$ are used in \cite{SV1}.
The mapping
$$\Phi: (\mathfrak u,\mathfrak v) \to (i_1, i_2, i_3)$$
gives a birational parametrization of $\mathcal L_2$. The dihedral invariants $\mathfrak u, \mathfrak v$ are given explicitly as rational
functions of $i_1, i_2, i_3$, see \cite{SV1}. For $g=2$, the curve $Y^2=X^6-X$ is the only genus 2 curve (up to
isomorphism) which has extra automorphisms and is not in $\mathcal L_2$. The automorphism group in this case is $\mathbb Z_{10}$.
Relations between elliptic subcovers of such $\mathcal X_2$ were studied in detail in \cite{SV1}.

\subsection{Invariants of binary forms}
In this section we define the action of $GL_2(k)$ on binary forms and discuss the basic notions of
their invariants. Let $k[X,Z]$ be the polynomial ring in two variables and let $V_d$ denote the
$(d+1)$-dimensional subspace of $k[X,Z]$ consisting of homogeneous polynomials
$$f(X,Z) = a_0X^d + a_1X^{d-1}Z + ... + a_dZ^d$$
of degree $d$. Elements in $V_d$ are called \emph{binary forms} of degree $d$. We let $GL_2(k)$ act
as a group of automorphisms on $ k[X, Z] $ as follows:
\begin{equation}
\mbox{for} \quad
\begin{pmatrix} a &b \\ c & d \end{pmatrix}
\in GL_2(k)\ , \quad M
\begin{pmatrix} X \\ Z \end{pmatrix} =
\begin{pmatrix} aX+bZ \\ cX+dZ \end{pmatrix}.
\end{equation}
This action of $GL_2(k)$ leaves $V_d$ invariant and acts irreducibly on $V_d$.

Let $A_0,\ A_1,\ldots,A_d$ be coordinate functions on $V_d$. Then the coordinate ring of $V_d$ can
be identified with $k[A_0,\ldots,A_d]$. For any $I \in k[A_0,\ldots,A_d]$ and $M \in GL_2(k)$, we
define $I^M \in k[A_0,\ldots,A_d]$ as follows:
$${I^M}(f):= I(M(f))$$
for all $f \in V_d$. Then this equation and $I^{MN} = (I^{M})^N$ define an action of $GL_2(k)$ on
$k[A_0,\ldots,A_d]$.
A homogeneous polynomial $I\in k[A_0,\dots,A_d,X,Z]$ is called a \emph{covariant} of index $s$ if
$$I^M(f)=\delta^s I(f)$$
where $\delta =\det(M)$. The homogeneous degree in $a_1,\dots,a_n$ is called the \emph{degree} of
$I$, and the homogeneous degree in $X,Z$ is called the \emph{order} of $I$. A covariant of order
zero is called \emph{invariant}. An invariant is a $SL_2(k)$-invariant on $V_d$.

We will use the symbolic method of classical theory to construct covariants of binary forms. Let
$$f(X,Z):=\sum_{i=0}^n \begin{pmatrix} n \\ i \end{pmatrix} a_i X^{n-i} \, Z^i, \qquad
\texttt{}g(X,Z):=\sum_{i=0}^m \begin{pmatrix} m \\ i \end{pmatrix} b_i X^{n-i} \, Z^i$$
be binary forms of degree $n$ and $m$ respectively with coefficients in $k$. We define the
\emph{r-transvection}
\begin{small}
$$(f,g)^r := d \ \sum_{k=0}^r (-1)^k\begin{pmatrix} r \\ k \end{pmatrix} \cdot
\frac {\partial^r f} {\partial X^{r-k}\ \partial Z^k} \cdot \frac {\partial^r g} {\partial X^k\
\partial Z^{r-k} }$$
\end{small}
where
$d=\frac {(m-r)! \, (n-r)!} {n! \, m!}.$
Then $(f,g)^r$ is a homogeneous polynomial in $k[X,Z]$ and therefore a covariant of order $m+n-2r$ and degree 2.
In general, the $r$-transvection of two covariants of order $m,n$ (resp. degree $p, q$) is a covariant of order
$m+n-2r$ (resp. degree $p+q$). See \cite{Al}, \cite{Cl}, \cite{H}, \cite{KS} for details.

\section{Hyperelliptic curves of genus three}
In this section we study hyperelliptic curves of genus 3 with extra involutions. Let $\mathcal X_3$ be such
a curve. Then, $\mathcal X_3$ has normal equation
$$Y^2 = X^8 + a_3\,X^6 + a_2\,X^4 + a_1\,X^2 + 1,$$
see \cite{GS}. The dihedral invariants of $\mathcal X_3$ are
$$u_1=a_1^4+a_3^4 \, \, \quad u_2=(a_1^2+a_3^2)\,a_2 \, \, \quad u_3=2\,a_1\,a_3.$$
If $a_1=a_3=0$, then $u_1=u_2=u_3=0$. In this case $w:=a_2^2$ is invariant. Thus, we define
\begin{equation} \label{def_u}
\begin{split}
\mathfrak u (\mathcal X_3) \, = \left\{ \aligned & w & \mbox{if} \quad a_1=a_3=0,\\
 & (u_1, w, u_3) & \mbox{if} \quad a_1^2+a_3^2=0\ \mbox{and} \, \, a_2\neq 0,\\
 & (u_1, u_2, u_3) & \mbox{otherwise}. \\ \endaligned
\right.
\end{split}
\end{equation}
To have an explicit way of describing a point in the moduli space of hyperelliptic curves of genus 3 we need the
generators of the field of invariants of binary octavics. These invariants are described in terms of covariants of
binary octavics. Such covariants were first constructed by van Gall who showed that the graded ring of covariants
is generated by 70 covariants and explicitly constructed them, see \cite{Shi1}.

