text:
\section{Introduction}\label{s1}
\subsection{The initial question}\label{s1.1}
Let $X$ be a non-singular real algebraic plane projective curve and let
another real algebraic plane projective curve $Y$ cut on $X$ real points
$P_1$, \dots, $P_n$.

{\em How can $P_1$, \dots, $P_n$
move along $X$, when $Y$ changes continuously?}

We assume that $Y$ remains transversal to $X$ and the
degree of $Y$ does not change.
Then the number of real intersection
points does not change.  In particular, the points do not collide
with each other and move continuously along $X$.
It looks like a motion of an ensemble of dancers on a curve.

If $X$ is a rational curve, then the answer is purely topological. Any
continuous isotopy of the intersection can be obtained in this way.
For curves of positive genus, there are obstructions. They appear both
infinitesimally (see section \ref{s1.3}),
and globally (see  sections \ref{s1.5} and \ref{s4}).

\subsection{Generalizations to linear equivalence classes of
divisors}\label{s1.2}
Intersections of a plane projective curve $X$ with other plane projective
curves $Y$ of a fixed degree form a single class of linear equivalent
divisors on $X$. This is  quite a special class.

The question that was posed in Section \ref{s1.1} makes sense for
any class of linear equivalent divisors. Let $X$ be a non-singular real
algebraic projective curve, and $D=P_1+\dots+P_n$ be a simple real divisor
on $X$.

{\em How can $D$ move on $X$ continuously in its linear equivalence class?}
We assume that $D$ remains simple and real during the movement.

 %

\subsection{The velocities of points}\label{s1.3}
Linear equivalence classes of divisors are described by the Abel-Jacobi
theorem. It provides also a complete
{\em infinitesimal answer\/} to the question on a movement:

If the movement is smooth, then each point of a moving divisor
has a velocity vector. It follows from Abel-Jacobi Theorem that

{\it a moving divisor does not leave its linear equivalence class
if and only if, for any holomorphic differential form on the curve,
the sum of values of the form on the velocity
vectors over all the points of the divisor vanishes.}

Global restrictions on motions of a simple real divisor, which we are going
to study, look quite differently.

\subsection{Topological choreography}\label{s1.4} If the
algebraic restrictions on a moving configuration of points are dismissed
(i.e., the points are not required to be cut by other curve or form a
divisor linear equivalent to the same divisor), then a complete
description of possible motions becomes easy.

Let $T$ be a closed 1-manifold.  Connected components of $T$ are
homeomorphic to the circle. Denote them by $T_1$, \dots, $T_r$.
Consider the set $\UConf_d(T)$ of unordered
configurations $\{P_1,\dots,P_d\}$ of $d$ pairwise distinct points in $T$.
The set $\UConf_d(T)$ has a natural topology.
A connected component of  $\UConf_d(T)$ is determined by a distribution of
points $P_1$, \dots, $P_d$ in components $T_1$, \dots, $T_r$ of $T$.

Consider a motion of  $\{P_1,\dots,P_d\}\in\UConf_d(T) $ which is closed in
the sense that the final configuration of points coincides with the initial
one (although the points may shuffle).
In other words, consider a loop $L$ in  $\UConf_d(T)$.

The paths which are traced by individual points of a configuration
constitute a 1-cycle on $T$. Its
homology class belongs to $H_1(T)$. We will call this homology class
the {\em tracing\/} of $L$ and denote it by $\tr(L)$. It does not change
under a homotopy of $L$.

The homology  $H_1(T)$ is a free abelian group generated by
the fundamental classes $[T_1]$, \dots, $[T_r]$ of the components.
Hence $\tr(L)=\sum_{k=1}^rc_k[T_k]$, for some $c_k\in\Z$.

{\bf Homotopy classification of loops in $\UConf_d(T)$.}
{\it  A loop $L$ \ in \  $\UConf_d(T)$ which begins at $\{P_1,\dots,P_d\}$
is defined up to homotopy by
the distribution of points \ $P_1$, \dots, $P_d$ \ in \ $T_1$, \dots,
$T_r$
and  tracing $\tr(L)=\sum_{k=1}^rc_k[T_k]\in H_1(T)$.
The only restriction on $\tr(L)$ is that if \
$T_k$ contains no points from $\{P_1,\dots,P_d\}$, then $c_k=0$.}

The proof is straightforward.\qed

\subsection{Choreography of divisors}\label{s1.5}
For loops in the space of real simple divisors linear equivalent
to each other, the tracing satisfies many extra restrictions, which
depend of the curve and divisor.  We postpone the detailed statements till section \ref{s4}. They are
the main results of this paper.

\subsection{Acknowledgements}\label{s1.6}
The first interesting special
case  emerges, when a plane projective M-cubic is cut by another cubic.
It was considered by Ay\c{s}eg\"ul \"Ozt\"urkalan
\cite{Ay}.
She relied on a completely different technique, elliptic addition on an
M-cubic. As is shown below, most of her results do not require
elliptic addition and hold true in much more general situation. However, it
was her results that have inspired and motivated the present work.

I am grateful to Grisha Mikhalkin, John Milnor, Kristin Shaw, Dennis
Sullivan and Julia Viro for valuable remarks and suggestions.

\section{Preliminary notions}\label{s2}

\subsection{A real curve}\label{s2.1}
In this paper almost everything
happens on an {\em algebraic curve\/} $X$.
We will focus on the situations
when this ambient curve is {\em non-singular, irreducible and
projective\/}.  In this section, we assume also that the ambient curve is
{\em real.\/}

We do not assume that $X$ is planar or embedded in any specific standard
algebraic variety. Although the curve is assumed to be projective, the
ambient projective space will not play any r\^ole.
We would rather think of $X$ as an abstract complete curve.
The set of its real points is denoted by $\R X$, the set of its complex
points, by $\C X$. Projectivity of $X$ implies that both $\C X$ and
$\R X$ are compact.

From analytical perspective, $\C X$ is a closed
non-singular Riemann surface with an anti-holomorphic involution $\conj$,
and $\R X$ is the  fixed point set of $\conj$.

\subsection{Complex topological characteristics of a real curve}\label{s2.2}
A curve $X$ is said to be of {\em type I\/} if $\R X$ bounds in $\C X$ (i.e.,
$\R X$ realizes $0\in H_2(\C X;\Z/{2\Z})$). Otherwise $X$ is said to be of
{\em type II\/}.

If $X$ is of type I, then $\C X\sminus \R X$ consists of two connected
components homeomorphic to each other and interchanged by $\conj$. The
orientation of a connected component of $\C X\sminus \R X$ defines an
orientation on $\R X$ (as on the boundary of its closure).
The other connected component of $\C X\sminus\R X$
defines the opposite orientation on $\R X$. These orientations are called
{\em complex orientations\/} because they come from the orientation of
$\C X$, which is determined by the complex structure of $\C X$.

A connected component $X_i$ of $\R X$ is homeomorphic to circle.
If $X$ is of type I, then a complex orientation of $\R X$ defines
orientations of $X_i$. With these orientations, the components define
the basis $[X_1],\dots,[X_r]$ of $H_1(\R X)$. The sum $[X_1]+\dots+[X_r]$
of these generators is called a {\em complex orientation class\/} of $X$.
It defines and is defined by a complex orientation. The
inclusion homomorphism $H_1(\R X)\to H_1(\C X)$ has an infinite cyclic
kernel generated by a complex orientation class.

