text:
\section{Introduction}
Let $K$ be a local field with ring of integers $R$,  normalised valuation $v$ and uniformiser $\pi$. For an element $r \in R$, we write $\bar{r}$ for the reduction modulo $\mathfrak{p}$. The separable closure of $K$ is denoted by $K^{sep}$ and the algebraic closure by $\bar{K}$. We assume that the residue field $k = R/(\pi)$ is perfect and has characteristic $p \neq 2$.

Let $C/K$ be a hyperelliptic curve given by a Weierstraß equation
\[C : y^2 =f(x).\]
Throughout this paper, we will always assume that the curve $C$ has semistable reduction and genus $g > 1$. Since $p \neq 2$, it is easy to read off from the polynomial $f$ whether $C$ is semistable and determine the semistable reduction. This is described in \cite{bouw2017computing} and \cite{dokchitser2018arithmetic}.

To a Weierstraß equation, we associate the differential forms
\begin{align*}
\omega_0 = \frac{dx}{2y},\; \omega_1 = \frac{xdx}{2y},\; \dots, \; \omega_{g-1} = \frac{x^{g-1}dx}{2y}.
\end{align*}
These form a basis of $H^0(C, \Omega_{C/K})$. Let $\regX \rightarrow \Spec R$ be the minimal regular model of $C$ and $\omega_{\regX/R}$ the relative dualising sheaf \cite[Definition 6.4.18.]{liu2002alggeo}. In our situation, the relative dualising sheaf is isomorphic to the canonical sheaf \cite[Theorem 6.4.32]{liu2002alggeo}. We have that $H^0(C, \Omega_{C/K}) = H^0(\regX,\omega_{\regX/R}) \otimes_R K$ and $H^0(\regX,\omega_{\regX/R})$ is a free $R$-module of rank $g$. So $\det H^0(\regX,\omega_{\regX/R}) := \bigwedge^g H^0(\regX,\omega_{\regX/R})$ is free of rank one over $R$.

In this paper, we are going to study the following two problems:
\begin{enumerate}
	\item Let $ \omega := \omega_0 \land \dots \land \omega_{g-1} \in  \det H^0(C,\Omega^1_{C/K})$. Determine $\lambda_C \in K$, such that $\lambda_C \cdot \omega$ generates  $\det H^0(\regX,\omega_{\regX/R})$ as an $R$-module.
	\item Explicitly determine a basis for the global sections of $\omega_{\regX/R}$.
\end{enumerate}

Our approach is based on results of \cite{kausz1999discriminant}. Under simplified hypotheses  a formula for $\lambda_C$ and a description of a basis for the global sections of $\omega_{\regX/R}$ is given in Proposition 5.5. of that paper.

\subsection{Motivation}
Our motivation for studying the differential forms on the minimal regular model comes from the Birch and Swinnerton-Dyer conjectures. Originally formulated for elliptic curves, the conjectures were later generalised to abelian varieties over number fields by Tate \cite{tate1966bsd}.

 Let $C$ be a hyperelliptic curve defined over the rational numbers $\QQ$ and let $J$ denote its Jacobian. The second Birch and Swinnerton-Dyer conjecture in this situation is

Here $L(J, s)$ is the $L$-series of $J$ and $r$ its analytic rank. Reg denotes the regulator of $J(\QQ)$. For a prime $p$, the Tamagawa number is denoted by $c_{p}$. The Shafarevich-Tate group is represented by $\Sh(J,\QQ)$ and $J(\QQ)_{\mathrm{tors}}$ is the torsion subgroup of $J(\QQ)$. The results of this paper can be applied to calculate the sixth quantity, that is the period $\Omega$. For the description of this quantity we follow the outline in \cite[Section 3]{van2017numerical} and \cite[Section 3.5]{flynn2001empirical}. Both papers provide numerical evidence for the Birch and Swinnerton-Dyer conjectures for hyperelliptic curves.

Let $\nerJ \rightarrow \Spec \ZZ$ denote the N\'eron model of $J$. We have that $H^0(\nerJ,\Omega_{\nerJ/\ZZ})$ is a free $\ZZ$-module of rank $g$. Let $(\mu_0, \dots, \mu_{g-1})$ be a basis for this module. Then $\mu:= \mu_0 \land \dots \land \mu_{g-1}$ is a generator for $\bigwedge^g H^0(\nerJ,\Omega_{\nerJ/\ZZ})$ and $\Omega$ is defined as
\[\Omega:= \int_{J(\RR)} |\mu|. \]
Finding a basis for $H^0(\nerJ,\Omega_{\nerJ/\ZZ})$ can be done locally. Let $p \in \ZZ$ be a prime. We write $R=\ZZ_p$ and use the notation introduced in the beginning. Since $H^0(\nerJ,\Omega_{\nerJ/R})$ and $H^0(\regX, \omega_{\regX/R})$ are isomorphic as $\ZZ_p$-modules, see \cite[Lemma 9]{van2017numerical}, it is enough to find a basis for the global sections of $\omega_{\regX/R}$.

For computational purposes it is not always necessary to know the basis of regular differentials. In \cite{van2017numerical} and \cite{flynn2001empirical} the authors first evaluate $\int_{J(\RR)} |\omega|$, where $\omega$ is the exterior product of the differentials $\omega_0, \dots, \omega_{g-1}$ associated to the Weierstraß equation for $C$ and then compute a correction term in order to get the value  $\Omega:= \int_{J(\RR)} |\mu|$.

\subsection{Results}

We give a brief overview of our main results and illustrate them by means of an example. The results are stated in terms of cluster pictures. A cluster picture is a combinatorial object associated to the Weierstraß equation of a curve. It encodes different invariants of the curve. The concept has been studied in \cite{dokchitser2018arithmetic}. For definitions we refer to \cite[Section 1.5.]{dokchitser2018arithmetic} or Section \ref{sec:cluster} of this paper. Here we only recall the necessary notation.
\begin{multicols}{2}
	\begin{itemize}
		\item[$c_f$] leading coefficient of $f$
		\item[$\cR$]  set of roots of $f$ in $K^{sep}$
		\item[$\cs$] a cluster
		\item[$z_{\cs}$] a centre of $\cs$
		\item[$d_{\cs}$] depth of $\cs$
		\item[$\delta_{\cs}$] relative depth of $\cs$
		\item[$\cs \land \cs'$] smallest cluster containing $\cs$ and $\cs'$
		\item[$\nu_{\cs}$]  $ = v(c_f)+\sum_{r \in \cR} d_{r \land \cs}$
	\end{itemize}
\end{multicols}

First, we show how to read off a basis for $H^0(\regX,\omega_{\regX/R})$ from the cluster picture of a curve.

\begin{thm}[Theorem \ref{thm:mainbasis}]
	\label{thm:mainbasis_int}
	Let $C/K$ be a semistable hyperelliptic curve defined by an integral Weierstraß equation $C:y^2=f(x)$ with cluster picture $\Sigma$.
	Let $\regX/ R$ be the minimal regular model. Assume that the residue field $k$ is algebraically closed.

	Choose clusters $\cs_0, \dots, \cs_{g-1}$  inductively such that
	\[e_i = \frac{\nu_{\cs_i}}{2} - \sum_{j=0}^{i}d_{\cs_j \land \cs_i} = \max \left( \frac{\nu_{\cs}}{2} - \sum_{j=0}^{i-1}d_{\cs_j \land \cs} -d_{\cs}\right),\]
	where the maximum is taken over all proper clusters in $\Sigma$.
	If the maximal value is obtained by two different clusters $\cs$ and $\cs'$ with $\cs' \subset \cs$, choose  $\cs_i = \cs$.

	Then an $R$-basis for the global sections of the relative dualising sheaf $\omega_{\regX/R}$ is given by $(\mu_0, \dots \mu_{g-1})$, where
	\[\mu_i = \pi^{e_i} \prod_{j=0}^{i-1}(x-z_{\cs_{j}}) \frac{dx}{2y}.\]

\end{thm}

Note that the sequence constructed in the theorem is not canonical, since it can happen that the maximum is obtained by two incomparable clusters, in which case any of the two clusters may be chosen. Before we illustrate the construction by an example, we make an observation concerning the assumptions in the theorem.

\begin{rem}
	\label{rem:k_closed}
	In Theorem \ref{thm:mainbasis_int}, it is assumed that the residue field is algebraically closed. This assures that every proper cluster has a $K$-rational centre. However, it should be pointed out that the construction of the minimal regular model commutes with taking the maximal unramified extension.
	More precisely, let $K$ be a local field with ring of integers $R$, perfect residue field $k$  and $K^{ur}$ the maximal unramified extension with ring of integers $R'$. Let $C$ be a smooth projective curve over $K$ and $\regX$ its minimal regular model. Then $\regX'= \regX \times_{\Spec R} \Spec R'$ is the minimal regular model of $C$ over $R'$. This follows from Lemma 10.3.33.(b) in \cite{liu2002alggeo}.

	As a consequence $H^0(\regX',\omega_{\regX'/R'}) = H^0(\regX,\omega_{\regX/R}) \otimes_{R} R'$.
	Moreover, if we find a basis for $H^0(\regX',\omega_{\regX'/R'})$ which is defined over $K$, it is also a basis for $H^0(\regX,\omega_{\regX/R})$.
	Translated to the situation of the theorem, this means that we can replace the assumption $k$ algebraically closed with the condition that every proper cluster admits a $K$-rational centre.

	 Note that for the same reason, the residue field $k$ does not need to be algebraically closed in the statement of  Theorem \ref{thm:main_lambda_int}.
\end{rem}

\begin{ex}
	\label{ex:basis}
	Let $p>3$ and $C/\QQ_p$ the hyperelliptic curve of genus $g = 5$ defined by
	\[y^2 = x(x-p^6)(x-2p^6)(x-p^4)(x-2p^4)(x-3p^4)(x-1)(x-1-p^8)(x-1-2p^8)(x-3p^8)(x-2)(x-3).\] The proper clusters are
	\[ \cR,\; \ct_1 = \{0,p^6, 2p^6, p^4,2p^4, 3p^4\},\; \ct_2 = \{0, p^6, 2p^6\},\; \ct_3 = \{1, 1+p^8, 1+2p^8,1+3p^8\}. \]
	These clusters have depths $d_\cR = 0, \; d_{\ct_1} = 4, \; d_{\ct_2} = 6$, $d_{\ct_3} = 8$ and relative depths $\delta_{\ct_1} = 4, \; \delta_{\ct_2} = 2$, $\delta_{\ct_3} = 8$. This information is contained in the cluster picture $\Sigma$:
	\begin{center}
		\clusterpicture
		\Root(0.00,2)A1(a1);
		\Root(0.40,2)A1(a2);
		\Root(.80,2)A1(a3);
		\Root(1.4,2)A1(b1);
		\Root(1.8,2)A1(b2);
		\Root(2.2,2)A1(b3);
		\Root(2.8,2)A1(c1);
		\Root(3.2,2)A1(c2);
		\Root(3.6,2)A1(c3);
		\Root(4.0,2)A1(c4);
		\Root(4.6,2)A1(d1);
		\Root(5.0,2)A1(d2);
		\ClusterL(C3){(a1)(a2)(a3)}{$\mathbf{2}$};
		\ClusterL(C2){(C3)(C3n)(b1)(b2)(b3)}{$\mathbf{4}$};
		\ClusterL(C4){(c1)(c2)(c3)(c4)}{$\mathbf{8}$};
		\ClusterL(C1){(C2)(C2n)(C4)(C4n)(d1)(d2)}{$\mathbf{0}$};

		\endclusterpicture
	\end{center}
	The subscript of the top cluster is its depth. The subscripts of the other clusters are their relative depths.