Let $f(X,Y)$ be the binary octavic
$$f(X,Y) = \sum_{i=0}^8 a_i X^i Y^{8-i}.$$
We define the following covariants:
\begin{equation}
\begin{split}
 & g=(f,f)^4, \quad k=(f, f )^6, \quad h=(k,k)^2, \quad m=(f,k)^4, \\
 & n=(f,h)^4, \quad p=(g,k)^4, \quad q=(g, h)^4. \\
\end{split}
\end{equation}

Then the following
\begin{equation}\label{J}
\begin{split}
&J_2=(f,f)^8,\quad J_3=(f,g )^8,\quad J_4=(k,k)^4,\quad \\ & J_5=(m,k)^4,\quad J_6 = (k,h )^4,
\quad J_7= (m,h)^4
\end{split}
\end{equation}
are $SL_2(k)$-invariants. Shioda has shown that the ring of invariants is a finitely generated module of
$k[J_2,\dots,J_7]$, see \cite{Shi1}. The expressions of these covariants are very large in terms of the
coefficients of the curve and difficult to compute. However, in terms of the dihedral invariants $u_1,u_2,u_3$
these expressions are smaller. Analogously, $J_{14}$ is the discriminant of the octavic. We define $M:=2u_1+u_3^2$
and assume $M\neq 0$.

\begin{small}

\[
\begin{split}
J_2 = & \, \frac 1 M \, \, (560u_1 + 280u_3^2 + 10u_3u_1 + 5u_3^3 + 2u_2^2)\\  \\
J_3 = & \, \frac {u_2 } {a_2 M^2} \, (12u_2^3 + 4200u_1^2 + 4200u_1u_3^2 + 1050 u_3^4 -110u_3u_2u_1-55u_3^3u_2 \\
&+ 7840u_2u_1 + 3920u_2u_3^2) \\  \\
J_4 = &\, \frac {32}{M^2}\,(2u_2^4-1568u_2^2u_1-784u_2^2u_3^2 + 1008u_2u_1^2 + 1008u_2u_1u_3^2+ 252u_2u_3^4  \\
 & + 8u_1^2u_3^2+307328u_1^2 + 307328u_1u_3^2 + 76832u_3^4+62u_3u_2^2u_1+ 31u_3^3u_2^2\\
 &-784u_3u_1^2+8u_1u_3^4+2u_3^6 -784u_3^3u_1-196u_3^5) \\ \\
 %
 %
J_5 = & \, - \frac {16 u_2}{ a_2 M^3}\,(104u_2u_1^2u_3^2-614656u_2u_1u_3^2-41160u_3^6-614656u_2u_1^2- 153664u_2u_3^4\\
& -246960u_1u_3^4-2296u_2^2u_1u_3^2 +104u_2u_1u_3^4-41552u_3u_2u_1^2-41552u_3^3u_2u_1\\
& +26u_2^3u_3u_1+ 1568u_2^3u_3^2 +13u_2^3u_3^3 + 26u_2u_3^6 - 2296u_2^2u_1^2-574u_2^2u_3^4 -10388u_3^5u_2 \\
& + 1120u_3u_1^3 + 840u_3^5u_1 -4u_2^5 -329280u_1^3  + 140u_3^7 -493920u_1^2u_3^2+3136u_2^3u_1 \\
& + 1680u_3^3u_1^2) \\  \\
J_6 &=  \, -\frac {256} {M^3} \,(2u_3u_1 + u_3^3-392u_1-196u_3^2+u_2^2) (-2u_2^4- 8u_1^2u_3^2-8u_1u_3^4-2u_3^6\\
&+ 154u_3u_2^2u_1+77u_3^3u_2^2 + 1568u_2^2u_1 + 784u_2^2u_3^2 + 3024u_2u_1^2 + 3024u_2u_1u_3^2\\
&+ 756u_2u_3^4 + 10192u_3u_1^2 + 2548u_3^5- 307328u_1^2-307328u_1u_3^2 \\
& -76832u_3^4+ 10192u_3^3u_1) \\  \\
J_7 &=  \, \frac {64 u_2} {a_2 M^4}\, (129077760 u_3^6 u_1 - 481890304 u_2 u_1^3 +516311040u_1^3u_3^2 \\
& +387233280u_1^2u_3^4 + 921984u_2^3u_3^4 + 7299040u_3^7u_2-90u_2^5u_3^3  -14896u_2^4u_1^2\\
& -3724u_2^4u_3^4+3360u_3^6u_1^2+1120u_3^8u_1+141120u_2u_1^4+4480u_1^3u_3^4+16134720u_3^8 \\
& + 2086u_3^7u_2^2 + 345u_2^3u_3^6+38u_3^9u_2  + 3687936u_2^3u_1^2 -68600u_3^9 -25480u_2u_3^8 \\
& + 5180672u_2^2u_1^3 + 647584u_2^2u_3^6-1097600u_3u_1^4+ 8u_2^7 + 140u_3^{10} + 258155520u_1^4 \\
 & -1646400u_3^5 u_1^2-548800u_3^7 u_1-9408u_2^5u_1 -4704u_2^5u_3^2 -722835456u_2u_1^2u_3^2\\
 & +16688u_3u_2^2u_1^3 +304u_3^3u_2u_1^3 + 456u_3^5u_2u_1^2  -199920u_2u_1^2u_3^4+7840u_2u_1^3u_3^2\\
 & -14896u_2^4u_1u_3^2+ 25032u_3^3u_2^2u_1^2 +43794240u_3^5u_2u_1 +87588480u_3^3u_2u_1^2 \\
 & + 228u_3^7u_2u_1  +1380u_2^3u_3^4u_1-78400u_2^3u_3u_1^2 + 58392320u_3u_2u_1^3 +1380u_2^3u_3^2u_1^2\\
 & -135240u_2u_1u_3^6 +3885504u_2^2u_1u_3^4+ 3687936u_2^3u_3^2u_1  -60236288u_2u_3^6\\
 & +12516u_3^5u_2^2u_1 -78400u_2^3u_3^3u_1-361417728u_2u_1u_3^4 + 2240u_1^4u_3^2 \\
 & + 7771008u_2^2u_1^2u_3^2-19600u_2^3u_3^5 -180u_2^5u_3u_1-2195200u_3^3 u_1^3 ) \\
 \end{split}
\]

\[
\begin{split}
J_{14} & =  \, \frac {16} {M^4} \, (-1024u_3^4-64u_2^4-4096u_1^2-4096u_1u_3^2 -2304u_2u_1u_3^2 +6u_3^6 + 384u_3^5\\
& + 1024u_2^2u_1+ 512u_2^2u_3^2 -2304u_2u_1^2-576u_2u_3^4 + 456u_1^2u_3^2+ 132u_1u_3^4\\
 &  + 160u_3^3u_2^2 + 1536u_3u_1^2 + 1536u_3^3u_1 -2u_2^2u_1u_3^2 + 320u_3u_2^2u_1 - 144u_3u_2u_1^2 \\
 & -144u_3^3u_2u_1 + 32u_2^3u_1 + 16u_2^3u_3^2-u_2^2u_3^4 - 36u_3^5u_2 + 8u_3^3u_1^2 + 8u_3^5u_1 \\
 & + 432u_1^3+ 2u_3^7 )^2 \\ \\
\end{split}
\]