If $X$ is of type II, then $\C X\sminus\R X$ is connected and the inclusion
homomorphism $H_1(\R X)\to H_1(\C X)$ is injective.

The types were introduced by Felix Klein \cite{Kl} in 1876. The complex
orientations of type I curves were introduced by V.~A.~Rokhlin \cite{R1},
\cite{R2}.

\subsection{Real divisors}\label{s2.3}
Let $D$ be a formal linear combination $n_1P_1+\dots+n_lP_l$ of points
$P_1,\dots,P_l\in \C X$ with {\em positive integer\/} coefficients
$n_1,\dots,n_l$. In other words, let $D$ be an effective divisor on $X$.
Recall that the natural number $n_1+\dots+n_l$ is called the {\em degree\/}
of divisor $D=n_1P_1+\dots+n_lP_l$ and denoted by $\deg D$.
The set $\{P_1,\dots,P_l\}$ is called
the {\em support\/} of $D$ and denoted by $\supp D$.

A divisor $D=n_1P_1+\dots+n_lP_l$ is said to be {\em real\/} if it is
invariant under $\conj$. This means that each $P_i$ is either real (i.e.,
$P_i=\conj P_i$), or $D$ contains together with the summand $n_iP_i$
also the conjugate summand $n_jP_j=n_i\conj P_i$.

A real effective divisor
$D$ admits a unique decomposition $D=D'+D''$ into an effective divisor
$D'$ with $\supp D'\subset\R X$
and an effective divisor $D''$ with
$\R X\cap\supp D''=\varnothing$. Then $D'$ is called the {\em real
part\/} of $D$ and denoted by $\R D$, and $D''$ is called the
{\em imaginary part\/} of $D$ and denoted by $\Im D$.
The degree of $\R D$ is called {\em real
degree\/} of $D$. If the real degree of $D$ equals degree (i.e., if
$D=\R D$ and $\Im D=0$), then $D$ is said to be {\em purely real\/}.

\subsection{Divisors form a space}\label{s2.4}
 An effective divisor of degree $d$
can be identified with a point of the {\em $d$-th symmetric power\/}
$\Sym^d(\C X)$. Recall that the $d$th symmetric power $\Sym^dT$ of a
topological space
$T$ is the quotient space of the Cartesian power $T^d$ under the natural
action of the symmetric group $\mathbb S_d$.

As is well-known, {\it the space $\Sym^d(\C X)$ is a manifold of
dimension $2d$
and can be equipped with a complex structure
such that the natural projection $(\C X)^d\to\Sym^d(\C X)$ is
holomorphic.\/}
This can be easily shown using
a well-known bijection $\Sym^d\C\to\C^d$ which maps
$\{x_1,\dots,x_d\}\in\Sym^d\C$ to $(a_1,\dots,a_d)\in\C^d$ such that
$(x-x_1)\dots(x-x_d)=x^d+a_1x^{d-1}+\dots+a_d.$

\subsection{Linear equivalence of divisors}\label{s2.5}
Remind that some divisors $D=\sum_{k=1}^ln_kP_k$ on $\C X$ can be
presented as $f^*(v)$ for a regular (i.e., holomorphic) map
$f:\C X\to\C P^1$ and $v\in\C P^1$. This means that
$f^{-1}(v)=\{P_1,\dots,P_l\}$ and
$n_k$ is the multiplicity of $P_k$ as a root of equation $f(x)=v$
for each $k$.

Recall that effective divisors $D$ and $E$ are said to be
{\em linear equivalent\/}
if there exists a regular map $f:\C X\to\C P^1$ such that
$f^*(v)+E=D+f^*(w)$  for some $v,w\in \C P^1$. Linear equivalence is indeed
an equivalence relation, see e.g., \cite{Sh}.

Denote by $\Leq(D)$ the set of
effective divisors on $X$ linearly equivalent to an effective
divisor $D$. The set $\Leq(D)$ is a subvariety of $\Sym^{\deg D}\C X$
isomorphic to a finite dimensional complex projective space, see e.g.,
\cite{Sh}.

\subsection{Linear equivalence of real divisors}\label{s2.6}
If $D$ is a real effective divisor on $X$,
then $\conj$ induces an anti-holomorphic involution on $\Leq(D)$ and turns
it into a real algebraic variety.

The subset of $\Leq(D)$ which is formed
by non-simple divisors is a hypersurface, it is called the {\em discriminant
hypersurface\/} and denoted by $\GS \Leq(D)$.

\subsection{Deformations of real divisors}\label{s2.7}
Let  $D$ be a simple real divisor on $X$.
The connected component of $\R \Leq(D)\sminus\GS \Leq(D)$ which contains $D$
is denoted by $\Def(D)$.
It can be described as the space of {\em simple real\/} divisors on $X$ which are
linearly equivalent to $D$ and can be connected to $D$ by {\em a continuous
family of simple real divisors.\/} A path in $\Def(D)$ gives rise to an
isotopy
of $\supp D$ in the process of which $\supp\R D$ moves in $\R X$, while
$\supp\Im D$ moves in
$\C X\sminus \R X$.\medskip

\noindent{\itshape\bfseries Example.}
Let $X$ be a real plane curve, and $Y$ another real plane curve transversal
to $X$. Let $D$ be a simple divisor on $X$ which is cut by $Y$ on $X$, that
is $\supp D=\C X\cap\C Y$.
Then for each $E\in \Def(D)$ the set $\supp E$ is a
transversal intersection of $\C X$ with $\C Y'$, where $Y'$ is a plane curve
of the same degree as $Y$ and $\C X\cap\C Y'$ is isotopic to $\C X\cap\C Y'$
in the space of finite subsets of $\C X$ invariant under $\conj$.

\section{Tracings of a real moving simple divisor}\label{s3}
\subsection{Real and imaginary tracing maps}\label{s3.1}
Let $D$ be a simple real divisor in $X$ with
$\R D=\sum_{k=1}^nP_k$ and $\Im D=\sum_{k=1}^{m}(P_{n+k}+\conj P_{n+k})$.

Let $L:[0,1]\to \Def(D)$ be a loop with $L(0)=L(1)=D$.
Then there are uniquely defined
paths $L_1,\dots,L_n:I\to\R X$ and $L_{n+1},\dots,L_{n+m}:I\to\C X\sminus\R
X$ such that $L(t)=\sum_{k=1}^{n}L_k(t)+\sum_{k=1}^{m}(L_{n+k}(t)+\conj
L_{n+k}(t))$
for any $t\in I$.

Each of these paths can be considered as a singular 1-simplex.
Notice that singular chains $L_1+\dots+L_n$ and
$(L_{n+1}+\conj\circ L_{n+1})+\dots+(L_{n+m}+\conj\circ L_{n+m})$ are
cycles, because each of
the paths starts at a point where exactly one of the paths finishes.

The cycle $L_1+\dots+L_n$ determines a homology class which is an
element of $H_1(\R X)$, the cycle
$(L_{n+1}+\conj\circ L_{n+1})+\dots+(L_{n+m}+\conj\circ L_{n+m})$
determines a homology class which is an element of $H_1(\C X\sminus\R X)$.
We will call these homology classes {\em real\/} and {\em imaginary
tracing\/} classes of $L$, respectively, and denote them
by $\Re\tr(L)$ and $\Im\tr(L)$.