	We construct a sequence of clusters as described in Theorem \ref{thm:mainbasis_int}. The theorem is applicable since every cluster has a centre in $\QQ_p$ (see the previous remark).
	First we choose $\cs_0$ to be the cluster that maximises $\frac{\nu_{\ct}}{2}-d_{\ct}$. The evaluation of this term for each cluster can be found in the first column (after the double line) of the table below. Next we choose $\cs_1$ to be the cluster that maximises $\frac{\nu_{\ct}}{2}-d_{\ct}- d_{\ct\land\cs_0}$, the cluster $\cs_2$ is the one that maximises $\frac{\nu_{\ct}}{2}-d_{\ct}- d_{\ct\land\cs_0}- d_{\ct \land\cs_1}$ and so on.

	\begin{center}
		\begin{tabular}{ l | c | c || c | c | c | c | c }
			&$\nu_{\ct}$&$d_{\ct}$&$\frac{\nu_{\ct}}{2}-d_{\ct}$&$\frac{\nu_{\ct}}{2}-d_{\ct} - d_{\ct \land \ct_2}$&\dots $- d_{\ct \land \ct_3}$&\dots$-d_{\ct \land \ct_1}$&$\dots - d_{\ct \land \cR}$\\ \hline
			$\cR$   & 0  & 0 & 0  & 0 & 0 & \numcirclemodi{0}&\numcirclemodi{0}\\ \hline
			$\ct_1$ & 24 & 4 & 8 & 4& \numcirclemodi{4}& 0 &0\\ \hline
			$\ct_2$ & 30 & 6 & \numcirclemodi{9} & 3&3& -1 &-1\\ \hline
			$\ct_3$ & 32 & 8 & 8 &  \numcirclemodi{8}&0& 0 &0\\
		\end{tabular}
	\end{center}
	In each column the maximal value is circled to indicate which cluster is chosen in the respective step. Three dots always represent the entire expression in the previous column. We can read off from the table

	\begin{center}
		\bgroup
		\def\arraystretch{1.5}
		\begin{tabular}{ c  c  c  c  c c }
			&$\cs_0 = \ct_2$,  & $\cs_1 = \ct_3$, & $\cs_2 = \ct_1$,&$\cs_3 = \cR$,&$\cs_4 = \cR$, \\
			and &$e_0 = 9,$ &$e_1 = 8$, & $e_2 = 4$, &$e_3 = 0$, &$e_4 = 0$.\\
		\end{tabular}
		\egroup
	\end{center}
	If we choose $z_{\cR} = z_{\ct_1} = z_{\ct_2} = 0 \textrm{ and } z_{\ct_3} = 1$ as centres for the clusters,
	we get the following basis for the global sections of $\omega_{\regX/R}$:
	\[ \left( \mu_0 = p^9 \,\frac{dx}{2y}, \;\mu_1 = p^8 x\,\frac{dx}{2y}, \; \mu_2 = p^4 x(x-1)\, \frac{dx}{2y}, \; \mu_3 = x^2(x-1) \,\frac{dx}{2y}, \; \mu_4 = x^3(x-1)\, \frac{dx}{2y} \right)\]
\end{ex}

As mentioned before, it is often not necessary to determine the basis for $H^0(\regX,\omega_{\regX/R})$ explicitly, but it suffices to know a generator for $\det H^0(\regX,\omega_{\regX/R})$. Theorem \ref{thm:main_lambda_int} gives a convenient formula for determining this generator.

\begin{thm}[Theorem \ref{thm:main_lambda}]
	\label{thm:main_lambda_int}
	Let $C/K$ be a semistable hyperelliptic curve defined by $C: y^2=f(x)$ with $f(x) = c_f\prod_{r\in \cR}(x-r)$. We write $\omega_0 = \frac{dx}{2y}, \dots, \omega_{g-1} = \frac{x^{g-1}\,dx}{2y}$ for the differentials associated to this equation and $\omega := \omega_0 \land \dots \land \omega_{g-1} \in  \det H^0(C,\Omega^1_{C/K})$. Let $\regX \rightarrow \Spec R$ be the minimal regular model of $C$. Suppose that $\lambda_C \cdot \omega$ is a basis for $\det H^0(\regX,\omega_{\regX/R})$. Then
	\begin{align*}
	\begin{split}
	8 \, v(\lambda_C) =  &\; 4\, g \cdot v(c_f) + \sum_{\substack{|\cs| \textrm{ even}\\ \cs \neq \cR}} \delta_{\cs}  (|\cs|-2)|\cs| + \sum_{\substack{|\cs| \textrm { odd}\\ \cs \neq \cR}} \delta_{\cs} (|\cs|-1)^2\\
	&~~~~+ d_{\cR}
	\begin{cases}
	(|\cR|-2)|\cR|, & \text{if $|\cR| = 2g+2$}\\
	(|\cR|-1)^2, & \text{if $|\cR| = 2g+1$}
	\end{cases}.
	\end{split}
	\end{align*}
\end{thm}

Let us revisit the above example.

\begin{ex}
	\label{ex:lambda}
	Let $p>3$ and $C/\QQ_p^{ur}$ be the hyperelliptic curve from Example \ref{ex:basis}. Recall the cluster picture $\Sigma$:
	\begin{center}
		\clusterpicture
		\Root(0.00,2)A1(a1);
		\Root(0.40,2)A1(a2);
		\Root(.80,2)A1(a3);
		\Root(1.4,2)A1(b1);
		\Root(1.8,2)A1(b2);
		\Root(2.2,2)A1(b3);
		\Root(2.8,2)A1(c1);
		\Root(3.2,2)A1(c2);
		\Root(3.6,2)A1(c3);
		\Root(4.0,2)A1(c4);
		\Root(4.6,2)A1(d1);
		\Root(5.0,2)A1(d2);
		\ClusterL(C3){(a1)(a2)(a3)}{$\mathbf{2}$};
		\ClusterL(C2){(C3)(C3n)(b1)(b2)(b3)}{$\mathbf{4}$};
		\ClusterL(C4){(c1)(c2)(c3)(c4)}{$\mathbf{8}$};
		\ClusterL(C1){(C2)(C2n)(C4)(C4n)(d1)(d2)}{$\mathbf{0}$};

		\endclusterpicture
	\end{center}
	Then the formula of Theorem \ref{thm:main_lambda_int} yields
	\begin{align*}
	8 \, v(\lambda_C) &=   4 \cdot g \cdot v(c_f) &+\;& \delta_{\ct_1}(|\ct_1|-2)|\ct| &&+\; \delta_{\ct_2}(|\ct_2|-1)^2 &+\;& \delta_{\ct_3}(|\ct_3|-2) |\ct_3| &+\;& d_{\cR}\, (|\cR|-2)|\cR|\\
	&= 4 \cdot 5 \cdot 0 &+\;& 4 \cdot 4 \cdot 6 &&+\; 2 \cdot 2^2 &+\;& 8 \cdot 2 \cdot 4 &+\;& 0 \cdot 10 \cdot 12\\
	&= 8 \cdot 21. &&&&&&&&
	\end{align*}
	So \[\mu: = p^{21} \, \frac{dx}{2y} \land \frac{xdx}{2y}\land \frac{x^2dx}{2y}\land \frac{x^3dx}{2y}  \land \frac{x^4dx}{2y} \]
	generates $\det H^0(\regX,\omega_{\regX/R})$. Note that the value $v(\lambda_C) = 21$ is equal to the sum over the $e_i$ determined by Theorem \ref{thm:mainbasis_int}.
\end{ex}

\subsection{Outline}
In Section \ref{sec:cluster}, we review some definitions and facts about cluster pictures. In Section \ref{sec:lambda} we translate Proposition 5.5. of \cite{kausz1999discriminant} into the language of cluster pictures and generalise it in order to prove Theorem \ref{thm:main_lambda_int}. The last section is dedicated to the proof of Theorem \ref{thm:mainbasis_int}. This is done by first showing that the differentials forms defined in the theorem are indeed global sections of the canonical sheaf and then applying Theorem \ref{thm:main_lambda_int}.

\subsection*{Acknowledgements}
I would like to thank Vladimir Dokchitser for proposing to work on this topic and his support throughout the creation of this paper. I would also like to thank Stefan Wewers for very helpful discussions and comments on earlier versions of this paper, as well as Adam Morgan who also suggested a proof of Proposition 3.3.

\section{Cluster Pictures}
\label{sec:cluster}
In this section, we describe the cluster picture associated to an equation defining a hyperelliptic curve and briefly introduce the notation used in the subsequent sections. All information is taken from \cite{dokchitser2018arithmetic}.

Let $C/K$ be a hyperelliptic curve defined by a Weierstraß equation $C:y^2 = f(x)$. We write $\cR$ for the set of roots of $f(x)$ in $K^{sep}$ and $c_f$ for its leading coefficient, so that
\[f(x)= c_f\prod_{r\in \cR}(x-r).\]

\begin{defs}
	\begin{enumerate}[(i)]
		\item (\cite{dokchitser2018arithmetic} Definition 1.1.) A \textit{cluster} is a non-empty subset $\cs \subset \cR$ of the form $\cs = D\cap \cR$ for some disc $D = \{x \in \bar{K} \,|\,v(x-z) \geq d\}$ for some $d\in \QQ$ and $z\in \bar{K}$. We say that $z$ is a \textit{centre} of the cluster and write $z = z_{\cs}$.
		\item 	(\cite{dokchitser2018arithmetic} Definition 1.1.) If $|\cs|>1$, then $\cs$ is called a \textit{proper} cluster and its \textit{depth} is defined to be $d_{\cs} = \min_{r,r' \in \cs} v(r-r')$.
		\item (\cite{dokchitser2018arithmetic} Definition 1.3.) If $\cs' \subsetneq \cs$ is a maximal subcluster, we write $\cs' < \cs$ and say that $\cs'$ is a child of $\cs$.
		For two clusters (possibly roots) $\cs,\; \cs'$ we write $\cs \land \cs'$ for the smallest cluster that contains them.
		\item (\cite{dokchitser2018arithmetic} Definition 1.4.) A cluster $\cs$ is \textit{principal} if $|\cs| \geq 3$,
		except if either $\cs = \cR$ is even and has exactly two children, or if $\cs$ has a child of size $2g$.
		\item (\cite{dokchitser2018arithmetic} Definition 1.5.) If $\cs \neq \cR$ is proper, the \textit{relative depth} of  $\cs$ is defined as the difference between the depth of $\cs$ and the depth of the smallest cluster strictly containing $\cs$.  It is denoted by $\delta_{\cs}$.
		\item (\cite{dokchitser2018arithmetic} Definition 1.6.) For a proper cluster $\cs$, set $\nu_{\cs} = v(c_f)+\sum_{r \in \cR} d_{r \land \cs}$.
	\end{enumerate}
\end{defs}

By the \textit{cluster picture} associated to an equation, we mean the collection of clusters $\{\cs \subset \cR\}$ together with their depths. See Example \ref{ex:basis} in the introduction for an illustration of these definitions.

\begin{rem}
	\label{rem:compute_vs}
	Note that for a root $r \in \cR$ and a cluster $\cs$, we have \[d_{r \land \cs} = d_{\cR} + \sum_{\substack{\cs' \neq \cR:\\ \cs \land r \subseteq \cs'  }} \delta_{\cs'}.\] So $\nu_{\cs}$ can also be calculated via \[\nu_{\cs} = v(c_f)+d_{\cR}|\cR| +  \sum_{\substack{\cs' \neq \cR:\\ \cs \subseteq \cs'  }} \delta_{\cs'} |\cs'|. \]
\end{rem}

There is a notion of equivalence for cluster pictures that respects isomorphisms of hyperelliptic curves. For a complete discussion of the topic we refer to  \cite[Section 14]{dokchitser2018arithmetic}. The following proposition is important for the proof of Theorem \ref{thm:main_lambda}. So we state it here for the convenience of the reader.