\end{small}

The next theorem is a direct corollary of Theorem \ref{thm1}. However, we provide a computational
proof.

\begin{theorem}\label{thm_3}
 $k(\mathcal L_3)=k(u_1, u_2 , u_3)$.
\end{theorem}

\begin{proof}
Notice that $J_3, J_5, J_7$ have $a_2$ as a factor. However, this does not contradict Theorem
\ref{thm1}. The function field $k(\mathcal L_3)$ is generated by absolute invariants. To define such
absolute invariants one must raise $J_3, J_5, J_7$ to some power and therefore absolute invariants
will have only $a_2^2=\frac {2u_2^2} {2u_1+u_3^2}$ as a factor. Hence, computationally we have
shown that $i_1, \dots , i_5 \in k(u_1, u_2, u_3)$, as expected. Let
$$i_j = \frac {p_j (u_1, u_2, u_3)} {q_j (u_1, u_2, u_3)}$$
for $j=1,\dots,5$ and certain polynomials $p_i$, $q_i$. Then, we have the system of equations
$$i_j \cdot {q_j (u_1, u_2, u_3)} - {p_j (u_1, u_2, u_3)} =0, \quad j=1, \dots , 5.$$
We can solve for $u_1, u_2, u_3$ and express them as rational functions in $i_1, i_2, i_3$.
Therefore, $\ k(u_1, u_2, u_3) \, =\, k(\mathcal L_3)$.
\end{proof}

\begin{remark}
The expressions of $u_1, u_2, u_3$ as rational functions in $i_1, i_2, i_3$ are rather large and we
don't display them. Using the above equations, one can find explicit equations of the locus $\mathcal L_3$
in terms of $i_1,\dots,i_5$. However, such equations are very large and not practical to use.
Instead, using the dihedral invariants $u_1, u_2, u_3$ is much more convenient.
\end{remark}

\subsection{The locus of genus 3 hyperelliptic curve with prescribed automorphism group}
In this section we describe the locus of genus 3 hyperelliptic curves in terms of dihedral invariants or classical
invariants. First we briefly describe the list of groups that occur as automorphism groups of genus 3
hyperelliptic curves. This list has been computed by many authors; we refer to \cite{MS} for the correct result
and a complete list of references.

We denote by $U_6$, $V_8$ the following groups:
$$U_6 := \< x, y \ | \ x^2, y^6, x\, y\, x\, y^4 \>, \qquad
V_8 := \< x, y \ | \ x^4, y^4, \, (x\,y)^2, \, (x^{-1}y)^2 \>.$$
In Table 1 we list the automorphism groups of genus 3 hyperelliptic curves. The first column is the
case number, in the second column the groups which occur as full automorphism groups are given, and
the third column indicates the reduced automorphism group for each case. The dimension $\delta$ of
the locus and the equation of the curve are also given in the next two columns. The last column is
the \textsc{GAP} identity of each group in the library of small groups in \textsc{GAP}.

\begin{table*}[ht!]
\begin{small}
\caption{$\mathrm{Aut}(X_3)$ for hyperelliptic $X_3$}
 \vskip 0.2cm
\begin{center}
\begin{tabular}{||c|c|c|c|c|c||}
\hline \hline & & & & & \\
 & $\mathrm{Aut}(\mathcal X_g)$ & $\overline G$ & $\,\delta\,$ & equation $y^2= f(x)$ & Id. \\
 & & & & & \\
\hline \hline & & & & & \\
1 & $\mathbb Z_2$ & $\{1\}$ & 5 & $x(x-1)(x^5+ax^4+bx^3+cx^2+dx+e)$ & $(2,1)$ \\
 & & & & & \\
2 & $\mathbb Z_2\times\mathbb Z_2$ & $\mathbb Z_2$ & 3 & $x^8+a_3x^6+a_2x^4+a_1x^2+1$ & $(4,2)$ \\
3 & $\mathbb Z_4$ & $\mathbb Z_2$ & 2 & $x(x^2-1)(x^4+ax^2+b)$ & $(4,1)$ \\
4 & $\mathbb Z_{14}$ & $\mathbb Z_7$ & 0 & $x^7-1$ & $(14,2)$ \\
 & & & & & \\
5 & $\mathbb Z_2^3$ & $D_4$ & 2 & $(x^4+ax^2+1)(x^4+bx^2+1)$ & $(8,5)$ \\
6 & $\mathbb Z_2\times D_8$ & $D_8$ & 1 & $x^8+ax^4+1$ & $(16,11)$ \\
7 & $\mathbb Z_2\times \mathbb Z_4$ & $D_4$ & 1 & $(x^4-1)(x^4+ax^2+1)$ & $(8,2)$ \\
8 & $D_{12}$ & $D_6$ & 1 & $x(x^6+ax^3+1)$ & $(12,4)$ \\
9 & $U_6$ & $D_{12}$ & 0 & $x(x^6-1)$ & $(24,5)$ \\
10 & $V_8$ & $D_{16}$ & 0 & $x^8-1$ & $(32,9)$ \\
 & & & & & \\
11 & $\mathbb Z_2\times S_4$ & $S_4$ & 0 & $x^8+14x^2+1$ & $(48,48)$ \\
 & & & & & \\
\hline\hline
\end{tabular}
\end{center}
\end{small}
\end{table*}

\begin{remark}
Note that $\mathbb Z_2$, $\mathbb Z_4$ and $\mathbb Z_{14}$ are the only groups which don't have extra involutions.
Thus, curves with automorphism group $\mathbb Z_2$, $\mathbb Z_4$ or $\mathbb Z_{14}$ do not belong to the locus $\mathcal L_3$.
\end{remark}