 It is easy to see that the tracing homology classes
depend only on the homotopy class of $L$ (a homotopy between loops in
$\Def(D)$ gives rise to a homology between the tracing cycles exactly in
the same way as $L$ turns into tracing cycles  $\Re\tr(L)$ and $\Im\tr(L)$).
The maps
\begin{align*}
&\Re\tr:\pi_1(\Def(D))\to H_1(\R X),\\
&\Im\tr:\pi_1(\Def(D))\to H_1(\C X\sminus\R X)
\end{align*}
defined by this construction will be called
{\em real\/} and {\em imaginary tracing maps\/}, respectively.
One can easily check that they are homomorphisms.

\subsection{Encoding of real tracing}\label{s3.2}
Let $X_1$, \dots, $X_r$ be connected components of $\R X$. Each of them is
homeomorphic to circle. An orientation of $\R X$ gives generators $[X_i]\in
H_1(X_i)$, which identify $H_1(X_i)$ with $\Z$.  The group $H_1(\R X)=
\oplus_{i=1}^rH_1(X_i)$ is identified with $\Z^r$.

Thus, an orientation of $\R X$ encodes the real tracing class
$\Re\tr(L)$ of a loop $L:I\to \Def(D)$ by a sequence of integers
$(c_1(L),\dots,c_r(L))$ such that  $\Re\tr(L)=c_1(L)[X_1]+\dots+c_r(L)[X_r]$.

Recall that if $X$ is a curve of type I, then a complex orientation
provides a natural set of generators of $H_1(X_i)$, see Section \ref{s2.2}.

\subsection{Monodromy permutations}\label{s3.3}
The paths $L_1,\dots,L_n:I\to\R X$ defined by a loop $L:I\to \Def(D)$
form an isotopy of the set $A=\supp\R D$ in $\R X$. The
isotopy defines a monodromy map  $\mu_L:A\to A$ that maps
$P_k=L_k(0)$ to $L_k(1)$ for $k=1,\dots,n$.

Of course, points $P_k$ and $\mu_L(P_k)$ belong to the same
connected component of $\R X$. Denote $A\cap X_i$ by $A_i$.
Thus, $\mu_L$ maps $A_i$ to itself.

An orientation of $X_i$ induces a cyclic order on $A_i$.
An isotopy of a finite subset $A_i$ of $X_i$ preserves the cyclic order
on $A_i$.  Therefore the monodromy map  $\mu_L:A\to A$ restricted to $A_i$
is a cyclic permutation.

The group of permutations preserving a cyclic order on a finite set
has a canonical generator which sends each
element to the next one. Let us denote this canonical generator by $\nu$.

The monodromy map $\mu_L:A\to A$ is completely determined by the
tracing class:
$$\mu_L|_{A_i}=\nu^{c_i(L)},$$
where $c_i(L)$ are the integers which encode
$\Re\tr(L)=\sum_{k=1}^rc_k(L)[X_k]$, see Section \ref{s3.2}.

\subsection{On a complex curve}\label{s3.4} Let $X$ be a non-singular
{\em complex\/} curve and $D$ be a simple divisor on $X$. As above, we
denote by $\Leq(D)$ the variety of divisors on $X$ which are linear
equivalent to $D$, and by $\GS\Leq(D)$ the discriminant hypersurface
of $\Leq(D)$ which consists of non-simple divisors.

The complement $\C\Leq(D)\sminus\C\GS\Leq(D)$ is a counter-part of
$\Def(D)$: it consists of simple divisors which are linearly equivalent to
$D$ and can be connected to $D$ by a continuous family of simple divisors.
The only difference is that divisors are not required to be real, cf.
section \ref{s2.7}.

Let $L:[0,1]\to \C\Leq(D)\sminus\C\GS\Leq(D)$ be a loop with $L(0)=L(1)=D$.
Then there are uniquely defined
paths $L_1,\dots,L_d:I\to\R C$ such that $L(t)=\sum_{k=1}^{d}L_k(t)$
for any $t\in I$.

Each of these paths can be considered as a singular 1-simplex.
Notice that singular chains $L_1+\dots+L_d$  is a
cycle, because each of
the paths starts at a point where exactly one of the paths finishes.

The cycle $L_1+\dots+L_n$ determines a homology class which is an
element of $H_1(\C X)$. We will call this homology class the {\em
tracing\/} class of $L$ and denote it by $\tr(L)$.

The tracing homology class $\tr(L)$
depends only on the homotopy class of $L$. A homotopy between loops in
$\C\Leq(D)\sminus\C\GS\Leq(D)$ gives rise to a homology between the tracing
cycles exactly in the same way as $L$ turns into the tracing cycle
$\tr(L)$.
The map
$$\tr:\pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\to H_1(\C X)$$
defined by this construction will be called
{\em tracing map\/}.
One can easily check that it is  a homomorphism.

\subsection{Relation among tracing maps}\label{s3.5} If $X$ is a
real non-singular curve, $D$ is a simple real divisor on $X$, then the
following diagram is obviously commutative
\begin{equation}\label{tr-diagram}
\begin{tikzcd}
\pi_1(\Def(D))\arrow[r,"\Re\tr\oplus\Im\tr"]\arrow[d,"\inc_*"]
& H_1(\R X)\oplus H_1(\C X\sminus\R X)\arrow[d,"\inc_*+\inc_*"]\\
\pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\arrow[r,"\tr"]&H_1(\C X)
\end{tikzcd}
\end{equation}
That is, $\tr\circ\inc_*=\inc_*\circ\Re\tr+\inc_*\circ\Im\tr$.

\section{Tracing theorems}\label{s4}
\subsection{Real tracing of a purely real divisor}\label{s4.1}

\begin{Th}\label{Th1}
Let $X$ be a non-singular projective real algebraic curve and let $D$ be a
simple purely real divisor on $X$.\\
\hspace*{21pt}{\bf(a) } If $X$ is of type I, then the image of real
tracing map is contained in the cyclic subgroup generated by a complex
orientation class.\\
\hspace*{20pt}{\bf(b) } If $X$ is of type I and $\R X$
has a connected component disjoint from $\supp D$, then the real tracing map
$\Re\tr:\pi_1(\Def(D))\to H_1(\R X)$ is trivial.\\
\hspace*{22pt}{\bf(c) } If $X$ is of type II, then
the real tracing map
$\Re\tr:\pi_1(\Def(D))\to H_1(\R X)$ is trivial.
\end{Th}

\begin{rem}\label{rem4.1.1}
The parts (b) and (c) of Theorem \ref{Th1} describe $\Re\tr$ completely,
while the part (a) gives only an upper bound for the image
of $\Re\tr$. In particular, the part (a) implies the following statement.
\end{rem}

\begin{cor}\label{cor}
Let $X$ be a non-singular projective real algebraic curve of type I, let
$\R X$ be equipped with a complex orientation and consist of connected
components  $X_1,\dots,X_r$.
Let $D$ be a simple purely real divisor on $X$, let $A=\supp D$ and
$A_i=X_i\cap A$ be cyclically ordered via the complex
orientation of $X_i$.
If $L:I\to \Def(D)$ is a loop, then:

{\bf(a)} $\Re\tr(L)=C_L[\R X]$ for some integer $C_L$ and

{\bf(b)} the induced monodromy $\mu_L:A\to A$ restricted to $A_i$ is $\nu^{C_L}$.
\end{cor}

\subsection{Real tracing of real, but not necessarily purely real divisors}\label{s4.2}
The assumptions of the following theorem are weaker than the assumptions
of Theorem \ref{Th1}: the divisor is not assumed to be purely real.

\begin{Th}\label{Th2} Let $X$ be a non-singular projective real algebraic
curve of type I and $D$ be a real divisor on $X$.