\begin{prop}[\cite{dokchitser2018arithmetic}, Proposition 14.6.]
	\label{prop:manipulateclusters}
	Let $f(x) \in K[x]$ be a separable polynomial with roots $\cR \subset K^{sep}$, such
	that the absolute Galois group $G_K = G(K^{sep}/K)$ acts tamely on $\cR$, and let $\Sigma$ be the associated cluster picture. Suppose $\Sigma'$ is a cluster picture obtained from $\Sigma$ by one of the following constructions:
	\begin{enumerate}
		\item Increasing the depth of all clusters by some $n\in \ZZ$;
		\item Adding a root to $\cR$, provided $|\cR|$ is odd, $d_{\cR} \in \ZZ$ and $|k|> \#\{\cs < \cR : \cs \textrm{ is } G_K-\textrm{stable}\}$;

		\item Redistributing the depth between $\cs$ and $\cR \backslash \cs$ by decreasing the depth of $\cs$ by $1$ and increasing the depth of $\cR \backslash \cs$ by $1$,
		provided $|\cR|$ is even, $\cs < \cR$ is $G_K$-stable with $d_{\cR}, \,d_{\cs} \in \ZZ$ and $|k| > \#\{\ct < \cs : \ct \textrm{ is } G_K- \textrm{stable} \}$.
		\footnote{If $\cR \backslash \cs$ is not contained in $\Sigma$, it is added to the cluster picture as a new cluster with relative depth $\delta_{\cR\backslash\cs} =0$. In accordance with \cite[Definition 14.1.]{dokchitser2018arithmetic} redistributing the depth between $\cs$ and $\cR \backslash \cs$ includes decreasing (respectively increasing) the depth of all proper clusters contained in $\cs$ (respectively $\cR \backslash \cs$).}

	\end{enumerate}
	Then there is a Möbius transformation $\phi(z) = \frac{az+b}{cz+d}$ with $a, b, c, d \in K$, such that $\Sigma'$	is the cluster picture of $\cR' = \{\phi(r): r \in \cR \} \backslash \{\infty\}$ if $|\cR|$ is even and of $\cR' = \{\phi(r): r \in \cR \cup \{\infty\}\}\backslash \{\infty\}$ if $|\cR|$ is odd.	Moreover, if $y^2 = f(x)$ is a hyperelliptic curve, then there is a $K$-isomorphic curve given by a Weierstraß model whose cluster picture is $\Sigma'$.
\end{prop}

\section{A Basis for $\det H^0(\regX,\omega_{\regX/R})$}
\label{sec:lambda}

Let $C/K$ be a semistable hyperelliptic curve of genus $g$ defined by $C: y^2=f(x)$ with $f(x) = c_f\prod_{r\in \cR}(x-r)$. We write $\omega_0 = \frac{dx}{2y}, \dots, \omega_{g-1} = \frac{x^{g-1}\,dx}{2y}$ for the differentials associated to this equation and $\omega := \omega_0 \land \dots \land \omega_{g-1} \in  \det H^0(C,\Omega^1_{C/K})$. Let $\regX \rightarrow \Spec R$ be the minimal regular model of $C$. The main result of this section is Theorem \ref{thm:main_lambda}, where we determine $\lambda_C \in K$, such that $\lambda_C \cdot \omega$ generates  $\det H^0(\regX,\omega_{\regX/R})$ as an $R$-module. Note that $\lambda_C$ is only well-defined up to a unit. Moreover it is not a curve invariant, but depends on the equation.

\begin{thm}
	\label{thm:main_lambda}
	Let $C/K$ be a semistable hyperelliptic curve of genus $g$ defined by $C: y^2=f(x)$ with $f(x) = c_f\prod_{r\in \cR}(x-r)$. We write $\omega_0 = \frac{dx}{2y}, \dots, \omega_{g-1} = \frac{x^{g-1}\,dx}{2y}$ for the differentials associated to this equation and $\omega := \omega_0 \land \dots \land \omega_{g-1} \in  \det H^0(C,\Omega^1_{C/K})$. Let $\regX \rightarrow \Spec R$ be the minimal regular model of $C$. Suppose that $\lambda_C \cdot \omega$ is a basis for $\det H^0(\regX,\omega_{\regX/R})$. Then
	\begin{align}
	\label{eqn:lambda}
	\begin{split}
	8 \, v(\lambda_C) =  &\; 4\, g \cdot v(c_f) + \sum_{\substack{|\cs| \textrm{ even}\\ \cs \neq \cR}} \delta_{\cs}  (|\cs|-2)|\cs| + \sum_{\substack{|\cs| \textrm { odd}\\ \cs \neq \cR}} \delta_{\cs} (|\cs|-1)^2\\
	&~~~~+ d_{\cR}
	\begin{cases}
	(|\cR|-2)|\cR|, & \text{if $|\cR| = 2g+2$}\\
	(|\cR|-1)^2, & \text{if $|\cR| = 2g+1$}
	\end{cases}.
	\end{split}
	\end{align}
\end{thm}

A result in \cite{kausz1999discriminant} shows that this formula is true under several additional assumptions (Lemma \ref{lem:kausz}). Our strategy for the proof of the theorem is to pass to a tamely ramified extension of the base field and apply several Möbius transformations until Lemma \ref{lem:kausz} can be applied. In the course of these transformations the valuation of $\lambda$ changes. Hence it is necessary to add correction terms to the original formula. Their computation can be reduced to computing the change of the discriminant of the equation under these Möbius transformations, which has already been done in \cite[Section 16]{dokchitser2018arithmetic}.  A key ingredient for this reduction step is the \textit{hyperelliptic discriminant}, see Definition \ref{def:hyperellipticDiscriminant}. This is a curve invariant which connects the valuation of $\lambda$ with the valuation of the discriminant of a Weierstraß equation.

\begin{rem}
	\label{rem:notation_kausz}
	The starting point for our proof is \cite[Proposition 5.5.2]{kausz1999discriminant}. The results in that article are phrased in a different language. For the convenience of the reader, we shortly introduce the necessary notation and explain how it compares to cluster pictures. This notation will only be used in the proof of the following lemma.

	Throughout, let $y^2=f(x)$ be an equation defining a semistable hyperelliptic curve. Assume that the set of roots $\cR$ of $f$ is contained in $R$.
	Hence for any $n \in \NN \cup \{0\}$, there is a natural map
	\[
	\rho_n: \cR \to R/\cp^n
	\] and one may define
	\[
	\mathcal{V}_n = \{V \in R/\cp^n \mid \#\rho_n^{-1}(V) \geq 2\}.
	\]
	The set
	\[
	\mathcal{V}(T) = \cup_{n\geq 0} \mathcal{V}_n
	\]
	naturally has the structure of a rooted tree.
	For each $V$ in $\mathcal{V}(T)$, one defines $\phi(V) = |\rho_n^{-1}(V)|$ and $\gamma(V) = \lfloor (\phi(V) - 1)/2 \rfloor$.

	Let us now compare the tree $\mathcal{V}(T)$ to the cluster picture of the equation. Clearly, $\cR$ corresponds to the vertex $V_0$. Let $\cs = \{r_1, \dots, r_m\} \subsetneq \cR$ be a proper cluster with depth $d_{\cs}$ and relative depth $\delta_{\cs}$. Then $\rho_n(\cs) \in \mathcal{V}_n$ if and only if $d_{\cs} -\delta_{\cs}< n \leq d_{\cs}$. Roughly speaking, for each proper cluster $\cs \subsetneq \cR$, the tree $\mathcal{V}(T)$ contains $\delta_{\cs}$ copies of this cluster. Note that $\delta_{\cs}$ is integral by assumption. On the other hand, it is easy to see that each vertex in $\mathcal{V}(T)$ corresponds to some proper cluster and the value $\phi(V)$ is the cardinality of that cluster.
\end{rem}

\begin{lem}
	\label{lem:kausz}
	Let $C/K$ be a hyperelliptic curve defined by $C: y^2=f(x)$ with $f(x) = c_f\prod_{r\in \cR}(x-r)$. Assume that
\begin{multicols}{2}
	\begin{enumerate}[(i)]
		\item $\cR \subset R$,
		\item $v(r-s) \in 2\ZZ$ for all $ r,\, s \in \cR$,
		\item $c_f$ is a unit in $R$,
		\item $\#\cR = 2g+2$,
		\item $\#\{\bar{r} \,|\, r\in \cR\} \geq 3$.
		\item[]
	\end{enumerate}
\end{multicols}
	Then $v(\lambda_C)$ can be computed using Equation \ref{eqn:lambda}.
\end{lem}

\begin{proof}
	Under the conditions in the lemma, Equation \ref{eqn:lambda} reduces to
	\begin{align}
	\label{eqn:simple}
	8 \,v(\lambda_C) = \sum_{\substack{|\cs| \textrm{ even}\\ \cs \neq \cR}} \delta_{\cs} (|\cs|-2)|\cs| + \sum_{\substack{|\cs| \textrm { odd}\\ \cs \neq \cR}} \delta_{\cs} (|\cs|-1)^2.
	\end{align}
	This formula is due to Kausz, see \cite[Proposition 5.5.2.]{kausz1999discriminant}. The formula in the original source reads as
	\[
	\sum_{i=0}^{g-1} e_i = \frac{1}{2} \sum_{\substack{V>V_0\\\phi(V) \textrm{ even}}} \gamma(V)(\gamma(V) + 1) + \frac{1}{2} \sum_{\substack{V>V_0\\\phi(V) \textrm{ odd}}} \gamma(V)^2
	\] with the notation introduced in the remark preceding this lemma.  Note that the $e_i$'s play the same role as the $e_i$'s defined in Theorem \ref{thm:mainbasis} and it directly follows from the first part of the proposition in \cite{kausz1999discriminant} that $\sum e_i = v(\lambda_C)$. Phrased in terms of cluster pictures, this is precisely  Formula \ref{eqn:simple}. Several conditions are imposed on the equation defining $C$ (cf. \cite[Lemma 4.1]{kausz1999discriminant}). It is easy to see that these are implied by Conditions (i)-(v).
	Moreover, it is assumed that the residue field $k$ is algebraically closed, but this has no effect on the valuation of $\lambda_C$, see Remark \ref{rem:k_closed}.
\end{proof}

Note that the conditions in the above lemma imply semistability, see for example \cite[Theorem 7.1.]{dokchitser2018arithmetic}. Conversely, after a tamely ramified extension of the base field, there always exists an equation for $C$ satisfying the conditions listed in the above lemma if $C$ is semistable.

\begin{prop}
	\label{prop:H0basechange}
	Let $C/K$ be a semistable hyperelliptic curve with minimal regular model $\regX \rightarrow \Spec R$. Let $K'/K$ be a finite field extension. Write $C'$ for the base-change of $C$ to $K'$, $R' = \mathcal{O}_{K'}$ and $\regX'\rightarrow \Spec R'$ for the minimal regular model of $C'$.
	Then
	\[H^0(\regX', \omega_{\regX'/R'}) = H^0(\regX, \omega_{\regX/ R}) \otimes_R R'\]
	inside $H^0(C', \Omega_{C'/K'}) = H^0(C, \Omega_{C/K}) \otimes_K K'$.
\end{prop}

\begin{proof}
	Let $\stY \rightarrow \Spec R$ and $\stY' \rightarrow \Spec R'$ be the stable models of $C/K$ and $C'/K'$ respectively.
	The stable model $\stY$ is obtained from $\regX$ by contraction of all components $\Gamma$ of the special fibre for which $K_{\regX/ R} . \Gamma = 0$. Write $f: \regX \rightarrow \stY$ for the contraction morphism. Since the intersection matrix of the contracted components is negative definite, it follows from \cite[Corollary 9.4.18.]{liu2002alggeo} that $\omega_{\regX/R} = f_*\omega_{\stY/R}$. Therefore $ H^0(\regX, \omega_{\regX/R}) = H^0(\stY,\omega_{\stY/R})$ inside $H^0(C, \Omega_{C/K})$. By the same reasoning $ H^0(\regX', \omega_{\regX'/ R'}) = H^0(\stY',\omega_{\stY'/ R'})$.
	So it suffices to show that $H^0(\stY',\omega_{\stY'/R'}) = H^0(\stY,\omega_{\stY/R}) \otimes_R R'$.