We want to describe each of the above subloci and study inclusions among them. In order to study
such inclusions the lattice of the list of groups needs to be determined. The lattice of the groups
for genus 3 is given in Fig. 1. Each group is presented by its \textsc{GAP} identity. Each level
contains cases with the same dimension (i.e., the bottom level correspond to the 0-dimensional
families). The boxed entries correspond to groups with extra involutions.

\begin{figure}[ht!]
\begin{tiny}
$$\xymatrix{
  &  &  (2,1) \ar@{-}[d] \ar@{-}[ddrr] \ar@{-}[ddddll]  \\
  &  &  *+[F]{(4,2)} \ar@{-}[dd] \ar@{-}[dr] \ar@{-}[ddll]  \\
  &  &  &  *+[F]{(8,5)} \ar@{-}[dr] &(4,1) \ar@{-}[d]  \\
*+[F]{(8,2)} \ar@{-}[dr]  &  &  *+[F]{(12,4)}\ar@{-}[dl] \ar@{-}[drr]  &  &  *+[F]{(16, 11)}\ar@{-}[dl] \ar@{-}[d]  \\
(14, 2)  &  *+[F]{(24, 5)}  &  &  *+[F]{(32, 9)}  &  *+[F]{(48, 48)}  \\
}$$
\end{tiny}
\caption{The lattice of automorphism groups}
\end{figure}

Next we determine the algebraic relations among dihedral invariants for each case in the diagram.

\begin{theorem}\label{thm_4}
The algebraic relations of dihedral invariants for each case of Figure 1 are presented in Figure 2.
\noindent \begin{tiny}
\begin{figure*}
\hspace{-0.5cm} $ \xymatrix{
  &  &  *+[F]{(u_1, u_2, u_3)} \ar@{-}[dd] \ar@{-}[dr] \ar@{-}[ddl]  \\
  &  &  &  *+[F]{2u_1-u_3^2=0} \ar@{-}[d]  \\
  &  *+[F]{2u_1+u_3^2=0} \ar@{-}[d]  &  *+[F]{Eq.~\eqref{d_u_12}}\ar@{-}[dl] \ar@{-}[dr]  &  *+[F]{a_1=a_3 }\ar@{-}[dl] \ar@{-}[d]  \\
  &  *+[F]{u_2=0}  &  *+[F]{a_1=a_2=a_3=0}  &  *+[F]{(0, 196, 0),\left(\frac {8192} {81}, - \frac {1280} {27}, \frac {128} 9 \right)}  \\
}$
\caption{Corresponding relations of dihedral invariants}
\end{figure*}
\end{tiny}
\end{theorem}

\begin{proof}
If $\overline {\mathrm{Aut}}(\mathcal X_3){\, \cong\, } D_4$, then from Theorem \ref{thm_u} we have $4\,u_1^2=u_3^4$. This can be
checked easily by computing the dihedral invariants. If $\,2u_1^2\, = \,u_3^2$ then $\mathrm{Aut}(\mathcal X_3){\, \cong\, }
\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. If $\,2 u_1^2\, = \,-u_3^2$ then $\mathrm{Aut}(\mathcal X_3){\, \cong\, } \mathbb Z_2\times \mathbb Z_4$, see
the remark after Theorem \ref{thm_u}.

Let $\mathcal X_3$ be a curve with equation $Y^2=X\,(X^6 + a X^3 +1)$, where $a\neq 0,\pm 2$. By a
transformation $X \to \frac {X+1} {X-1}$, $\mathcal X_3$ has equation
$$Y^2= X^8 + (9\lambda -7) X^6 + 15 (\lambda-1) X^4 + (7 \lambda -9) X^2 -\lambda$$
where $\lambda=\frac {a-2}{a+2}$, $\lambda\neq 0, \pm 1$. Then, by another transformation $X \to
\sqrt[8]{-\lambda}\,X$, we compute the dihedral invariants:
\begin{small}
\begin{equation}
\begin{split}
u_1 & = - \frac 1 {\lambda^3} \, (6561\lambda^6-20412\lambda^5+26215 \lambda^4 - 24694 \lambda^3 \\
    & \, +26215\lambda^2 -20412\lambda +6561), \\
u_2 & = 15 \frac {(\lambda-1)^2 (81\lambda^2 - 94 \lambda +81)} {\lambda^2}, \\
u_3 & = -2 \frac {(7\lambda -9) (9\lambda -7)} {\lambda}.
\end{split}
\end{equation}
\end{small}
Eliminating $\lambda$, we get $\lambda = -126 \frac 1 {u_3 - 260}$ and
\begin{equation}\label{d_u_12}
\begin{split}
u_2 & = \frac 5 {588} \, (u_3 -8) (9 u_3 -1024),\\
u_1 & = \frac 9 {2744} u_3^3 - \frac {873} {686} u_3^2 + \frac {149504} {3087} u_3 - \frac
{1048576} {3087}.
\end{split}
\end{equation}
Notice that $u_3\neq 260$, otherwise $a=2$.

If $u_2=0$ then we have $81\lambda^2-94\lambda+81=0$ and
$$(u_1, u_2, u_3) \, = \, (8, 0, -32) \mbox{ or } \left( - \frac  {524288} {81}, 0, \frac {1024} 9 \right).$$
Both of these triples satisfy the relation $\, 2u_1\, = \, - u_3^2$ so both $D_{12}$ and
$\mathbb Z_2\times\mathbb Z_4$ are embedded in $\mathrm{Aut}(\mathcal X_3)$. Hence, $\mathrm{Aut}(\mathcal X_3){\, \cong\, } U_6$.