{\bf(a) } The image of real tracing map is
contained in the cyclic subgroup generated by a complex
orientation class and classes divisible by two.

{\bf(b) }  If $\R X$ has a connected
component disjoint from $\supp D$,  then the image of real tracing map
$\Re\tr:\pi_1(\Def(D))\to H_1(\R X)$ consists of classes divisible by two.
\end{Th}

Naturally, the conclusions in Theorem \ref{Th2} are also weaker than their
counter-parts in Theorem \ref{Th1}.
Namely, in the parts (a) and (b)
the subgroup, where the image of $\Re\tr$ is allowed to be contained,
increases by the set of all even elements. In other words, the conclusions
stay the same, but in $\Z_{/2}$-homology instead of
$\Z$-homology.

The part (c) of Theorem \ref{Th1} has no counter-part in
Theorem \ref{Th2}: Theorem \ref{Th2} does not contain any restriction
on the real tracing map, if the curve is of type II and the divisor is
not purely real.
 As we will see later, for a curve of type II the image
of $\Re\tr$ can be the whole $H_1(\R X)$.

\subsection{Tracing of a simple divisor on a complex curve}\label{s4.3}
Theorems \ref{Th1} and \ref{Th2} are deduced below in section \ref{s6}
from the well-known facts about relation of homology classes realized
in $H_1(\C X)$ by components of $\R X$ (see Section \ref{s2.2}), the
relation among tracing homomorphisms described in section \ref{s3.5}, and
the following Theorem \ref{Th3}.

\begin{Th}\label{Th3}
Let $X$ be a complex non-singular projective curve and $D$ be a simple
divisor on $X$. Then the tracing map
$\tr:\pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\to H_1(\C X)$
is trivial.
\end{Th}

\section{Tracing for non-simple divisors}\label{s5}

\subsection{Why and how to extend tracing to non-simple divisors}\label{s5.0}
Tracing homomorphisms appeared in a number of situations above, see sections
\ref{s1.4}, \ref{s3.1} and \ref{s3.4}. They have been defined for
spaces of {\em simple divisors}. However, they can
be defined for {\em symmetric powers\/}, like the space $\C\Leq(D)$,
which consists of divisors that are {\em not necessarily simple}.

The restriction to simple divisors makes the constructions easier.
On the other hand, it suffices for stating the main results of this paper.

However, for the proof of Theorem \ref{Th3} we have to factor the tracing map
$\tr:\pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\to H_1(\C X)$
through $\pi_1(\C\Leq(D))$. This is done below by defining tracing
$\pi_1(\Sym^d(\C X))\to H_1(\C X)$, that is by
allowing non-simple divisors.

As a matter of fact, tracing homomorphism $\pi_1(\Sym^d(T))\to H_1(T)$
can be defined for any ``non-pathological'' topological space $T$ (say, a
cw-complex). Here is a sketch for the construction:

As in the constructions above, it starts with taking a loop
$L:I\to\Sym^d(T)$ which represents an element of $\pi_1(\Sym^d(T))$ and
splitting it into $d$ paths $I\to T$. They are not uniquely
defined by $L$, since the natural projection $p:\Sym^d(T)\to T$ is not a
covering. In order to overcome this difficulty, let us make $L$ nice with
respect to the natural stratification of $\Sym^d(T)$, so that one can
decompose $I$ into cells such that, on the interior of each of the 1-cells,
$L$ would go in one of the strata.
Then on the restrictions of $L$ to each of the 1-cells can be split
to paths in $T$. The paths considered as singular 1-simplexes
would form a 1-cycle, and we take the homology class of this 1-cycle.

Then we would need to prove that the homology class does not depend on the
choices made in this construction and that homotopy of $L$ does not change
the homology class. The former is obvious. The latter can be made
by choosing the homotopy of $L$ nice  with respect to the natural
stratification of $\Sym^d(T)$ and subdividing it to nice pieces, and
proceeding as above, when we built the 1-cycle.

This is how the author convinced himself that a required tracing exists,
but writing down all the details seemed be quite cumbersome and encouraged
to look for simplifications.

We need this construction only for $T=\C X$.
Since $\Sym^d(\C X)$ is a manifold and the discriminant
$\GS$  has a real codimension 2, the loop $L$ which represents an
element of $\pi_1(\Sym^d(\C X))$ can be chosen in the complement of $\GS$.
Then it splits into paths $L_1,\dots,L_d\to\C X$ uniquely.

A homotopy between loops in $\Sym^d(T)$ may hit $\GS$.
However, it can be chosen meeting $\GS$ in a generic point
transversally. We will do this in section \ref{s5.6} below.
As a by-product,  a more
conceptual view on relation between maps of arbitrary space to $\Sym^d(T)$
and $T$ will be developed in  sections \ref{s5.4} and \ref{s5.5}.

There exists yet another construction for tracing
$\pi_1(\Sym^d(T))\to H_1(T)$ based on the Dold-Thom
isomorphism. It is presented in section \ref{s5.7} for the sake of
completeness. The construction is
elegant and conceptual, but difficult to relate to the constructions that
we use in configuration spaces.

\subsection{Configuration spaces}\label{s5.1}
Recall that spaces of effective divisors on a curve $X$ were introduced
above as symmetric powers $\Sym^d(\C X)$ of $\C X$ equipped with additional
structures, and spaces $\Leq(D)$ and $\Def(D)$ were introduced as subspaces
of $\Sym^d(\C X)$, see sections \ref{s2.4}, \ref{s2.6} and \ref{s2.7}.

Let $T$ be a topological space.
Points of $T^d$ are ordered $d$-tuples $(x_1,\dots,x_d)$ of points of $T$.
Points of $\Sym^dT$ are unordered $d$-tuples $\{x_1,\dots,x_d\}$ of points
of $T$. A point
can appear several times, so the number of pairwise different points in
a $d$-tuple may be less than $d$. A $d$-tuple of pairwise different points
of $T$ is called a {\em configuration\/} of $d$ points.

The subset of $T^d$ which consists of points $(x_1,\dots,x_d)$,
where $x_i=x_j$ for some $i\ne j$, is called the {\em big diagonal\/} of
$T^d$ and is denoted by $\GD$. Denote by $\GS$ the image of $\GD$ under the natural
projection $p:T^d\to\Sym^dT$, that is $\GS=p(\GD)$.

The space $T^d\sminus\GD$ is the {\em $d$-th ordered configuration space\/}
 of $T$ and denoted $\Conf_d(T)$. The space $\Sym^dT\sminus\GS$ is called
the {\em $d$-th unordered configuration space\/} and denoted by
$\UConf_d(T)$.
The map $\Conf_d(T)\to\UConf_d(T)$ defined as a submap of $p:T^d\to\Sym^dT$
is a regular $\mathbb S_d$-covering.

\subsection{Tracing in configuration spaces}\label{s5.2}
In this section we construct a homomorphism
$\Tr:\pi_1(\UConf_d(T))\to H_1(T)$ for any positive integer $d$
and a topological space $T$. We follow almost literally the constructions
of section \ref{s3.4}.