	 We have $\stY' = \stY \times_R R'$ and by \cite[Theorem 6.4.9.b]{liu2002alggeo} $\omega_{\stY'/ R'} = p^* \omega_{\stY/R}$, where $p : \stY \times_R R' \rightarrow \stY$ is the first projection. So the result follows from \cite[Corollary 5.2.27]{liu2002alggeo}.
\end{proof}

\begin{lem}
	\label{lem:tame}
	Let $C/K$ be a hyperelliptic curve with semistable reduction defined by $C: y^2=f(x)$. Let $K'/K$ be a finite extension and write $C'$ for the base-change of $C$ to $K'$. Then Equation \ref{eqn:lambda} holds for $C/K$ if and only if it holds for $C'/K'$.
\end{lem}

\begin{proof}
	 Let $e_{K'/K}$ denote the ramification degree of the extension $K'/K$. We write $\Sigma$ for the cluster picture associated to $C:y^2 = f(x)$ and $\Sigma'$ for the cluster picture associated to the equation $C':y^2=f(x)$ over $K'$. The clusters themselves do not change under a finite extension, but their depths do. More precisely we have $\delta_{\cs}' = e_{K'/K} \cdot \delta_{\cs}$ for each proper cluster $\cs\neq \cR$  and $d_{\cR}' = e_{K'/K} \cdot d_{\cR}$.
	 We write $R' := \mathcal{O}_{K'}$ and $v'$ for the normalised valuation. That is $v'(r) = e_{K'/K}\cdot v(r)$ for all $r\in R$.

	 For $\lambda_{C'}$, Equation \ref{eqn:lambda} yields
	 \begin{align*}
	 8\,v'(\lambda_{C'})
	 = &e_{K'/K} \cdot \Bigg(g \cdot v(c_f) +  \sum_{\substack{|\cs| \textrm{ even}\\ \cs \neq \cR}}  \delta_{\cs} (|\cs|-2)|\cs| + \sum_{\substack{|\cs| \textrm { odd}\\ \cs \neq \cR}}  \delta_{\cs}(|\cs|-1)^2 \\
	 &~~~~~~~~~+ d_{\cR} \begin{cases} (|\cR|-2)|\cR|, & \text{if $|\cR| = 2g+2$}\\
	(|\cR|-1)^2, & \text{if $|\cR| = 2g+1$}
	 \end{cases}\Bigg).\\
	 \end{align*}
	 From Proposition \ref{prop:H0basechange} it follows that $v'(\lambda_{C'})=e_{K'/K} \cdot v(\lambda_C)$. So the above calculation shows that Formula \ref{eqn:lambda} is true for $C/K$ if and only if it is true for $C'/K'$.
\end{proof}

\begin{defn}
	Let $C/K$ be a hyperelliptic curve of genus $g$, defined by some Weierstraß equation $y^2 =f(x)$. We denote by $c_f$ the leading coefficient of $f$. Then the \textit{discriminant} $\Delta$ of the equation is defined as \[\Delta :=  2^{4g}\, c_f^{4g+2} \disc \left(\frac{1}{c_f} \,f(x)\right).\]
\end{defn}

While the discriminant defined above is not a curve invariant, but depends on the equation, there exists a more natural definition of discriminant. See also the paragraph before Proposition 2.2. in \cite{kausz1999discriminant}.

\begin{defn}
	\label{def:hyperellipticDiscriminant}
	Let $C/K$ be a hyperelliptic curve of genus $g$, defined by some Weierstraß equation $y^2 =f(x)$ with discriminant $\Delta$. We associate to this equation the differential forms $\omega_0, \dots, \omega_{g-1}$ and write $\omega = \omega_0 \land \dots \land \omega_{g-1} \in H^0(C,\Omega^1_{C/K})$.

	Then the element
	\[\Lambda:=\Delta^g \cdot \omega^{\otimes 8g+4} \in (\det H^0(C,\Omega^1_{C/K}))^{\otimes 8g+4}\]
	is called \textit{hyperelliptic discriminant} of $C$.
\end{defn}

The following proposition shows that $\Lambda$ is well-defined.

\begin{prop}
	\label{fct:disc}
	Let $C/K$ be a hyperelliptic curve with hyperelliptic discriminant $\Lambda$. Let $y^2 =f(x)$ be some Weierstraß equation for $C$. We associate to this equation the elements $\Delta$ and $\lambda$.
	Then the following statements are true.
	\begin{enumerate}
		 \item The element $\Lambda$ is independent of the choice of equation.
		 \item  Viewed as a rational section of $(\det H^0(\regX,\omega_{\regX/R}))^{\otimes 8g+4}$, the order of vanishing in $\mathfrak{p}$ is given by
		\[\ord_{\mathfrak{p}}(\Lambda) = g\cdot v(\Delta) - (8g+4) \cdot v(\lambda_C). \]
		\item Let $y'^2 = g(x')$ be another equation defining the same curve with $\Delta'$ and $\lambda_{C'}$ the corresponding quantities. Then
	\[\frac{v(\Delta') - v(\Delta)}{8g+4} = \frac{v(\lambda_{C'})-v(\lambda_C)}{g}.\]
	\end{enumerate}
\end{prop}

\begin{proof}
	\begin{enumerate}
		\item This is Proposition 2.2.1. in \cite{kausz1999discriminant}.
		\item This follows from the definition of $\Lambda$. See also \cite[Section 5, Formula 1]{kausz1999discriminant}.
		\item This is a direct consequence of the first two statements.
	\end{enumerate}
\end{proof}

\begin{lem}
	\label{lem:cf}
	Let $C/K$ be a hyperelliptic curve defined by $C:y^2 =f(x)$ with $f(x)=c_f\prod_{r\in \cR}(x-r)$ and $v(c_f)$ in $2\ZZ$. Then the curve $C'/K$ defined by  $y'^2 = \prod_{r\in \cR}(x-r)$ is isomorphic to $C$ over $K^{nr}$ and
	\[v(\lambda_C) = v(\lambda_{C'}) +g \cdot \frac{v(c_f)}{2}.\]

\end{lem}

\begin{proof}
	Since $v(c_f)$ is even, $c_f$ is a square in $K^{nr}$ and the two equations define isomorphic curves over $K^{nr}$.

	By definition, the discriminant of the equation $y^2=c_f\prod_{r\in \cR}(x-r)$ is \[\Delta = 2^{4g}\, c_f^{4g+2}\,\disc(\prod_{r\in \cR}(x-r)).\] So
	\[v(\Delta) = v(\Delta') + (4g+2)\, v(c_f).\]
	Using part 3 of Proposition \ref{fct:disc}, we get
	\[v(\lambda_C) = v(\lambda_{C'}) +g \cdot \frac{v(c_f)}{2}. \]
\end{proof}

\begin{lem}
	\label{lem:manipulatesigma}
	Let $C/K$ be a hyperelliptic curve and $y^2 = f(x)$ a Weierstraß equation defining this curve. Let $\Sigma$ be its cluster picture and $\lambda_C$ the quantity associated to this equation.

	Let $y'^2=g(x')$ be a different equation for $C$. Denote by $\Sigma'$ and $\lambda_{C'}$ the corresponding elements.
	\begin{enumerate}
		\item \label{lem:increasedepth} If $\Sigma'$ is obtained from $\Sigma$ by increasing the depths of all clusters by some $t\in \ZZ$, then
		\[v(\lambda_C) = v(\lambda_{C'}) - \frac{t}{8} \cdot \begin{cases} (|\cR|-2)|\cR|, & \text{if $|\cR| = 2g+2$}\\
		(|\cR|-1)^2, & \text{if $|\cR| = 2g+1$}
		\end{cases}\\ \]
		\item \label{lem:addroot}If $\Sigma'$ is obtained from $\Sigma$ by adding a root to $\cR$, then
		\[v(\lambda_C) = v(\lambda_{C'}) - \frac{d_{\cR} (|\cR|-1)}{4}.\]
		\item \label{lem:redistributedepth} If $\cR$ has even size and $\Sigma'$ is obtained from $\Sigma$ by redistributing the depth between $\cs <\cR$ and $\cR \backslash \cs$ to $d_{\cs}' = d_{\cs} - t$ and  $d_{\cR \backslash \cs}' = d_{\cR \backslash \cs} +t$, then
		\[v(\lambda_C) = v(\lambda_{C'}) - t \cdot \frac{(|\cR|-2)(|\cR|-2|\cs|)}{8}.\]
	\end{enumerate}
\end{lem}

\begin{proof}
	From \cite[Lemma 16.6.]{dokchitser2018arithmetic}, we know how the discriminant changes under the above modifications of the cluster picture. We will combine these results with Proposition \ref{fct:disc}, part 3.
	\begin{enumerate}
		\item By  \cite[Lemma 16.6.(i)]{dokchitser2018arithmetic}
		\[v(\Delta') - v(\Delta) = t |\cR|(|\cR|-1).\]
		Now Proposition \ref{fct:disc} implies
		\begin{align*}
		v(\lambda_{C'})-v(\lambda_C) = &\,g\cdot \frac{ t\, |\cR|(|\cR|-1)}{8g+4}\\
		=&\,\frac{t}{8} \cdot \begin{cases} (|\cR|-2)|\cR|, & \text{if $|\cR| = 2g+2$}\\
		(|\cR|-1)^2, & \text{if $|\cR| = 2g+1$}
		\end{cases}\\
		\end{align*}
		\item By \cite[Lemma 16.6.(ii)]{dokchitser2018arithmetic}
		\[v(\Delta') - v(\Delta) = 2\, d_{\cR}\, |\cR|.\]
		Now Proposition \ref{fct:disc} implies
		\begin{align*}
		v(\lambda_{C'})-v(\lambda_C) = &g\cdot \frac{2\,d_{\cR}\, |\cR|}{8g+4}\\
		=& \frac{|\cR|-1}{2} \cdot \frac{d_{\cR}}{2}.
		\end{align*}
		\item By \cite[Lemma 16.6.(iv)]{dokchitser2018arithmetic}
		\[v(\Delta') - v(\Delta) = t \cdot (|\cR|-2|\cs|)(|\cR|-1).\]
		Now Proposition \ref{fct:disc} implies
		\begin{align*}
		v(\lambda_{C'})-v(\lambda_C) = &g\cdot \frac{t (|\cR|-2|\cs|)(|\cR|-1)}{8g+4}\\
		=& t\cdot \frac{|\cR|-2}{2} \cdot \frac{|\cR|-2|\cs|}{4}.
		\end{align*}
	\end{enumerate}
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:main_lambda}]
	Let $C/K$ be a hyperelliptic curve with semistable reduction defined by $C: y^2=f(x)$ with $f(x) = c_f\prod_{r\in \cR}(x-r)$. In Lemma \ref{lem:tame} we have seen that it suffices to prove that the formula holds after a finite extension. So we may assume that $\cR \subset K$,  $v(r-s) \in 2\ZZ$ for all $r,s \in \cR$, and $v(c_f) \in 2 \ZZ$. Subtracting the correction term $4g \cdot v(c_f)$ from the right hand side, we may even assume that $v(c_f) = 0$. This follows from Lemma \ref{lem:cf}.

	Further we can perform a Möbius transformation such that the cluster picture corresponding to the new equation has outer depth $d_\cR=0$ (see Proposition \ref{prop:manipulateclusters}, part 1). By Lemma \ref{lem:manipulatesigma} this corresponds to subtracting $ d_{\cR}  (|\cR|-2) |\cR|$ if $|\cR| = 2g+2$ or $d_{\cR} (|\cR|-1)^2$ if $|\cR| = 2g+1$ from $8v(\lambda_C)$. Note that decreasing the absolute depths does not change any relative depths.