If $2\,u_1^2=u_3^2$ then $81\,\lambda^2-1568\lambda+81=0$. Then,
$$(u_1, u_2, u_3)\, = \, (0, 196, 0) \mbox{ or } \left(\frac {8192} {81}, - \frac {1280} {27}, \frac {128} 9\right).$$
Both triples determine the same isomorphism class of curves and correspond to the case
$\mathrm{Aut}(\mathcal X_3){\, \cong\, }\mathbb Z_2\times S_4$. The proof is complete.
\end{proof}

The above loci can be easily determined in terms of $J_2,\dots,J_7$. This would be beneficial
because we don't have to find the normal decomposition form of the curve in order to determine the
automorphism group. However, we don't display such equations in terms of $J_2,\dots,J_7$. It is
worth mentioning that, if $\mathrm{Aut}(\mathcal X_3){\, \cong\, }\mathbb Z_2\times\mathbb Z_4,\ U_6$ or $V_8$, then $J_3=J_5=J_7=0$.

\section{Field of moduli of genus 3 hyperelliptic curves}
In this section we study the field of moduli of genus 3 hyperelliptic curves with extra
automorphisms. Let $\mathcal X$ be a curve defined over $\mathbb C$. A field $F\subset\mathbb C$ is called a \emph{field
of definition} of $\mathcal X$ if there exists $\mathcal X'$ defined over $F$ such that $\mathcal X'$ is isomorphic to $\mathcal X$
over $\mathbb C$.

\begin{definition}
The \emph{field of moduli} of $\mathcal X$ is a subfield $F\subset\mathbb C$ such that for every automorphism
$\sigma\in\mathrm{Aut}(\mathbb C)$ the following holds: $\mathcal X$ is isomorphic to $\mathcal X^\sigma$ if and only if $\ \sigma_F = id$.
\end{definition}

We will use $\mathfrak p=[\mathcal X]\in\mathcal M_g$ to denote the corresponding \emph{moduli point} and $\mathcal M_g(\mathfrak p)$ the residue field of
$\mathfrak p$ in $\mathcal M_g$. The field of moduli of $\mathcal X$ coincides with the residue field $\mathcal M_g(\mathfrak p)$ of the point $\mathfrak p$ in
$\mathcal M_g$. The notation $\mathcal M_g(\mathfrak p)$ (resp. $M(\mathcal X)$) will be used to denote the field of moduli of $\mathfrak p\in\mathcal M_g$ (resp.
$\mathcal X$). If there is a curve $\mathcal X^\prime$ isomorphic to $\mathcal X$ and defined over $M(\mathcal X)$, we say that $\mathcal X$ has a
\emph{rational model over its field of moduli}. As mentioned above, the field of moduli of curves is not
necessarily a field of definition, see \cite{Shi1} for examples of such families of curves.

\begin{lemma}
Let $\mathfrak u_0\in\mathcal L_3(k)$ such that $|\mathrm{Aut}(\mathfrak u_0)| > 4$. Then, there exists a genus 3 hyperelliptic curve
$\mathcal X_3$ defined over $k$ such that $\mathfrak u(\mathcal X_3)=\mathfrak u_0$ as defined in Eq (\ref{def_u}). Moreover, the
equation of $\mathcal X_3$ over its field of moduli is given by:

\medskip

i) If $|\mathrm{Aut}(\mathcal X_3)|=16$ then $$Y^2=w X^8+w X^4+1.$$

\smallskip

ii) If $\mathrm{Aut}(\mathcal X_3){\, \cong\, } D_{12}$ then
\begin{small}
\begin{equation}
\begin{split}
Y^2 = & \ (u_3-260) X^8 - 7(u_3-98)X^6 + 15(u_3-134)X^4 \\
  & - 9(u_3-162)X^2 + 126.
\end{split}
\end{equation}
\end{small}

iii) If $\mathrm{Aut}{\, \cong\, }\mathbb Z_2\times\mathbb Z_4$ then $$Y^2=u_3^4 X^8+u_3^4X^6+8 u_3X^2-16.$$

iv) If $\mathrm{Aut}(\mathfrak u){\, \cong\, }\mathbb Z_2^3$ then $$Y^2= u_1 X^8 +u_1 X^6 + u_2 X^4 + u_3 X^2 +2.$$
\end{lemma}

\begin{proof}
The proof in all cases consists of simply computing the dihedral invariants. It is easy to check
that these dihedral invariants satisfy the corresponding relations for $\mathrm{Aut}(\mathcal X_3)$ given above.
\end{proof}

\begin{corollary}
Let $\mathfrak p\in\mathcal H_3$ such that $|\mathrm{Aut}(\mathfrak p)|>2$. Then the field of moduli of $\mathfrak p$ is a field of
definition.
\end{corollary}

\begin{proof}
There is only one hyperelliptic curve of genus 3 which has no extra involutions and whose automorphism group has
more than $4$ elements, see \cite{MS}. This curve is $Y^2= X^7-1$ and its field of moduli is $\mathbb Q$. The result
follows from the above Lemma for all groups of order $>4$. Let $\mathcal X_3$ be a curve such that its automorphism group
$\mathrm{Aut}(\mathcal X_3)$ has order 4. Then $\mathrm{Aut}(\mathcal X_3)$ is isomorphic to $\mathbb Z_4$ or $V_4$. In both cases the quotient curve
$\mathcal X_3 \ \mathrm{Aut}(\mathcal X_3)$ is a conic which contains a non-trivial rational point. The result follows from \cite{DE}.
\end{proof}

\begin{remark}
An interesting problem would be to find an algorithm which finds a rational model over the field of moduli for
curves with automorphism group $\mathbb Z_2$. In the case of genus 2 this has been done by Mestre; see \cite{Me}.
\end{remark}

For all the computations in this paper we have used Maple; see \cite{Ma}. The above results have been implemented
in a Maple package which is available upon request. In this package we can compute the field of moduli of any
genus 3 hyperelliptic curve and provide a rational model for curves which have more than 4 automorphisms. Further,
the automorphism group can be computed and all invariants of binary octavics defined in Eq.\eqref{J}.

redpajama_set_name: RedPajamaArXiv


text:
Q: Control index.xml for Atom/RSS (hugo / blogdown generates feed with relative links) Which parameters in config.toml or which files in /layouts control the generation of index.xml?
When I validate my Atom feed, I get a number of errors, pretty much all of them due to links not being complete links.
https://validator.w3.org/feed/check.cgi?url=https%3A%2F%2Ftaraskaduk.com%2Findex.xml
Example:
 <title>My first shiny app: calculating your hourly rate as a consultant</title>
 <link>/post/2018-01-shiny-rate/shiny_rate/</link>
 <pubDate>Sat, 13 Jan 2018 00:00:00 +0000</pubDate>

My understanding that a full link is requested, i.e. https://taraskaduk.com/post/2018-01-shiny-rate/shiny_rate/
How do I control this?
Also, is there a way to pass the entire post into Atom, not just description?