Shortly speaking, $\Tr$ assigns to a homotopy class
of a loop in the configuration space $\UConf_d(T)$ the homology class
of the sum of paths
which are traced by all the points of the configuration while the
configuration traces the loop. Here are more
details.

Let  $L:I\to\UConf_d(T)$  be a loop. By the Lifting Path Theorem,
there exists a path   $\widetilde L:I \to\Conf_d(T)$ covering $L$.
It can be presented as $\widetilde L:t\mapsto (L_1(t),\dots,L_d(t))$,
where $L_k:I\to T$
with $k=1,\dots,d$ are paths in $T$. Since $L$ is a loop, i.e.,
$L(0)=L(1)$,  we have
$\{L_1(0),\dots,L_d(0)\}=\{L_1(1),\dots,L_d(1)\}$.
It follows that the singular chain $L_1+\dots+L_d\in C_1(T)$ is a cycle.
Denote by $\Tr(L)$ the homology class of this cycle in $H_1(T)$.

In the construction of $\Tr(L)$, we have chosen a covering path
$\widetilde L$ of $L$. A covering path is determined by a choice of its
initial point. The initial point of $L$ is a $d$-element set
$\{x_1,\dots,x_d\}\subset T$. An initial point of the covering path is a
point of $T^d$ obtained by any ordering of this set.
The ordering of the initial points is nothing but an ordering of paths
$L_k$. It does not affect the homology class of $L_1+\dots+L_d$. Thus,
$\Tr(L)$ does not depend on the choice.

For $\Gl\in\pi_1(\UConf_d(T))$, define $\Tr(\Gl)$ to be $\Tr(L)$
for a loop $L$ which belongs to $\Gl$. It does not depend on the choice of
$L$. Indeed, if $L$ and $L'$ are homotopic loops in $\UConf_d(T)$,
then homotopy between them  can be lifted to $ \Conf_d(T)$ by
the Lifting Homotopy Theorem and gives rise to a homology between the
cycles $\Tr(L)$ and $\Tr(L')$ in $T$.
\hfill$\square$

\subsection{Relation between the tracing maps}\label{s5.3}
If $X$ is a non-singular complex projective curve and $T=\C X$ and $D$ is a
simple real divisor of degree $d$ on $X$, then $\C\Leq(D)\sminus\C\GS\Leq(D)$ is a subspace of
$\UConf_d(T)$. The tracing map
$\tr:\pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\to H_1(\C X)$
defined in section \ref{s3.4} is factored through $\Tr$ via the
following commutative diagram.

\begin{equation}\label{eq1}
\begin{tikzcd}
\pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\arrow[rd,"\tr"]\arrow[r,"\inc_*"]&\pi_1(\UConf_d(\C X))\arrow[d,"\Tr"]\\
&H_1(\C X)
\end{tikzcd}
\end{equation}

\subsection{Spaces of pointed unordered $d$-tuples}\label{s5.4} An
unordered $d$-tuple of points, in which one of the points is
distinguished (we will say also {\em pointed\/}), is called {\em pointed.\/}
The space of pointed unordered
$d$-tuples of points of a space $T$ can be identified with
$(\Sym^{d-1}T)\times T$. Indeed,
$(\{x_1,\dots,x_{d-1}\},x)\in(\Sym^{d-1}T)\times T$ can be identified
with the unordered $d$-tuples $\{x_1,\dots,x_{d-1},x\}\in \Sym^dT$ in
which $x$ is distinguished.

 Like $\Sym^dT$, the space $(\Sym^{d-1}T)\times T$
can be obtained as a quotient space of $T^d$. Indeed,
$(\Sym^{d-1}T)\times T$ is the orbit space of the action of $\mathbb S_{d-1}$
in $T^d$ by permuting the first $d-1$ components and keeping the last
component fixed.
Each orbit of this  action is contained in an orbit of the natural action
of $\mathbb S_d$ in $T^d$. This gives a map $q:(\Sym^{d-1}T)\times T\to\Sym^dT$.
In terms of pointed $d$-tuples, $q$ forgets which element of $d$-tuple was
pointed. For an unordered $d$-tuple $\{x_1,\dots,x_d\}\in\Sym^dT$,
the preimage under $q$ consists of the same $d$-tuple in which pointing is
made in all the ways.

Thus the space of pointed unordered $d$-tuples of points of $T$ occupies
an intermediate position between $T^d$ and $\Sym^dT$: there are natural
surjections $T^d\to(\Sym^{d-1}T)\times T\to\Sym^dT$.

Over $\UConf_d(T)=\Sym^dT\sminus\GS$, the map $q$ defines a covering
$$\left((\Sym^{d-1}T)\times T\right)\sminus q^{-1}\GS\to\Sym^dT\sminus\GS$$
The space $\left((\Sym^{d-1}T)\times T\right)\sminus q^{-1}\GS$ can be
interpreted as the space of {\em pointed\/} unordered $d$-element subsets
of $T$.  Denote this space by $\PConf_d(T)$ and the covering
$\PConf_d(T)\to\UConf_d(T)$ forgetting the pointing, by $r$.

\subsection{Generalizations of tracing map}\label{s5.5}
In this section
for any topological space $Y$ and its continuous map $f:Y\to\UConf_d(T)$
we will construct a $d$-fold covering space $\widetilde Y$ of $Y$ and a
continuous map $\widetilde f:\widetilde Y\to T$.

The covering $\widetilde Y\to Y$ is the pullback $f^*r$ of the covering
$r:\PConf_d(T)\to\UConf_d(T)$
by $f$. By the definition of pullback, the total space $\widetilde Y$
consists of pairs $(y,z)\in Y\times\PConf_d(T)$ such that $f(y)=r(z)$.
 The map $\widetilde f$ maps $(y,z)\in\widetilde Y$ to the distinguished
point of $z$. (Recall that $z\in\PConf_d(T)$ is a pointed unordered
configuration of points of $T$.)

All these maps are gathered in the following
commutative diagram:
\begin{equation}\label{cd1}
\begin{tikzcd}
\widetilde Y\arrow[d,"f^*r"]\arrow[r,"r^*f"]\ar[rrr,bend
left=30,"\widetilde f"]
&\PConf_d(T)\arrow[d,"r"]\arrow[r,hook]
&(\Sym^{d-1}T)\times T\arrow[d,"q"]\arrow[r,"pr_{\scriptscriptstyle
 T}"]&T\\
Y\arrow[r,"f"]&\UConf_d(T)\arrow[r,hook]&\Sym^dT&
\end{tikzcd}
\end{equation}

The construction has several obvious nice properties:
\begin{enumerate}\setlength\itemsep{0pt} \setlength\topsep{0pt}
\item If $Y$ is a manifold, then $\widetilde Y$ is a manifold.
\item An orientation of a manifold $Y$ defines an orientation of
$\widetilde Y$.
\item If $f$ is homotopic to $g$, then $\widetilde f$ is homotopic to $\widetilde g$.
\item If $Y$ is an oriented manifold and $f:Y\to\UConf_d(T)$ is bordant to
zero, then $\widetilde f$ is bordant to zero.
\end{enumerate}
These properties obviously imply the following theorem.

\begin{Th}\label{Th-tr}
For any topological space $T$ and any natural numbers $d$ and $k$, there is
a natural homomorphism $\Omega_k(\UConf_d(T))\to \Omega_k(T)$ which maps
a bordism class of a map $f:M\to\UConf_d(T) $ to a bordism class of
$\widetilde f:\widetilde M\to T$.
\end{Th}

Since the 1-dimensional bordism group is naturally isomorphic the
1-dimensional homology group, Theorem \ref{Th-tr} provides a homomorphism
$$H_1(\UConf_d(T))\to H_1(T)$$ It is easy to see that its composition
$\pi_1(\UConf_d(T))\to H_1(\UConf_d(T))\to H_1(T)$ with
the Hurewicz homomorphism coincides with homomorphism $\Tr$ introduced in
Section \ref{s5.2}.