	In case that  $|\cR| =2g+1$, we can perform a Möbius transformation that corresponds to adding one root to the cluster picture (as in Proposition \ref{prop:manipulateclusters}, part 2). Since $d_\cR=0$, this does not change the valuation of $\lambda_C$.
	After these two steps we are left with proving the simplified formula that already appeared in Lemma \ref{lem:kausz}. That is
	\[
		8 \,v(\lambda_C) = \sum_{\substack{|\cs| \textrm{ even}\\ \cs \neq \cR}} \delta_{\cs} (|\cs|-2)|\cs| + \sum_{\substack{|\cs| \textrm { odd}\\ \cs \neq \cR}} \delta_{\cs} (|\cs|-1)^2.
	\]
	with Conditions (ii), (iii) and (iv) of the lemma being satisfied and $d_{\cR} =0$.

	Without loss of generality we may always assume $G_K$-stability for clusters because the formula for  $\lambda_C$ behaves well under tamely ramified extension (see Lemma \ref{lem:tame}). So we can apply Part 3 of Proposition \ref{prop:manipulateclusters}, that is redistribute depth between a ($G_K$-stable) cluster $\cs$ and $\cR\backslash\cs$. This allows us to manipulate the cluster picture such that $\cR$ has at least three children in the following way. Assume that $\cR$ has exactly two children, $\cs_1$ and $\cs_2$. Without loss of generality assume that $\cs_1$ is not a root, hence it has at least two children on its own, say $\ct_1$ and $\ct_2$. After redistributing depth between $\cs_1$ and $\cs_2$ in such a way that $d_{\cs_1} = 0$, $\cs_1$ is no longer a cluster but $\cR$ has at least three children given by $\ct_1, \ct_2,\cs_2$. Together with the fact that $d_{\cR}=0$, this implies Condition (v) of Lemma \ref{lem:kausz}.

	Let us show that the formula behaves well when redistributing depths. Let $\cs^* < \cR$ be a cluster in $\Sigma$. Let $\Sigma'$ be the cluster picture obtained after redistributing depth between $\cs^*$ and $\cR \backslash \cs^*$. That is $d_{\cs^*}' = d_{\cs^*} - t$ and  $d_{\cR \backslash \cs^*}' = d_{\cR \backslash \cs^*} +t$ for some $t\in \ZZ$. We have already seen in Lemma \ref{lem:manipulatesigma}, part 3 that
	\[ 8\,\left(v(\lambda_{C'}) - v(\lambda_C)\right) =t \cdot (|\cR|-2)(|\cR|-2|\cs|).\]
	The calculation below shows that this equals the change on the right hand side of the equation.
\begin{align*}
\sum_{\substack{|\cs| \textrm{ even}\\ \cs \neq \cR}}& \delta_{\cs}' (|\cs|-2)|\cs| + \sum_{\substack{|\cs| \textrm { odd}\\ \cs \neq \cR}} \delta_{\cs}'(|\cs|-1)^2\\
=& \sum_{\substack{|\cs| \textrm{ even}\\ \cs \neq \cs^*,\cR \backslash \cs^*}} \delta_{\cs} (|\cs|-2)|\cs| +  \sum_{\substack{|\cs| \textrm { odd} \\ \cs \neq \cs^*,\cR \backslash \cs^*}} \delta_{\cs}(|\cs|-1)^2 \\
&~~~~~+ \begin{cases} (\delta_{\cs^*}-t) (|\cs^*|-2) |\cs^*| + (\delta_{\cR\backslash\cs^*}+t) (|\cR\backslash\cs^*|-2) |\cR\backslash\cs^*|, & \text{ $|\cs^*|$ even}\\
(\delta_{\cs^*}-t)  (|\cs^*|-1)^2 + (\delta_{\cR\backslash\cs^*}+t) (|\cR\backslash\cs^*|-1)^2, & \text{ $|\cs^*|$ odd}
\end{cases}\\
=&\sum_{\substack{|\cs| \textrm{ even}\\ \cs \neq \cR}} \delta_{\cs} (|\cs|-2)|\cs| + \sum_{\substack{|\cs| \textrm { odd}\\ \cs \neq \cR}} \delta_{\cs}(|\cs|-1)^2\\ &~~~~~~ +
\begin{cases} -t  (|\cs^*|-2)|\cs^*|+ {t}(2g- |\cs^*|)(2g+2-|\cs^*|), & \text{$|\cs^*|$ even}\\
-t(|\cs^*|-1)^2 + {t}(2g + 1 - |\cs^*|)^2, & \text{$|\cs^*|$ odd}
\end{cases}\\
 =& \sum_{\substack{|\cs| \textrm{ even}\\ \cs \neq \cR}} \delta_{\cs} (|\cs|-2)|\cs| + \sum_{\substack{|\cs| \textrm { odd}\\ \cs \neq \cR}} \delta_{\cs}(|\cs|-1)^2 + t (|\cR|-2)(|\cR|-2|\cs|)
\end{align*}

Now the only condition missing in order to apply Lemma \ref{lem:kausz} is $\cR \subset R$. We already have $\cR \subset K$. Since $d_{\cR} =0$ and $v(c_f) = 0$, it follows from \cite[Theorem 13.3]{dokchitser2018arithmetic} that there exists $z \in K$ such that $f(x-z) \in R[x]$. Clearly, such a shift changes neither the cluster picture nor the valuation of $\lambda$. Hence we may assume $\cR \subset R$ without adding a further correction term.
\end{proof}

\section{A Basis for $H^0(\regX,\omega_{\regX/R})$}
\label{sec:base}

Let $C/K$ be a semistable hyperelliptic curve defined by a Weierstraß equation $C: y^2=f(x)$ with $f(x) = c_f\prod_{r\in \cR}(x-r)$. To this equation we associate the cluster picture $\Sigma$. Let $\regX \rightarrow \Spec R$ be the minimal regular model of $C$.
In this section we show how to read off a basis for the global sections of the canonical sheaf $\omega_{\regX/R}$ from the cluster picture $\Sigma$.
Our result is valid in a more general setting than \cite[Proposition 5.5.1.]{kausz1999discriminant} and at the same time simplifies the construction of the basis described therein.

\begin{thm}
	\label{thm:mainbasis}
Let $C/K$ be a semistable hyperelliptic curve defined by an integral Weierstraß equation $C:y^2=f(x)$ with cluster picture $\Sigma$.
Let $\regX/ R$ be the minimal regular model. Assume that the residue field $k$ is algebraically closed.

Choose clusters $\cs_0, \dots, \cs_{g-1}$  inductively such that
\[e_i = \frac{\nu_{\cs_i}}{2} - \sum_{j=0}^{i}d_{\cs_j \land \cs_i} = \max \left( \frac{\nu_{\cs}}{2} - \sum_{j=0}^{i-1}d_{\cs_j \land \cs} -d_{\cs}\right),\]
where the maximum is taken over all proper clusters in $\Sigma$.
If the maximal value is obtained by two different clusters $\cs$ and $\cs'$ with $\cs' \subset \cs$, choose  $\cs_i = \cs$.

Then an $R$-basis for the global sections of the relative dualising sheaf $\omega_{\regX/R}$ is given by $(\mu_0, \dots \mu_{g-1})$, where
\[\mu_i = \pi^{e_i} \prod_{j=0}^{i-1}(x-z_{\cs_{j}}) \frac{dx}{2y}.\]
\end{thm}

Note that the same cluster can appear multiple times in the sequence $\cs_0, \dots, \cs_{g-1}$. Moreover the sequence is not canonical, since it can happen that the maximum is obtained by two incomparable clusters, in which case any of the two clusters may be chosen.
Since $k$ is algebraically closed, one can find a centre $z_{\cs} \in K$ for every proper cluster $\cs$. This follows from \cite[Lemma 4.2.]{dokchitser2018arithmetic}. For an illustration of the theorem, we refer to Example \ref{ex:basis} in the introduction.

The strategy for the proof is a follows. First, we show that the sum over the $e_i$ is exactly the valuation of $\lambda_C$ as defined in the previous section. For this we need Lemma \ref{lem:gamma}. Then, we show that the differential forms are indeed global sections by computing their order of vanishing along different components of the special fibre of the minimal regular models, see Lemma \ref{lem:e_i}. The theorem then follows as a corollary of Theorem \ref{thm:main_lambda}.

\begin{lem}
	\label{lem:gamma}
	Let $C/K$ be a hyperelliptic curve defined by an integral Weierstraß equation $C:y^2=f(x)$ and $\Sigma$ the associated cluster picture. Let the clusters $\cs_0, \dots, \cs_{g-1} \in \Sigma$ be chosen according to Theorem \ref{thm:mainbasis}.
	Then for every cluster $\cs \in \Sigma$:
	\[\gamma(\cs) := \# \{\cs_i \,|\, \cs_i \subset \cs\} = \left \lfloor{\frac{|\cs|-1}{2}}\right \rfloor\]
\end{lem}

\begin{proof}
 Since $\gamma(\cR) = g$, the statement is true for $\cs = \cR$. Let $\cs \neq \cR$ and let $\cs'$ be the parent of $\cs$, that is $\cs < \cs'$. Then
	\begin{align*}
		d_{\cs} =& d_{\cs'} + \delta_{\cs},\\
		\nu_{\cs} =& \nu_{\cs'}+\delta_{\cs}|\cs|,\\
		d_{\cs_j\land\cs} =& \begin{cases}
			d_{\cs_j \land \cs'} + \delta_{\cs} &\textrm{ if } \cs_j \subset \cs\\
			d_{\cs_j \land \cs'}  &\textrm{otherwise}
		\end{cases} ~~\textrm{ for }j \in \{0, \dots, i-1\}.
	\end{align*}
	So for any $i \in \{0, \dots ,g-1\}$
	\[\frac{\nu_{\cs}}{2} - \sum_{j=0}^{i-1}d_{\cs_j \land \cs} -d_{\cs} =  \frac{\nu_{\cs'}}{2} - \sum_{j=0}^{i-1}d_{\cs_j \land \cs'} -d_{\cs'} + \delta_{\cs} \left(\frac{|\cs|}{2} - 1 -\# \{\cs_j \,|\, \cs_j \subset \cs, \;0 \leq j \leq i-1\}\right).\]

	Assume that $\cs_i = \cs$, then by construction of the sequence $(\cs_j)$, it must hold that $\frac{\nu_{\cs}}{2} - \sum_{j=0}^{i-1}d_{\cs_j \land \cs} -d_{\cs} >  \frac{\nu_{\cs'}}{2} - \sum_{j=0}^{i-1}d_{\cs_j \land \cs'} -d_{\cs'}$. The equation above then implies $ \# \{\cs_j \,|\, \cs_j \subset \cs, \;0 \leq j \leq i-1\} < \frac{|\cs|}{2} - 1$. Hence $\{\cs_j \,|\, \cs_j \subset \cs, \;0 \leq j \leq i\} \leq \frac{|\cs| -1}{2}$. Since this is true for every $i$, we may conclude $\gamma(\cs) \leq \lfloor \frac{|\cs|-1}{2} \rfloor$.