A: *

*You need a full baseURL in config.toml (see Section 2.2 of the blogdown book);

*I recommend you to publish your website via Github and Netlify (Section 3.1). Don't use the public/ directory build locally on your computer. Build it on the server side instead. If you must build your site locally, every time before you publish the public/ folder, make sure that you run blogdown::hugo_build() (Section 1.7).

For the reason why your RSS feed contains relative URLs, see Appendix D.3 (short answer: baseurl = "/").

redpajama_set_name: RedPajamaStackExchange


text:
Brands during COVID-19
SHARE | Thursday, 12 November 2020	| By Gabriel Tlagae
Gabriel Tlagae
The COVID-19 pandemic has certainly been a something anyone would be hard-pressed not to mention in review of the year 2020, for its presence and impact is, quite frankly, unmissable. Societies, communities and economies have felt the brunt of the negative impact, and even retail brands have had no option but to shift in business practice. Botswana has not been an exception in this regard. We continue to see an increase in retail brands collapsing or re-engineering their strategies to adapt or die. So how are brands faring during this pandemic, and how might brand positioning need to change during and post the COVID-19 pandemic?
As many consumers choose to prioritise essential goods and services and rethink personal finances, brands and retail companies operating in Botswana are faced with the challenge of how to survive and operate in the 'new normal'. Certain industries offering secondary services, such as entertainment, hospitality and transportation are feeling this pinch more acutely, and therefore many have had to realign their business structures as a whole to ensure the safety of customers while also ensuring their survival in the long-term.
In 2013, Nokia held a press conference to announce the mobile phone branch of the company was to be sold to Microsoft, in which CEO Stephen Elop said, "We did nothing wrong, and still we lost". Nokia was then a leading mobile giant. However, at the time of this press conference, their market share had dropped drastically and was overrun by new players on the market. Elop was right in stating that they had done nothing wrong, unfortunately, external factors can often prove to be detrimental to the survival of a brand.
This was one of many instances where brands and corporates would fare better with contingency strategy in place to ensure survival or at least adaptation of market shift dynamics. This year alone, numerous corporates and businesses received a wakeup call to sharpen their communication channels.
Currently, many brands across industries are realigning their brand positioning and overall tone in their messages to meet the needs of the consumer. We saw this realignment with one of the most popular fast-food chicken franchises KFC. The brand recently did away with their popular slogan that promoted 'finger licking' because it was not a 'COVID-19 friendly' message. Does this mean permanent scrapping of brand collateral? What does this mean for the business's bottom-line?
While these questions are to be considered for the long-time survival of brands and retail businesses, such actions need to be considered. The pandemic has forced a lot of brands to take a step back and consider their role during this moment in history. Consumers no longer expect frivolous messaging but instead action that is authentic, helpful and empathetic.
Marketing and advertising budgets have been set aside to support business' core strategies and to make way for the financial downturn, as well as to plough resources into providing relief to the general public - as is the duty of brands. Marketers have been forced to take a step back and rethink their relationship with customers. Creativity and brand positioning are now being swiveled to meet the demands of the crisis and with great reason. However, should we really sacrifice communications when it may be needed now more than ever? Communications teams have an even greater role to play today, in this brave new world, perhaps just away from shiny TV spots selling product, and more consumer-centric, or simply "human."
Profits need not take a further hit, but this does not simply need to be the brand's primary concern. It is bigger than that, and the brands that are making an even bigger mark are those that show their more human purpose.
*Botswana Life's PR, Comms & Marketing Manager
You might also want to read..
BIHL celebrates 45 Years
BEMA upbeat on export sector growth
BIHL operating profit up 12%

redpajama_set_name: RedPajamaCommonCrawl


MARILYN BERNICE DOUGLAS "Niecie" (Age 85) Transitioned peacefully into the loving arms of our Lord and Savior, Jesus Christ on Wednesday, August 29, 2018. She was predeceased by her husband John Douglas, Sr., and sister, Gwendolyn "Bunny" Malloy. Marilyn is survived by her devoted daughters, Lisa Douglas-Knight (Joseph) and Julie Douglas-Wilkins (David); son, John Douglas, Jr. (Pam); grandchildren, John Douglas III (Asha), Asha Douglas, Storm Douglas, and Lake Wilkins; great-grandson, Luke Douglas; sisters, Thelma Byrd and Inga Russell; and brother, Tony Russell (Claudia). Marilyn is affectionately known as "Grandma" and "Mother Douglas". She will be deeply missed by a host of loving nieces, nephews, other family members and friends where her legacy of love and devotion will live on. The viewing will be held on Thursday, September 6, 2018 from 1:30 pm to 2:30 pm, with a celebration of life beginning at 2:30 pm at the First Baptist Church of Glenarden (Ministry Center), 3600 Brightseat Road, Glenarden, MD 20706. Repass immediately following Benediction.