\subsection{Extension of tracing}\label{s5.6}
\begin{Th}\label{Th5}
If $X$ is a non-singular complex projective curve  and $d\ge2$ is an
integer, then there exists a homomorphism
$\TR:\pi_1(\Sym^d(\C X))\to H_1(\C X)$ such that the following diagram
is commutative
\begin{equation}\label{eq2}
\begin{tikzcd} \pi_1(\UConf_d(\C
X))\arrow[r,"\inc_*"]\arrow[d,"\Tr"]&
\pi_1(\Sym^d(\C X))\arrow[ld,"\TR"]\\H_1(\C X)&
\end{tikzcd}
\end{equation}
\end{Th}

\begin{proof}  Since $\GS$ is a complex hypersurface in
$\Sym^d(\C X)$, its real codimension is 2 and the inclusion homomorphism
$$\inc_*: \pi_1(\UConf_d(\C X))(=\pi_1(\Sym^d(\C
X)\sminus\GS))\to\pi_1(\Sym^d(\C X))$$
is surjective. Consider the composition
$$
\pi_1(\Sym^d(\C X))\xrightarrow{\inc_*^{-1}}\pi_1(\UConf_d(\C X))
\xrightarrow{\Tr}H_1(\C X)
$$
A priori, this is a multivalued map, since $\inc_*$ is not injective.
In order to prove that this multivalued map is univalued, we have
to verify that $\Tr(\Ker\inc_*)=0$. Down to loops, the latter means
that for any continuous map $f:S^1\to\Sym^d(\C X)\sminus\GS$, which admits a
continuous extension $F:D^2\to\Sym^d(\C X)$, the map
$\widetilde f:\widetilde S^1\to\C X$ is bordant to zero in $\C X$.

By a small deformation make both $f$ and $F$ differentiable,
make the image of $F$ intersecting only the highest dimensional
stratum of the discriminant $\GS$ and make $F$ transversal to it.

Now we apply a construction similar to the construction of section
\ref{s5.5}.
Pull back the ramified
covering $\Sym^{d-1}(\C X)\times\C X\to\Sym^d(\C X)$ via $F$. This gives rise
to the diagram
$$\begin{tikzcd
\widetilde{D^2}\arrow[d,swap,"F^*q"]\arrow[r,"\widetilde F"]& \Sym^{d-1}(\C
X)\times\C X\arrow[d,"q"]\\
D^2\arrow[r,"F"]&\Sym^d(\C X)
\end{tikzcd}
$$
The map $F^*q$, as a pull back of a ramified covering via a map transversal
to the ramification locus, is a
covering ramified over the preimage of the ramification locus of
$q$, that is over a finite set $F^{-1}\GS$. Therefore, $\widetilde{D^2}$ is a
compact oriented surface with boundary.
\end{proof}

\begin{rem}
Compare the proof of Theorem \ref{Th5} to the proof of Lemma 2.6 in
Ozsv\'ath and Szab\'o \cite{OS}.
Substantially, this is the same construction, but presented here more
formally and explicitly. We could not limit ourselves to referring to
\cite{OS}, because we deal with a slightly more general situation:
in \cite{OS} the number $d$ coincides with the genus of $\C X$.
\end{rem}

\subsection{Dold-Thom theorem and tracing}\label{s5.7}
Let us point out a connection of the tracing map to the Dold-Thom
isomorphism.

Symmetric powers $\Sym^d(T)$ of a space $T$ with a base point $t_0\in T$ are
embedded to each other
$$\Sym^d(T)\to\Sym^{d+1}(T):\{x_1,\dots,x_d\}\mapsto\{x_1,\dots,x_d,t_0\}.$$
The union $\cup_d\Sym^d(T)$ is called {\em the infinite symmetric
power\/} of $T$ and denoted by $\Sym(T)$.

According to the Dold-Thom theorem \cite{DT}, for a reasonable space $T$
(say, a cw-complex), {\em there is a natural isomorphism
$H_n(T)\to\pi_n(\Sym(T))$}.

The tracing homomorphism $\TR$, which was introduced and used above only
for $T=\C X$, can be
obtained in a more general situation as the composition
$$\pi_1(\Sym^d(T))\xrightarrow{\inc_*}\pi_1(\Sym(T))\xrightarrow{DT^{-1}}H_1(T),$$ where
$\inc_*$ is the inclusion homomorphism and $DT^{-1}$ is the isomorphism
inverse to the isomorphism from the Dold-Thom theorem.

Unfortunately, this elegant construction is less convenient for our
purposes than the more explicit construction used above.

\section{Proofs of tracing theorems}\label{s6}
\subsection{Proof of Theorem \ref{Th3}}\label{s6.1}
Recall that Theorem \ref{Th3} claims that the tracing map
$$\tr:\pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\to H_1(\C X)$$
is trivial for any simple divisor $D$ on a complex non-singular projective
curve $X$.

The diagrams \eqref{eq1} and \eqref{eq2} can be united
as follows:
\begin{equation}\label{eq3}
\begin{tikzcd}
 \pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\arrow[rd,"\tr"]\arrow[r,"\inc_*"]&\pi_1(\UConf_d(\C
X))\arrow[d,"\Tr"]\arrow[r,"\inc_*"]&\pi_1(\Sym^d(\C X))\arrow[dl,"TR"]\\
&H_1(\C X)&
\end{tikzcd}
\end{equation}
On the top row of this diagram, we have two inclusion homomorphism. Replace
these two homomorphisms by their composition, which is also
 an inclusion homomorphism:
\begin{equation}\label{eq4}
\begin{tikzcd}
 \pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\arrow[rd,"\tr"]\arrow[rr,"\inc_*"]&
&\pi_1(\Sym^d(\C X))\arrow[dl,"TR"]\\ &H_1(\C X)&
\end{tikzcd}
\end{equation}
Observe that the space $\C\Leq(D)\sminus\C\GS\Leq(D)$ of simple divisors
linear equivalent
to $D$ is contained in   $\C\Leq(D)$ of all effective divisors
linear equivalent to $D$. Furthermore, $\C\Leq(D)\subset\Sym^d(\C X)$.
Therefore we can insert $\C\Leq(D)$ into the top row of diagram \eqref{eq4}:
\begin{equation}\label{eq5}
\begin{tikzcd}
 \pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\arrow[rd,"\tr"]\arrow[r,"\inc_*"]&
\pi_1(\C\Leq(D)))\arrow[r,"\inc_*"]&\pi_1(\Sym^d(\C X))\arrow[dl,"TR"]\\
&H_1(\C X)&
\end{tikzcd}
\end{equation}
The space $\C\Leq(D)$ is known to be homeomorphic to a complex projective
space. Hence $\pi_1(\C\Leq(D))=0$. Therefore the
composition of the horizontal homomorphisms of diagram \eqref{eq5} is
trivial, as well as their composition
$\pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\xrightarrow{\inc_*}\pi_1(\Sym^d(\C X))$. Hence,
the homomorphism
$$\tr:\pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\to H_1(\C X)$$ is trivial. \qed