	To show equality, we proceed by induction. Again let $\cs \neq \cR$ and assume that the statement holds for every cluster strictly containing $\cs$. If $\gamma(\cs) < \lfloor \frac{|\cs|-1}{2} \rfloor$, then at any step $i$, we have $\frac{\nu_{\cs}}{2} - \sum_{j=0}^{i-1}d_{\cs_j \land \cs} -d_{\cs} >  \frac{\nu_{\cs'}}{2} - \sum_{j=0}^{i-1}d_{\cs_j \land \cs'} -d_{\cs'}$, hence $\cs_i \neq \cs'$. This implies $\gamma(\cs') = \sum_{\ct < \cs'}\gamma(\ct)$. Using that $|\cs'| = \sum_{\ct < \cs'}|\ct|$, $\gamma(\ct)\leq  \lfloor \frac{|\ct|-1}{2} \rfloor$ for every $\ct < \cs'$ and $\gamma(\cs) <\lfloor \frac{|\cs|-1}{2} \rfloor$, we get
	\[
	\gamma(\cs') = \sum_{\ct < \cs'} \gamma(\ct)
	< \left\lfloor \frac{|\cs|-1}{2} \right\rfloor + \sum_{\cs \neq \ct < \cs'} \left\lfloor \frac{|\ct|-1}{2} \right\rfloor
	\leq  \left\lfloor \frac{\sum_{\ct < \cs'} |\ct| - 1}{2} \right\rfloor
	= \left\lfloor \frac{|\cs'|-1}{2} \right\rfloor.
	\]
	 This yields a contradiction. So $\gamma(\cs) = \lfloor \frac{|\cs|-1}{2} \rfloor$.

\end{proof}

In order to prove Theorem \ref{thm:mainbasis}, we have to show that the differential forms defined in the theorem are global sections of $\omega_{\regX/R}$. In particular, we need to be able to compute the order of vanishing along components of the special fibre of the minimal regular model $\regX$. For that purpose we use the description of $\regX$ given in [Theorem 8.5]\cite{dokchitser2018arithmetic}. Broadly speaking, each principal cluster gives rise to one or possibly two components in the special fibre. These components are connected by chains of projective lines. In the following, we will write $\Gamma_{\cs} \subset \regX_s$ for the component(s) corresponding to the principal cluster $\cs$.
On the other hand, a disc $D(\cs) \subset \bar{K}$ is associated to a principal cluster $\cs$. Equations for $\Gamma_{\cs}$ are now given in function of the disc $D(\cs)$ in \cite[Definition 5.4]{dokchitser2018arithmetic}. From this, we can conclude that there is an open neighbourhood $U_{\cs}$ of the generic point of $\Gamma_{\cs}$ which is contained in
\[
\Spec\left(R[x_{\cs},y_{\cs}]/ (y_{\cs}^2 - f_{\cs}(x_{\cs})) \right)
\]
with local coordinates
\[
x_{\cs} = \frac{x-z_{\cs}}{\pi^{d_\cs}}, \;
y_{\cs} = \frac{y}{\pi^{\nu_{\cs}/2}}.
\]

\begin{lem}
	\label{lem:e_i}
	Let $C/K$ be a hyperelliptic curve defined by an integral Weierstraß equation $C:y^2=f(x)$ and $\Sigma$ the associated cluster picture. Let the clusters $\cs_0, \dots, \cs_{g-1} \in \Sigma$ be chosen according to Theorem \ref{thm:mainbasis}. Let $\regX/R$ be the minimal regular model.

	Then for any principal cluster $\cs \in \Sigma$, we have that  $-\left(\frac{\nu_{\cs}}{2} - \sum_{j=0}^{i-1}d_{\cs_j \land \cs} -d_{\cs}\right)$ is a lower bound for the order of vanishing of the element $\prod_{j=0}^{i-1}(x-z_{\cs_{j}}) \frac{dx}{2y}$ along the component of the special fibre of $\regX$ that corresponds to $\cs$.
\end{lem}

\begin{proof}
	Let $\cs$ be a principal cluster in $\Sigma$ and denote by $\Gamma_{\cs}$ the component (or possibly the two components) of the special fibre of $\regX/R$ corresponding to this cluster.

	It suffices to compute the order of vanishing of $\prod_{j=0}^{i-1}(x-z_{\cs_{j}}) \frac{dx}{2y}$ in an open neighbourhood $U$ of the generic point of the component $\Gamma_{\cs}$. We choose $U = U_{\cs}$ as described in the paragraph preceding this lemma.

	We can write
	\begin{align*}
	\prod_{j=0}^{i-1}(x-z_{\cs_{j}}) \frac{dx}{2y} &= \prod_{j=0}^{i-1}(\pi^{d_{\cs}} x_{\cs}+z_{\cs}-z_{\cs_{j}}) \cdot \frac{d(\pi^{d_{\cs}}x_{\cs} +z_{\cs})}{2\pi^{\nu_{\cs}/2}y_{\cs}}\\
	&=\prod_{j=0}^{i-1}\pi^{d_{\cs \land \cs_j}} \cdot (\pi^{d_{\cs}-d_{\cs \land \cs_j}} x_{\cs} - \frac{z_{\cs}-z_{\cs_{j}}}{\pi^{d_{\cs \land \cs_j}}}) \cdot \frac{\pi^{d_{\cs}}}{\pi^{\nu_{\cs}/2}} \frac{dx_{\cs}}{2y_{\cs}}\\
	&=\pi^ {\sum_{j=0}^{i-1}d_{\cs \land \cs_j} + d_{\cs} - \frac{\nu_{\cs}}{2}} \cdot \prod_{j=0}^{i-1} (\pi^{d_{\cs}-d_{\cs \land \cs_j}} x_{\cs} - \frac{z_{\cs}-z_{\cs_{j}}}{\pi^{d_{\cs \land \cs_j}}}) \cdot \frac{dx_{\cs}}{2y_{\cs}}.
	\end{align*}
	Since  $d_{\cs} \geq d_{\cs \land \cs_j}$ and $\frac{z_{\cs}-z_{\cs_{j}}}{\pi^{d_{\cs \land \cs_j}}} \in R$ for all clusters $\cs, \cs_j \in \Sigma$, the element $\prod_{j=0}^{i-1} (\pi^{d_{\cs}-d_{\cs \land \cs_j}} x_{\cs} - \frac{z_{\cs}-z_{\cs_{j}}}{\pi^{d_{\cs \land \cs_j}}}) \frac{dx_{\cs}}{2y_{\cs}}$ is an integral section on $\omega_{\regX/R|_{U}}$. The statement of the lemma follows.

\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:mainbasis}]
	Let $\cs_0, \dots, \cs_{g-1}$ be a sequence constructed as described in the theorem and denote by $\mu_0, \dots, \mu_{g-1}$ the differential forms associated to this sequence.

	\textbf{Claim 1:} The differentials $\mu_0, \dots \mu_{g-1}$ are  global sections of $\omega_{\regX/R}$.\\
	Here, we are going to make use of the fact that inside $H^0(C, \Omega_{C/K})$ the global sections of the minimal regular model are equal to those of the stable model (cf. the proof of Proposition \ref{prop:H0basechange}). So by \cite[Theorem 5.24.]{dokchitser2018arithmetic}, it suffices to check that the differentials $\mu_0, \dots ,\mu_{g-1}$ are regular on the components of the special fibre corresponding to principal clusters.
	Let $\Gamma_{\cs}$ be a component of the special fibre corresponding to a principal cluster $\cs \in \Sigma$. From Lemma \ref{lem:e_i}, we know that  $-\left(\frac{\nu_{\cs}}{2} - \sum_{j=0}^{i-1}d_{\cs_j \land \cs} -d_{\cs}\right)$ is a lower bound for the order of the element $\prod_{j=0}^{i-1}(x-z_{\cs_{j}}) \frac{dx}{2y}$ on $\Gamma_{\cs}$. Since $e_i = \max_{\cs \in \Sigma} \left( \frac{\nu_{\cs}}{2} - \sum_{j=0}^{i-1}d_{\cs_j \land \cs} -d_{\cs}\right)$, the order of $\mu_i = \pi^{e_i} \cdot \prod_{j=0}^{i-1}(x-z_{\cs_{j}}) \frac{dx}{2y}$ is non-negative on every such component.

	For the horizontal part, we have to consider the restriction of $\mu_i$ to the generic fibre. Clearly $\mu_i \otimes 1 \in H^0(\regX,\omega_{\regX/R}) \otimes_R K = H^0(C,\Omega_{C/K})$. This proves the first claim.

	\textbf{Claim 2:} Let $\lambda_C$ be the quantity defined in the previous section, then
	$\sum_{i=0}^{g-1} e_i = v(\lambda_C)$.
	By definition we have
	\[2 \sum_{i=0}^{g-1} e_i~
	= \sum_{i=0}^{g-1}\left(\nu_{\cs_{i}}- 2 \sum_{j=0}^{i} d_{\cs_j \land \cs_i}\right).
	\]
	We divide this expression into two parts. For the first part, we use the formula from Remark \ref{rem:compute_vs} and get
	\begin{align*}
	 \sum_{i=0}^{g-1}\nu_{\cs_{i}}
	=&\; \sum_{i=0}^{g-1} \left(v(c_f) +
	d_{\cR} |\cR| +  \sum_{\cs \supset \cs_i} \delta_{\cs} |\cs|\right)
	=g  (v(c_f) + d_{\cR} |\cR|) + \sum_{\cs \neq \cR} \delta_{\cs} |\cs| \gamma(\cs).
	\end{align*}
	The second part gives
	\begin{align*}
	2 \sum_{i=0}^{g-1}\sum_{j=0}^{i} d_{\cs_j \land \cs_i}
	= 2\sum_{i\geq j} \left(\sum_{\cs \supset \cs_j\land \cs_i} \delta_{\cs} +d_{\cR} \right)
	=  \sum_{\cs \neq \cR} \delta_{\cs} \gamma(\cs)(\gamma(\cs)+1) + d_{\cR} \gamma(\cR)(\gamma(\cR) + 1).
	\end{align*}
	For the first equality, we used the fact that the depth of a proper cluster $\ct$ may be written as $d_{\ct} = \sum_{\cs \supset \ct} \delta_{\cs} + d_{\cR}$.  For the second equality, we used that  $ \#\{\cs_j\land \cs_i \subseteq \cs~|~ j\leq i\} = \binom{\gamma(\cs)+1}{2}$ for every proper cluster $\cs$. By Lemma \ref{lem:gamma}, we have $\gamma(\cs) = \lfloor \frac{|\cs|-1}{2}\rfloor$.
	Combining the two parts, we obtain
	\begin{align*}
	8  \sum_{i=0}^{g-1} e_i~
	=&~~ 4g \cdot v(c_f) + \sum_{\substack{|\cs| \textrm{ even}\\ \cs \neq \cR}} \delta_{\cs} \cdot (|\cs|-2)|\cs| + \sum_{\substack{|\cs| \textrm { odd},\\ \cs \neq \cR}} \delta_{\cs}(|\cs|-1)^2 \\
	&~~~~~~ + d_{\cR} \begin{cases} (|\cR|-2)|\cR|, & \text{if $|\cR| = 2g+2$},\\
	(|\cR|-1)^2, & \text{if $|\cR| = 2g+1$}.
	\end{cases}
	\end{align*}
	The claim now follows from Theorem \ref{thm:main_lambda}.

	\textbf{Claim 3:}
	$\mu: = \mu_0 \land \dots \land \mu_{g-1}$ is a basis for $\det H^0(\regX,\omega_{\regX/R})$.\\
	Let $\omega_0, \dots ,\omega_{g-1}$ denote the differentials associated to the Weierstraß equation $C: y^2 = f(x)$. By construction, we have that
	\[(\mu_0,\dots,\mu_{g-1}) = (\omega_0, \dots ,\omega_{g-1}) \cdot A, \]
	where $A$ is a matrix of the form
	\(  A = \begin{pmatrix}
	\pi^{e_0}	&  & \ast    \\
	& \ddots  & \\
	0 	& 	 & \pi^{e_{g-1}}
	\end{pmatrix}.\)\\
	Therefore \[\mu = \det(A) \cdot  \omega_0 \land \dots \land \omega_{g-1} =  \pi^{v(\lambda_C)} \cdot \omega_0 \land \dots \land \omega_{g-1}.\] So by Theorem \ref{thm:main_lambda}, $\mu$ is a basis for $\det H^0(\regX,\omega_{\regX/R})$.\\

	We have seen that $\mu_0,\dots,\mu_{g-1} \in H^0(\regX,\omega_{\regX/R})$ and that $\mu: = \mu_0 \land \dots \land \mu_{g-1}$ is a basis for $\det H^0(\regX,\omega_{\regX/R})$. Therefore $(\mu_0,\dots,\mu_{g-1})$ is a basis for $H^0(\regX,\omega_{\regX/R})$.