text:
Flood Resilience Measurement for Communities
Post-Event Review Capability
You are here: Home Blogs Towards a step-change on Loss & Damage
Select which of the newsletter you wish to subscribe
Learning Report
Foundations for Change
Zurich Flood Resilience Alliance
From Grey to Green infrastructure: a paradigm shift needed to deliver on climate action
Michael Szoenyi, Zurich Insurances
Towards a step-change on Loss & Damage
Reinhard Mechler Teresa M. Deubelli IIASA
Assessing the impacts of climate change on flood displacement risk
by Teresa M. Deubelli | Reinhard Mechler |
Making use of mutuality-solidarity-accountability-transparency principles
The Warsaw International Mechanism (WIM) for Loss and Damage and its ongoing review were hot topics at COP25. Hopes for a step-change on the issue of finance to scale up action and support have not been translated into action. Negotiating Parties remain divided over the way forward and the question of what kind of finance and for whom. We suggest to build on principles of risk governance, including insurance, and international cooperation – mutuality, solidarity, accountability and transparency – and to combine these in novel ways in order to upscale action on both averting and minimizing as well as addressing loss and damage under the WIM in a manner that truly shows responsibility for responding to the climate crisis.
Why do we need a step-change on Loss and Damage finance?
Hard limits to adaptation are creating situations beyond adaptation; think for example of communities fleeing desertification or sea-level rise that can only retreat so far. There is an urgent need to scale up efforts to address their loss and damage under the WIM. The WIM has made substantial strides on its objectives to advance knowledge and exchange since its creation, but now the time is ripe to take the next steps on addressing loss and damage from climate change.
So far, this third WIM priority has mostly been addressed through insurance approaches, such as the Fiji Clearing House. While there is value in scaling up risk transfer options, insurance comes with drawbacks: insurance premium costs often exceed financial capacities of vulnerable groups or may result in a false sense of protection that undermines further resilience-building action. Additionally, risk transfer options remain focused on sudden onset events.
Loss and damage from climate change is not just linked to sudden events; sea-level rise, desertification, and glacial melting take years to unfold, but once these tipping points are reached, recovery and reconstruction, and thus the typical logic of humanitarian assistance, are out of the question. As the climate crisis spirals forward tipping points may be reached sooner than expected, also challenging the sustainability of resilience building actions within the framework of development cooperation.
How to make a principled case for funders and affected to generate support for addressing loss and damage?
Most vulnerable countries agree that the WIM needs to advance on enhancing action and support for addressing loss and damage from climate change. Discussions at COP25 focused heavily on the issue of mobilizing finance for addressing loss and damage, but little headway was made, as views on the exact modalities of finance and its access differ vastly amongst Parties. Unfortunately, this means that the ongoing WIM review faces a certain risk of replicating the stalemate that characterised the Paris Agreement negotiations on the question of liability, when notions of compensatory justice were crossed out from Article 8 at the request of several developed, high emitting countries.
In order to propel the discourse forward in future rounds of climate talks and in the WIM review, we have suggested to recall the principles of international cooperation and insurance, mutuality, solidarity, accountability as well as the transversal principle transparency, and combine them in novel ways:
As the underlying insurance principle, mutuality is found in risk pooling and sharing – several parties pool funding to mobilise finance for offsetting losses in times of crisis, essentially spreading out and mutualising risks across participants.
Solidarity describes the shared responsibility for supporting others in times of hardship. As a principle it underpins development cooperation and humanitarian assistance and is at the heart of the Agenda 2030.
Accountability links actions with outcomes in a mutually responsible relationship and is motivated by a perceived ethical or legal obligation for supporting each other in addressing climate-attributed loss and damage.
As a transversal principle, transparency adds itself as a critical enabler of a finance architecture that expands the WIM to support those who are suffering loss and damage in ways that cannot be addressed through business-as-usual.
All principles lend themselves to the WIM as a ground for advancing on its priority to enhance action and support for addressing loss and damage from climate change, but also offer inspiration for thinking out novel ways to advance further.
What could this mean concretely?
These deliberations are not merely theoretical in nature but are seeing attention. For example through the further development of the African Risk Capacity (ARC) pool, a regional drought pool established in 2012 as a specialised agency of the African Union to help member states improve their planning, preparation and response capacities. Disbursements from the pool support participating governments' drought relief efforts, with requirements on how these are used (transparency and accountability).
Initial donor funding (solidarity) and ARC member annual premium payments (mutuality) capitalise the ARC. The pool is currently preparing for the launch of an additional capitalization mechanism, the Extreme Climate Facility (ARC-XCF). This would issue climate catastrophe bonds, resulting in pay-outs whenever the index tracking frequency and magnitude of droughts and extreme temperature exceeds a predefined threshold (transparency). While using the capital markets to access additional funding needs, accountability for climate change is factored in to some extent through the international support divested to setting up the mechanism.
The ARC-XCF is one way to address loss and damage and offers practical inspiration for setting up facilities for addressing loss and damage under the WIM. Especially where hard limits are pushed beyond adaptation and traditional insurance is no longer feasible, drawing on the experience from such risk pooling facilities can be useful input for setting up a specific facility under the WIM that supports those at the frontiers of climate change.
In doing so, it will be important to keep the principles of international cooperation and insurance - mutuality, solidarity, accountability and transparency – in mind to equitably address loss and damage, especially where risks are increasingly intolerable and beyond adaptation.
For more on the topic of Loss and Damage financing have a look at these resources:
Deubelli, T. and Mechler, R. (2019). Finance for Loss & Damage: Towards a comprehensive principled approach, unpublished.
Linnerooth-Bayer, J. Surminski, S., Bouwer, L., Noy, I., Mechler, R., McQuistan, C. (2018). Insurance as a Response to Loss and Damage? Mutuality – Solidarity – Accountability. Policy brief COP 24, December 2018
Mechler R, Bouwer L, Schinko T, Surminski S, Linnerooth-Bayer J (2018). Loss and Damage from climate change. Concepts, methods and policy options. Springer
Previous BlogNext Blog
Authors of this blog
Teresa M. Deubelli
Researcher at International Institute for Applied Systems Analysis
Reinhard Mechler
Deputy Program Director Risk and Resilience, IIASA
Global Food Security Guidelines:A step-by-step guide for...
This guide aims to assist National Society staff and volunteers throughout the world in undertaking food security assessments. It does not require...
How to conduct a food security assessment A step-by-step...
This guide is intended for National Society staff and volunteers in Africa who want to undertake food security assessments, but have no background...
Participatory Vulnerability Analysis, A step by step guide...
This guide is developed to assist field workers and communities to analyze people's vulnerability, draw action plans, mobilize resources and enact...
How to do a VCA A practical step-by-step guide for Red...
This 'How to do a VCA' second publication is intended as a practical, how to guide the National Society staff and volunteers to undertake VCA. This...
Measures to address loss and damage including insurance for...
This policy brief discusses the needs and methods for addressing the loss and damage climate change and natural hazards are likely to have on fishing...
Loss and Damage in IPCC's report on Global Warming of 1.5C...
Loss and Damage in IPCC's report on Global Warming of 1.5C (SR15). A summary by Reinhard Mechler and colleagues.
We have the data, and now what?
The Zurich Flood Resilience Alliance has launched a new guidance document to help practitioners use the data gathered through the Flood Resilience Measurement for Communities to plan appropr...
Making use of mutuality-solidarity-accountability-transparency principles The Warsaw International Mechanism (WIM) for Loss and Damage and its ongoing review were hot topics at COP25. Hope...
Foundations for change: what we've learned in year 1
The Zurich Flood Resilience Alliance entered its second phase in July 2018. Since then we have worked hard at creating an effective, collaborative, and innovative approach to disaster resili...
Invest in long-term resilience to prepare for a 1.5 degree plus world
Global Resilience Partnership Press Release 05 December COP25, Madrid, Spain. The world has made significant development progress over the past decades, but these gains have been uneven a...
Tell us about the resource that you think should be here. Send us the relevant links, or you can also upload relevant documents.
Upload Resources!
Do you need practical advice on something not currently included in the flood solutions catalogue?
Submit a Question!
© Copyright 2020. Zurich Flood Resilience Program - All rights reserved.
Developed by Smart Solutions