\subsection{Proof of Theorem \ref{Th1}}\label{s6.2}
Let $D$ be a simple purely real divisor on a non-singular projective
real algebraic curve $X$. Since $D$ is purely real, the imaginary tracing
$\Im\tr$ vanishes.
Therefore the diagram \eqref{tr-diagram} reduces to
\begin{equation}\label{purely-tr-diagram}
\begin{tikzcd}
\pi_1(\Def(D))\arrow[r,"\Re\tr"]\arrow[d,"\inc_*"]
& H_1(\R X)\arrow[d,"\inc_*"]\\
\pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\arrow[r,"\tr"]&H_1(\C X)
\end{tikzcd}
\end{equation}
By Theorem \ref{Th3}, the homomorphism
$\pi_1(\C\Leq(D)\sminus\C\GS\Leq(D))\xrightarrow{\tr}H_1(\C X)$
is trivial. Hence, the composition
$\pi_1(\Def(D))\xrightarrow{\Re\tr} H_1(\R X)\xrightarrow{\inc_*}H_1(\C X)$
is trivial.

If $X$ is of type II (as in the part (c)), then the inclusion homomorpism
$\inc_*: H_1(\R X)\to H_1(\C X)$ is injective (see section \ref{s2.2}).
Since $\inc_*\circ\Re\tr:\pi_1(\Def(D))\to H_1(\C X)$ is trivial and
$\inc_*: H_1(\R X)\to H_1(\C X)$ is injective, commutativity of
\eqref{purely-tr-diagram} implies that
$\Re\tr:\pi_1(\Def(D))\to H_1(\R X)$ is trivial.

If $X$ is of type I (as in the parts (a) and (b)), then the kernel of
$\inc_*: H_1(\R X)\to H_1(\C X)$ is an infinite cyclic group generated
by the class $[\R X]$ defined by a complex orientation (see section
\ref{s2.2}). Since the homomorphism
$$\inc_*\circ\Re\tr:\pi_1(\Def(D))\to H_1(\C X)$$ is
trivial, the image of $\Re\tr$ must be contained in the kernel of
 $\inc_*: H_1(\R X)\to H_1(\C X)$. This kernel is generated by
$[\R X]=\sum_i[X_i]$,
where $X_i$ runs over all connected components of $\R X$.
This is what is claimed in the part (b).

If $\R X$ has a connected component $X_j$ disjoint from $\supp D$
(as in part (a)), then the image of $\Re\tr$ does not fit to the kernel of
the inclusion homomorphism $\inc_*: H_1(\R X)\to H_1(\C X)$ unless
$\Re\tr=0$. \qed

\subsection{Proof of Theorem \ref{Th2}}\label{s6.3}
Since the curve $X$ is assumed to be of type I, $\C X\sminus\R X$
consists of two connected components, which are homeomorphic to each
other by $\conj$. Denote the closures in $\C X$ of these components
by $X_+$ and $X_-$.

Represent the real simple divisor $D$ as $\Re D+\Im D$, where
 $\Re D=\sum_{k=1}^nP_k$ and $\Im D=\sum_{k=1}^{m}(P_{n+k}+\conj P_{n+k})$
with $P_{n+k}\in X_+$.

Let $L:[0,1]\to \Def(D)$ be a loop with $L(0)=L(1)=D$.
Then there are uniquely defined
paths $L_1,\dots,L_n:I\to\R X$ and $L_{n+1},\dots,L_{n+m}:I\to X_+$ such that $L(t)=\sum_{k=1}^{n}L_k(t)+\sum_{k=1}^{m}(L_{n+k}(t)+\conj
L_{n+k}(t))$ for any $t\in I$.

According to definition, the cycle $\sum_{k=1}^nL_k$ defines
$\Re\tr(L)\in H_1(\R X)$. Denote the homology class of the cycle
$\sum_{k=1}^mL_{n+k}$ in $X_+$ by $\Im_+\tr(L)$.

\begin{lem}The homology class  $\Im_+\tr(L)\in H_1(X_+)$ belongs to the
image of $H_1(\R X)$ under the inclusion homomorphism $H_1(\R X)\to
H_1(X_+)$.
\end{lem}

\begin{proof}
The cycle
$$\sum_{k=1}^nL_k+\sum_{k=1}^mL_{n+k}+\sum_{k=1}^m\conj L_{n+k}$$
realizes $\Tr(\Gl)\in H_1(\C X)$.
 By Theorem \ref{Th3}, $\Tr(\Gl)=0$.
The relativization homomorphism $H_1(\C X)\to H_1(\C X, X_-)$
maps $\Tr(\Gl)$ to the homology class realized by $\sum_{k=1}^mL_{n+k}$.
It is zero, as the image of $0=\Tr(\Gl)$.
By excision, $H_1(\C X,X_-)=H_1(X_+,\R X)$. Hence, the cycle
$\sum_{k=1}^mL_{n+k}$ realizes $0\in H_1(X_+,\R X)$. By exactness of the
homology sequence of pair $(X_+,\R X)$, it means that the homology class
realized by $\sum_{k=1}^mL_{n+k}$ in $X_+$ (that is $\Im_+\tr(\Gl)$)
belongs to the image of inclusion homomorphism $H_1(\R X)\to
H_1(X_+)$.
\end{proof}

Let $\tau\in H_1(\R X)$ be a homology class such that its image under the
inclusion homomorphism $H_1(\R X)\to H_1(X_+)$ equals $\Im_+\tr(\Gl)$.
Then the image of $\Re\tr(\Gl)+2\tau\in H_1(\R X)$ under the inclusion
homomorphism $H_1(\R X)\to H_1(\C X)$ is realized by the same cycle
$\sum_{k=1}^nL_k+\sum_{k=1}^mL_{n+k}+\sum_{k=1}^m\conj L_{n+k}$ as
$\Tr(\Gl)$ which equals zero.
Hence, $\Re\tr(\Gl)+2\tau\in H_1(\R X)$ belongs to the
kernel of  the inclusion homomorphism $H_1(\R X)\to H_1(\C X)$. The kernel
is generated by a class of complex orientation $[\R X]$.
Therefore, $\Re\tr(\Gl)+2\tau=c[\R X]$ for some $c\in\Z$. It follows
that $\Re\tr(\Gl)$ belongs to the subgroup generated by $[\R X]$ and even
elements. This is the statement (a) of Theorem \ref{Th2}.

If $\R X$ has a connected component disjoint from $\supp D$, then the
fundamental class of this connected component is not involved in
$\Re\tr(\Gl)$. Therefore in the formula $\Re\tr(\Gl)+2\tau=c[\R X]$
obtained above, $c$ must be even. Hence,  $\Re\tr(\Gl)=-2\tau+c[\R X]$ is
divisible by two. This is the statement (b) of Theorem 2.
\qed

\section{Examples}\label{s7}

\subsection{Divisors of degree 2 on a plane M-cubic}\label{s7.1}
 Let $X$ be a plane projective
non-singular cubic curve with $\R X$ consisting of two components: an oval
$X_0$ and a one-sided component $X_1$. Let $A\in X_0$ and $B\in X_1$ be
arbitrary points. Consider divisor $D=A+B$. This is a simple purely real
divisor of degree 2.