\end{proof}


text:
Home Israel Government Facing Criminal Hearings, Netanyahu Appoints Himself Justice Minister
Facing Criminal Hearings, Netanyahu Appoints Himself Justice Minister
2 Sivan 5779 – June 4, 2019
Photo Credit: Hadas Parush/Flash90
Attorney-General Avichai Mandelblit and outgoing Justice Minister Ayelet Shaked at her farewell ceremony, June 4, 2019.
Prime Minister Benjamin Netanyahu on Tuesday informed Justice Minister Ayelet Shaked and Education Minister Naftali Bennett that he is dismissing them from their positions in the government and their presence is also no longer welcome in meetings of the political-security cabinet.
The decision to remove the two from their posts, which was approved by Attorney General Avichai Mandelblit, comes two days before the first meeting of the political-security cabinet in more than a month.
As the dismissal took effect, Prime Minister Netanyahu became the acting justice and education minister. He is also suspected of bribery, fraud and breach of trust in three pending criminal indictments, and the AG has announced his intention to prosecute him, subject to an October 2 hearing.
Netanyahu's office said at the beginning of the week that the PM planned to appoint replacements for the two positions within 48 hours. But on Monday the Likud Party announced that the assignments of the portfolios had been postponed to next week, while Netanyahu is discussing potential replacements with other coalition partners.
In the meantime, Netanyahu will hold both portfolios, as well as the defense and health portfolios.
On Tuesday, Shaked said farewell to the justice ministry, promising Douglas MacArthur-style she would return. She expressed her hope that her replacement "would continue the same line I instituted in the ministry."
According to Shaked, the era of justice ministers whose main occupation was to show up at ribbon cutting ceremonies and leave the real work to the career employees – is over.
"The new era that began in 2015 is characterized by justice ministers who lead the ministry," Shaked said. "They set goals, and when necessary, they also set boundaries."
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Printed from: https://www.jewishpress.com/news/politics/facing-criminal-hearings-netanyahu-appoints-himself-justice-minister/2019/06/04/

redpajama_set_name: RedPajamaCommonCrawl


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Welcome back to the third, and final, installment on stop-motion animation. When planning a stop motion animation project with students teachers need to be aware of the tools necessary, the steps for creating a project like this, some tips for success, and additional resources to ensure students learn, have fun, and create quality products.
Determine Frames Per Second (# of images needed to make a second = # of images needed to make the full length feature) Frames Per Second for video are normally 30 but programs can let you adjust that) – let your students do the calculations when planning their projects.
Plan carefully – Make a storyboard before you animate.
Small movements – A little goes a long way; use the onion-skinning option to see where the last frame was placed.
Assign roles – Modellers, clickers, spotters, movers; each student can be responsible for a task.
Leave a comment…..We'd love your thoughts and ideas about stop motion animation as authentic products in the classroom. Share links to your student's stop-motion projects. Tell us about all of your experiences using stop motion animation including implementation ideas, tips, additional resources, etc.
Brooke Higgins is an instructional specialists for the eMINTS National Center and Allison Byford is the Instructional Technology Coordinator with the Springdale Public Schools in Arkansas and is an eMINTS PD4ETS graduate.
Recently, Allison Byford shared the basic tools and steps for helping students to create stop motion animation movies in her post Stop Motion Animation Made Easy – Part 1 . Today we'll take that a step further and share more examples, tools needed, step-by-step how to instructions, and some tips for a successful implementation.
Many of you have seen the OK GO video for their song "Here it Goes", but have you seen the LEGO "Here it Goes". It's an example of how a student with a creative mind and a camera recreated the video. If you are an eMINTS teacher you have surely seen some of the great Common Craft videos covering topics such as "GoogleDocs in Plain English" or "Electing a US President in Plain English". While common craft videos aren't purely stop motion they include some content recorded in stop motion animation as well as video.
When thinking about how stop motion animation could be connected to curriculum teachers can use stop motion video to present new concepts, record what happens during an extended Science experiment, or to present learned content.
Stop motion animation can be used in creative writing lessons to tell story with or without words or retell stories.
Students can present their understandings of math concepts by creating videos that may be used to teach newly learned skills to others.
Students can represent historical concepts, create documentaries, or reenact historical events to show their understanding of the impact of history.
Not only can curriculum content be covered but students can also focus in on using 21st Century Skills when creating stop motion animation projects. These kinds of projects can require students to Think and Work Creatively with Others, Communicate Clearly, Collaborate with Others, Adapt to Change, Be Flexible, Manage Goals, Time, and Projects, Produce Results, Create Media Products, and Apply Technology Effectively.
Tune in tomorrow to learn the steps for creating successful projects like these.
Have you ever watched a movie like Wallace and Grommit or The Nightmare Before Christmas or even films like Toy Story or Up and wished that you were good enough to do that? Have you ever marveled in amazement in what it takes to animate and the frames and frames of footage that have to be created and put in place to produce even one minute of video. The intricacies of this kind of project have to be mind-boggling, right?
Well, I've got good news! Animation and creating animated projects is easier that one might think. In fact, not only is it easy to do for one person….but better yet, it's even easier if you use a team approach with interdependent roles! In fact, it is proving to be, in our district, one of the most engaging ways for students to create a culminating project. They can create movies fast and they love it so much they can't STOP!!
Stop-motion animation is essentially collecting a series of photographs and rendering them together at a rate of anywhere from 8 frames a second to 30 frames a second. Thanks to some amazing web 2.0 tools, kids (even our youngest) can do this with ease.
At the most basic level, for stop-motion animation projects to be created, you need a storyboard/script, camera, some props, and software that "sticks" all the photographs together. If you are using a PC, a good option is Jellycam. Jellycam has a quick and simple tutorial that will walk a learner through the specifics of the software in just a couple of minutes. It allows the user to easily capture and manipulate the pictures taken with a web cam within the program. Users determine how many images per second and can play back the video as it is being created. It also allows for adding credits and music within the software. JellyCam is a completely free download.
Creating a stop-motion animation project can also be easily created on an iPad2. One favorite App for doing this is StoMo. In this App, you capture images directly into your iPad using the capture button. You can set the rate of pictures being seen per second, arrange and re-arrange the images captured, playback projects as you work on them and use either the front or rear camera. When you export the project to the library, the images are rendered together and the "film" is put in the iPhotos library to be viewed. StoMo does not allow users to add music, voice over, or text to the film but once a project is finished it can be exported to iMovie, Moviemaker, or some other movie editing software to add music, voice over, or text.
Both of these applications have a key feature that is critical for stop motion creators – onion skinning. If you think about the skin of an onion, it is somewhat transparent. In stop-motion animation it is critical to be able to see the last image taken and be able to compare it to the one about to be taken. Onion skinning allows for the creator to move objects in the films as much or little as desired. Another classroom benefit of onion skinning allows for students, who's "filming" schedule is often interrupted by the bell, to come back the next day and pick up where left off. They can see the last image captured and begin from there on a new day.
As a teacher, the best part of stop-motion animation projects is that the students REALLY need each other to complete a quality project quickly! In a group of four students each has a vital role. One person serves as the project manager, or director of the project. That person focuses on the vision and directs the project as it progresses. One person needs to focus completely on the software. That person knows exactly how the software works, captures images, and keeps the program running. The other two members of a group are moving manipulatives (characters and props) to create the animated sequences. It takes everyone in the group doing their parts to be successful.
Check out some examples of student stop-motion animation projects below and start planning a project for your students. Stay tuned for the next Stop Motion Animation installment sharing more classroom examples, how-to's, tips, and resources for classroom stop motion projects.
Social Studies – Confederacy or Union?
How might your students express what they have learned with a stop motion animation project in your classroom?
Allison Byford is the Instructional Technology Coordinator with the Springdale Public Schools in Arkansas and is an eMINTS PD4ETS graduate.


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"The Suicide Collectors" by David Oppegaard (Reviewed by Robert Thompson)
Official David Oppegaard Website
Official The Suicide Collectors Myspace
Order "The Suicide Collectors" HERE
Read An Interview via The Soulless Machine Review
Listen To An Excerpt HERE (MP3)
AUTHOR INFORMATION: David Oppegaard holds a B.A. in English Literature from St. Olaf College and an M.F.A. in Writing from Hamline University, and was a finalist for the Indiana Review Fiction Award and the Iowa Fiction Award. "The Suicide Collectors" is his first novel. David's second book, "Wormwood, Nevada", was recently picked up by St. Martin's Press.
PLOT SUMMARY: The Despair has plagued the earth for five years. Most of the world's population has inexplicably died by its own hand, and the few survivors struggle to remain alive. A mysterious, shadowy group called the Collectors has emerged, inevitably appearing to remove the bodies of the dead. But in the crumbling state of Florida, a man named Norman takes an unprecedented stand against the Collectors, propelling him on a journey across North America. It's rumored a scientist in Seattle is working on a cure for the Despair, but in a world ruled by death, it won't be easy to get there…
CLASSIFICATION: "The Suicide Collectors" blends horror with drama and traces of the fantastic (futuristic technology, supernatural elements), and adds a unique twist to the post-apocalyptic blueprint.
FORMAT/INFO: Page count is 320 pages divided over five Parts, twenty-nine numbered chapters and an Epilogue. Narration is in the third-person, almost exclusively via Norman, except for the Epilogue. "The Suicide Collectors" is self-contained, but does leave room for possible sequels. December 9, 2008 marks the North American Hardcover publication of "The Suicide Collectors" via St. Martin's Press. Cover designed by Michael Collica.
ANALYSIS: Post-apocalyptic tales—whether you're talking about novels, comic books, television, movies or videogames—are, for the most part, a dime a dozen. So when I heard about David Oppegaard's "The Suicide Collectors", I was immediately intrigued by the book's original concept: a near future Earth where ninety percent of mankind has been wiped out by suicide. Of even more interest were the Collectors, an enigmatic group of dark-robed, seemingly inhuman people who collect the bodies of the deceased for some unknown purpose…
Of course, what's a concept without substance? Fortunately, David's debut novel is much more than a neat idea. The prose, while not flashy, is consistent and accessible. The characters—which included Norman, his neighbor and friend Pops, an eleven-year-old girl named Zero, the Mayor, etc—are endearing backed by strong dialogue and banter. And the novel is excellently fast-paced, propelled by a number of riveting questions such as: What horrors will Norman and Pops encounter on their trek across America? What caused the Despair? Is there really a cure in Seattle? Who or what are the Collectors? What do they do with the bodies they collect? What is the Source? And can the Despair be stopped?
On a personal note, I was particularly moved by some of the book's more shocking and profound moments including two little girls who viciously murder an old woman, a house covered in sheets of paper which is basically one massive suicide note, a cult that worships the Source, this one very horrific scene in a church, and the desperate acts (post-Despair) that people are forced to commit in the name of compassion. There were also a couple of intriguing ideas discussed about what caused the Despair—one revolved around the dead of the world during the past million years unifying in its desire for revenge against the living:
"They'd suck all the life out of the world, greedy for its warmth, and they'd laugh as remaining husks killed themselves one by one, until the entire world was dead, together again at last."
Negatively, there is an unlikely coincidence in the novel and much of the story's ambiguous answers might leave the reader unfulfilled, but these are minor issues compared to the book which, as a whole, is striking and memorable.
CONCLUSION: "The Suicide Collectors" reminded me somewhat of The Stand, Robert R. McCammon's "Swan Song", The Road, and various other post-apocalyptic tales in the way that it observes such familiar themes as life, death, the resiliency of the human spirit, religion, etc., but possesses enough distinguishing features of its own to ensure a worthwhile reading experience. In short, "The Suicide Collectors" is an impressive first novel by an author in David Oppegaard who I expect to see a lot more of in the future…
Great review. I have this book on my wish list to get.
Thanks Cheryl! I was really impressed with the book and I hope you enjoy it :D
Great review. This sound like it's right up my alley.
Have a great Christmas!!
Thanks! Happy Holidays to you as well :D
I am about halfway through this book now. Your right about it being similar to The Stand. This is a whole lot darker of a story than what it was though. Not sure what I think of it yet. Will have to see how it ends. Great review you wrote up. Hope you don't mind if when I get mine written up I add a link to your review.
Thanks for the comment Kathy! I'd be more than happy if you linked my review with yours :) Can't wait to see your final thoughts on the book!