You've got to watch this video. Why? Because when was the last time you saw 101 Russian female skydivers break the record for the world's largest flower-pattern freefall?
As someone who is petrified of heights, it took a bit of willpower for me to even watch this video (my legs got a little shaky, no lie), but it's pretty awe-inspiring, right? Personally, there's nothing that could ever ever ever (ever) get me to skydive for any reason, but this is still incredible. How do they move through the air horizontally in order to join hands with one another? It's amazing to see, almost like they're flying.
The jump took place in the sky over Kolomna, Russia. To make the whole thing even more touching, the jump was dedicated to the group's former captain, Irina Sinitsina, who died while skydiving last year in California. If you look closely, you can see that there's one space left open in the middle, in honor of Irina.


I don't remember much from being a kid before Kindergarten. But Gavin says that he does. So either he has a great imagination or I just have a very poor memory!
Here are a few pics from Layton's adventure before bed. It sure looks like he's having fun. He couldn't stop giggling!
The roll I'm really talking about is quilting- of course! I should have 2 new quilts listed in my shop by this weekend; the Look and Learn Playmat and the Pretty Plain Jane. I just finished the quilt top for my Fall Splendor, Very Hungry Caterpillar and 30's Tumbler quilts and finished a strand of, completely hand sewn, yo-yo Christmas garland. I cut the binding for my Mumm's the Word on Santa quilt; just need to sew it on. Plus I started another Christmas quilt with the Moda Isn't Christmas Jolly? fabric. Wowsa, if you're not out of breath from reading all of that, I sure am from actually sewing all of it!
Remember- quilts make GREAT Christmas gifts; one that won't be stuffed into a closet and forgotten about 1 month after receiving it!
This is what came out of our garden on Friday! Yum!
when we do and keep you updated on its progress!
Well, it's official- Timber has a tooth! YIKES! No wonder she's been biting me while nursing! She's only 5.5 months old so she must be an early teether. Layton didn't get his first tooth until around 7 months so this one took me by surprise! It's on her lower gums in the front. I may be able to snap a photo but I'm not positive. She's sleeping right now so I won't even bother her.
There isn't a whole lot new around here with the exception of the tooth. I have almost finished my Fall Splendor quilt top. I just need to add a border to it. Here's the fabric line that it is from.
I ordered some of the "Very Hungry Caterpillar" fabric last week for a new quilt. I don't have a pattern in mind yet so we'll have to see what I can come up with. Layton is addicted to that book lately. My cousin bought it for him last Christmas along with 2 other Eric Carle books. We read them daily, if not multiple times a day.
My At Home In The Country quilt is kind of on hold for now. I do have some of it sewn but for some reason I'm not as into it as some of my others, so it always gets put on the back burner.
I also have started a 30's tumbler quilt. It's simple but I think it will really turn out fantastic!
Well, that's it for baby and quilting news! Don't forget to stop in to my shop once in a while to see what's new!
Two little bundles in one metal basket.........too cute.
Every August our church, Smith River Baptist, holds "Family Camp" for 4 days where we camp, eat and worship! We aren't camping this year because of many things, but we did go last night for the pot luck and service. The food was great, the praise and worship was wonderful and our guest speaker was fabulous! Wyatt missed out that evening due to football practice but he was able to be there during the day. Everyone had so much fun. I added a little video at the end as well of Timber. It's pretty cute- check it out! Enjoy!
ME, ON THE FRONT PAGE!! You can imagine how ecstatic I was!! I've never been on the front page before! Monika of Quilt Lover put together this fabulous treasury with items created by members of the Quiltsy team. Way to go!
So already today I have nearly completed my Look and Learn Playmat and will now go and work some more on my Fall Splendor quilt.
I've kind of been in a slump lately when it comes to quilting. Life just got extremely hectic and I don't know where the time has gone.
Wyatt is playing football this year and has practice every night from 5-7pm. We had a garage sale on Saturday and got rid of a whole bunch of stuff from my FINALLY cleared out storage unit that I had before Gavin and I got married.
My goals then this week are to finish up a play-mat that I am working on, which is very low stress, to work a little more on a fall quilt that I have coming out and to always remember that nothing is more important than my kids and my husband! And that's a good thing!


Are you recycling as much household waste as you can and is there anything that would make it easier for you?
Those are some of the questions local people are being urged to answer as the Council launches a survey into the county's recycling provision for householders.
The Household Waste and Recycling survey can be filled out online.
Additionally, staff from the Council's waste team will be out and about at Household Waste and Recycling Centres (HWRCs) over coming weeks speaking to members of the public about opening hours, kerbside collections, and encouraging them to fill out a survey.
The survey takes around five minutes to complete.
Everyone who completes the survey can enter a draw for a chance to win a £20 voucher.
Paper copies of the survey are available from the Council's Customer Services receptions in School Place, Kirkwall and Warehouse Buildings, Stromness.
The survey closes on 31 October.