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Monday Musings….More Books to Look Forward To!
Happily Ever After …Or Happily Nevermore? Gisela's childhood was filled with laughter and visits from nobles such as the duke and his young son. But since her father's death, each day has been filled with nothing but servitude to her stepmother. So when Gisela learns the duke's son, Valten—the boy she has daydreamed about for years—is throwing a ball in hopes of finding a wife, she vows to find a way to attend, even if it's only for a taste of a life she'll never have. To her surprise, she catches Valten's eye. Though he is rough around the edges, Gisela finds Valten has completely captured her heart. But other forces are bent on keeping the two from falling further in love, putting Gisela in more danger than she ever imagined.
The year is 1838, and seventeen-year-old Julia Elliston's position has never been more fragile. Orphaned and unmarried in a time when women are legal property of their fathers, husbands, and guardians, she finds herself at the mercy of an anonymous guardian who plans to establish her as a servant in far-off Scotland.
With two months to devise a better plan, Julia's first choice to marry her childhood sweetheart is denied. But when a titled dowager offers to introduce Julia into society, a realm of possibilities opens. However, treachery and deception are as much a part of Victorian society as titles and decorum, and Julia quickly discovers her present is deeply entangled with her mother's mysterious past. Before she knows what's happening, Julia finds herself a pawn in a deadly game between two of the country's most powerful men. With no laws to protect her, she must unravel the secrets on her own. But sometimes truth is elusive and knowledge is deadly.
After years of watching his children and grandchildren wander from their faith, Iddo's prayers are answered: King Cyrus is allowing God's chosen people to return to Jerusalem. Jubilant, he joyfully prepares for their departure, only to learn that his family, grown comfortable in the pagan culture of Babylon, wants to remain.
Zechariah, Iddo's oldest grandson, feels torn between his grandfather's ancient beliefs and the comfort and success his father enjoys in Babylon. But he soon begins to hear the voice of God, encouraging him to return to the land given to his forefathers.
Bringing to life the biblical books of Ezra and Nehemiah, Return to Me tells the compelling story of Iddo and Zechariah, the women who love them, and the faithful followers who struggle to rebuild their lives in obedience to the God who beckons them home.
Patrick Bowers has pursued the nation's fiercest serial killers—and now one elusive foe is back for revenge.
Settling into a new post at the FBI academy, Patrick and his fiancée, Lien-hua Jiang, are planning their future together with his stepdaughter, Tessa.
But just when his life seems normal, a demon from the past returns to draw him down a dark road he hoped had closed forever. Forced into a desperate hunt to save the two women he loves most, Patrick is in a race against time to stop an international conspiracy from becoming the most widespread act of terrorism in U.S. history.
When his friend is murdered, illusionist Jevin Banks is determined to find out what really happened. Drawn into a web of conspiracy and top-secret research on human consciousness, Jevin won't stop digging until the truth is revealed. Soon he uncovers a dark secret–one that could change the very fabric of human life on the planet.
Bristling with mystery, suspense, and intrigue, Singularity is the second riveting book in The Jevin Banks Experience. Readers will devour this scientific thriller, flipping pages late into the night until the final shocking page.
Don't you just love the cover of Melanie Dickerson's The Captive Maiden? It just screams fairytale. Anyone looking forward to any of these books in particular?
Previous Post Morgan Busse's Son of Truth
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2 thoughts on "Monday Musings….More Books to Look Forward To!"
Kandra says:
definitely the Steven James ones! I have been eagerly awaiting their release.
I know! And two in one year just makes my heart happy! lol


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ARMY BATTLE CASUALTIES WELFARE FUND (ABCWF)
Related Topics: Defence Sector, Welfare Schemes
Raksha Mantri Rajnath Singh has given in principle approval to enhancement of monetary assistance to Next of Kin (NoK) of all categories of Battle Casualty (BC) from Rs 2 lakh to Rs 8 lakh.
The amount will be granted under Army Battle Casualties Welfare Fund (ABCWF).
Earlier Financial Assistance
Earlier, there was a provision of financial assistance of Rs 2 lakh to the Battle casualties which are fatal, 60 per cent & above disability, battle casualties invalided out due to disability and Rs 1 lakh to disability less than 60 per cent.
It was in addition to liberalised family pension, financial assistance from Army Group Insurance, Army Welfare Fund and ex-gratia amount.
About Army Battle Casualties Welfare Fund (ABCWF)
It has been set up under Department of Ex-Servicemen Welfare (ESW), Ministry of Defence after a large number of people offered to provide monetary assistance to the families of battle casualties following an incident in February 2016 at Siachen, wherein 10 soldiers were buried in an avalanche.
It was instituted in July 2017 and was implemented retrospectively with effect from April 2016.
The fund was created under the Charitable Endowments Act, 1890.
This fund is in addition to the various existing schemes for the welfare of NoK and children of battle casualties in form of additional ex gratia.
[Source: PIB]
OCTOBER 7: WORLD COTTON DAY
Related Topics: Agriculture, Cotton Technical Assistance Programme
The Ministry of Textiles, Government of India, is participating in the World Cotton Day being observed from 7th October to 11th October 2019 in Geneva.
World Cotton Day
It is organized by World Trade Organisation (WTO) in collaboration with the Secretariats of the United Nations Food and Agriculture Organization (FAO), the United Nations Conference on Trade and Development (UNCTAD), the International Trade Centre (ITC) and the International Cotton Advisory Committee (ICAC).
WTO is hosting the event at the request of the Cotton – 4 countries, Benin, Burkina Faso, Chad and Mali.
The day will celebrate the many advantages of cotton, from its qualities as a natural fibre, to the benefits people obtain from its production, transformation, trade and consumption.
It will also serve to shed light on the challenges faced by cotton economies around the world because cotton is important to least developed, developing and developed economies worldwide.
To give exposure and recognition to cotton and all its stakeholders in production, transformation and trade.
To engage donors and beneficiaries and strengthen development assistance for cotton
Seek new collaborations with the private sector and investors for the cotton related industries and production in developing countries
Promote technological advances, as well as further research and development on cotton.
Cotton is a global commodity that is produced all over the world and a single tonne of cotton provides year-round employment for five people on average.
It is a drought – resistant crop ideal for arid climates.
It occupies just 2.1 % of the world's arable land, yet it meets 27% of the world's textiles need.
In addition to use in textiles and apparel, food products are also derived from cotton like edible oil and animal feed from the seed.
Cotton Technical Assistance Programme (Cotton TAP-I)
Between 2011 and 2018, India implemented Cotton Technical Assistance Programme (Cotton TAP-I) of about USD 2.85 million for seven African countries namely Benin, Burkina Faso, Mali, Chad, Uganda, Malawi and Nigeria.
The technical assistance focused on improving the competitiveness of the cotton and cotton-based textiles and apparel industry in these countries through a series of interventions which had significant outcomes leading to a demand for a follow on project.
ACCEPTANCE DEVELOPMENT FUND (ADF)
Related Topics: Indian Economy, Payment Systems Vision 2021
In an effort to promote widespread use of card payments in the country, the Reserve Bank of India (RBI) has proposed to set up an 'Acceptance Development Fund'.
What is Acceptance Development Fund (ADF)?
The Fund will be used to ensure growth of card acceptance infrastructure such as swipe machines across the country particularly in Tier III and Tier VI cities.
This means that using credit and debit cards in smaller towns and cities will become easier once the infrastructure expands as planned.
The government and the banking regulator have been taking several steps to increase digital payments and reduce the dependence of cash, a move that is aimed at making payments transactions transparent.
RBI's Payment System Vision, 2021 report released on May 2019, had suggested creation of an ADF to ensure acceptance more PoS (point of sales) or card acceptance infrastructure in tier III to tier VI centers.
This would to help facilitating innovation and reducing cost of such transactions.
Setting up ADF framework was also recommended by Nandan Nilekani headed panel on Deepening of Digital Payments.
The framework will be operationalised by December 2019.
Need for the Fund
Currently, while card acceptance infrastructure is well set up in large cities, it has not penetrated as much into smaller, Tier IV towns.
Consequently, people living there and those travelling to those towns find it difficult to use cards for their transactions.
Once infrastructure to accept and enable card payments (i.e., digital payments) becomes widely available even in smaller towns, it will become easier for individuals to adopt cards/digital methods as their main means of payment.
[Source: Livemint, Economic Times]
FACTS OF THE DAY
GREEN CRACKERS
The Government has launched green firecrackers, in a bid to resolve the crisis of air pollution. Council of Scientific and Industrial Research (CSIR) labs have been successful in developing various environment-friendly fireworks such as sound emitting crackers, flowerpots, pencils, chakkar and sparklers. The emissions testing facilities for the new fire crackers have been set up at CSIR-NEERI as well as their approved National Accreditation Board for Testing and Calibration (NABL) facilities. A Raw Materials Compositional Analysis (RACE) facility has been launched in Sivakasi to facilitate manufacturers for testing their raw materials and chemicals. Eight labs participated in the efforts to develop reduced emission/green fireworks, with CSIR NEERI coordinating the entire exercise. A green logo as well as a Quick Response (QR) coding system has been developed for differentiation of green crackers from conventional crackers. QR code is a novel feature incorporated on the fire crackers to avoid manufacture and sale of counterfeit products. This will also help the consumers to track the cracker using smart phones and other devices.
INCOME THRESHOLD TO AVAIL MICROFINANCE INCREASED
Reserve Bank of India has decided to raise the household income limit of borrowers to ₹2 lakh for urban and semi urban areas from ₹1.6 lakh. For rural areas, the limit has been increased from ₹1 lakh to ₹1.25 lakh. Loans by MFIs to such borrowers are uncollateralised. This will mean that all the borrowers having a household income of ₹2 lakh in urban/semi urban areas and ₹1.25 lakh in rural areas will be eligible for loans from MFIs, which are uncollateralised. The lending limit per borrower has also been hiked from ₹1 lakh to ₹1.25 lakh. The decision was taken after taking into consideration; the important role played by MFIs in delivering credit to those at the bottom of the economic pyramid and enables them to play their assigned role in a growing economy.
SR9009 – A POTENTIAL DRUG TO TREAT HEART ATTACK
The researchers have developed a potential drug to treat heart attack and prevent heart failure — for which no cure currently exists. When people suffer heart attacks, the resulting scarring of the heart tissue often leads to chronic heart failure. Researchers at the University of Guelph In Canada have discovered a drug that they believe can prevent that scarring and the progression to chronic heart failure. The drug, SR9009, works by capitalizing on the circadian clock inside the heart—the genes that turn on and off at specific times to regulate heart rate, blood pressure and other functions. The drug works by disrupting genes that would normally activate immune responses after a heart attack. In mice, it tamped down the production of the NLRP3 inflammasome, a protein complex that promotes scarring.
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