Assume that the points are chosen generically, so that
the line $l$ passing through $A$ and $B$ is transversal to $X$. Then it
intersects $X$ in three points, two of them are $A$ and $B$. Denote the
third intersection point by $C$. It belongs to $X_0$. For any other real
line $l'$ passing through $C$, the divisor $l'\cap\R X$ is of degree 3
and has a form $C+A'+B'$, where $A'\in X_0$ and $B'\in X_1$. For one of the
lines, namely for the tangent line to $X$ at $C$, $A'=C$.
\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[scale=1.0,>=latex]
\filldraw(-.04,.18)circle(1.5pt);
\filldraw(1.17,.67)circle(1.5pt);
\node at(.15,.5){{\color{blue!70!black}$A$}};
\node at(.95,.85){{\color{blue!70!black}$B$}};
\draw[-](-1.5,-.4)--(2.0,1);
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\draw[->,very thick,blue!70!black
](-0.02,-.05)--(-0.002,.07);
\draw[<-,very thick,blue!70!black](-.99,-.05)--(-.99,.07);
\draw[->,very thick,blue!70!black](1.01,-.05)--(1.01,.07);
\node at(-1.4,.2){{\color{blue!70!black}$X_0$}};
\node at(1.13,1.4){{\color{blue!70!black}$X_1$}};
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{((\x)^3-\x)^.5});
\draw[domain=1:1.4,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\draw[domain=1:1.6,smooth=200,very thick,blue!70!black,->] plot({\x},{((\x)^3-\x)^.5});
\end{tikzpicture}
\hspace*{.7cm}
\begin{tikzpicture}[scale=1.0,>=latex]
\filldraw(-.98,-.19)circle(1.5pt);
\filldraw(-.04,.18)circle(1.5pt);
\filldraw(1.17,.67)circle(1.5pt);
\node at(-1.2,-.6){{\color{blue!70!black}$C$}};
\node at(.15,.5){{\color{blue!70!black}$A$}};
\node at(.95,.85){{\color{blue!70!black}$B$}};
\draw[-](-1.5,-.4)--(2.0,1);
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\draw[->,very thick,blue!70!black](-0.02,-.05)--(-0.002,.07);
\draw[<-,very thick,blue!70!black](-.99,-.05)--(-.99,.07);
\draw[->,very thick,blue!70!black](1.01,-.05)--(1.01,.07);
\node at(-1.4,.2){{\color{blue!70!black}$X_0$}};
\node at(1.13,1.4){{\color{blue!70!black}$X_1$}};
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{((\x)^3-\x)^.5});
\draw[domain=1:1.4,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\draw[domain=1:1.6,smooth=200,very thick,blue!70!black,->] plot({\x},{((\x)^3-\x)^.5});
\end{tikzpicture}
\hspace*{.7cm}
\begin{tikzpicture}[scale=1.0,>=latex]
\filldraw(-.98,-.19)circle(1.5pt);
\filldraw(-.04,.18)circle(1.5pt);
\filldraw(1.17,.67)circle(1.5pt);
\node at(-1.2,-.6){{\color{blue!70!black}$C$}};
\node at(.15,.5){{\color{blue!70!black}$A$}};
\node at(.95,.85){{\color{blue!70!black}$B$}};
\draw[-](-1.5,-.4)--(2.0,1);
\draw[-](-1.5,-.17)--(2.0,-.37);
\filldraw(-.05,-.24)circle(1.5pt);
\filldraw(1.05,-.31)circle(1.5pt);
\node at(.10,-.5){{\color{blue!70!black}$A'$}};
\node at(0.8,-0.53){{\color{blue!70!black}$B'$}};
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\draw[->,very thick,blue!70!black](-0.005,+.01)--(-0.001,.06);
\draw[<-,very thick,blue!70!black](-.99,-.05)--(-.99,.07);
\draw[->,very thick,blue!70!black](1.01,-.05)--(1.01,.07);
\node at(-1.4,.2){{\color{blue!70!black}$X_0$}};
\node at(1.13,1.4){{\color{blue!70!black}$X_1$}};
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{((\x)^3-\x)^.5});
\draw[domain=1:1.4,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\draw[domain=1:1.6,smooth=200,very thick,blue!70!black,->] plot({\x},{((\x)^3-\x)^.5});
\end{tikzpicture}
\end{center}
\end{figure}

The divisor $A'+B'$ is linear equivalent to $D$. Rotating $l$ about $C$
by $\pi$ gives rise to a loop in $\Def(D)$. The real tracing of this loop
equals $[\R X]$. The monodromy is trivial.

If on the same curve points $A$ and $B$ are selected either both on the
same component (both on $X_0$ or $X_1$ - does not matter), then the divisor
$D=A+B$ has trivial tracing by Theorem \ref{Th1} (b).

\subsection{Divisors of degree 3 on a plane M-cubic}\label{s7.2}
Let $X$, $A,B,C$ and $l$ be as in the preceding example. Let $D=A+B+C$.
\begin{minipage}{\textwidth}
\begin{minipage}{4.5cm}
\begin{tikzpicture}[scale=1.0,>=latex]
\filldraw(-.5,0)circle(1pt);
\filldraw(-.98,-.19)circle(1.5pt);
\filldraw(-.04,.18)circle(1.5pt);
\filldraw(1.17,.67)circle(1.5pt);
\node at(-.3,-.1){$c$};
\node at(-1.2,-.6){{\color{blue!70!black}$C$}};
\node at(.15,.5){{\color{blue!70!black}$A$}};
\node at(.95,.85){{\color{blue!70!black}$B$}};
\draw[-](-1.5,-.4)--(2.0,1);
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\draw[->,very thick,blue!70!black](-0.01,-.05)--(-0.01,.07);
\draw[<-,very thick,blue!70!black](-.99,-.05)--(-.99,.07);
\draw[->,very thick,blue!70!black](1.01,-.05)--(1.01,.07);
\node at(-1.4,.2){{\color{blue!70!black}$X_0$}};
\node at(1.13,1.4){{\color{blue!70!black}$X_1$}};
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{((\x)^3-\x)^.5});
\draw[domain=1:1.4,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\draw[domain=1:1.6,smooth=200,very thick,blue!70!black,->] plot({\x},{((\x)^3-\x)^.5});
\end{tikzpicture}
\end{minipage}
\begin{minipage}{9.2cm}\vspace*{3pt}
Choose a point $c\in l$ inside  the open disk
bounded by $X_0$. Then the divisors which are cut on $X$ by
real lines passing through $c$ form a subspace $S$ of $\Def(D)$
homeomorphic to circles. A generator of its fundamental group has
tracing $[\R X]$. The monodromy is the transposition of $A$ and
$B$
\end{minipage}
\end{minipage}

\subsection{Divisors of degree 9 on a plane M-cubic}\label{s7.3}
One of the first non-trivial special cases of the problems under
consideration appears, when the curve is a plane projective M-cubic and
the divisor is cut on it by another cubic curve. This
special case was considered by Ay\c{s}eg\"ul \"Ozt\"urkalan \cite{Ay}.

Let $X$, $X_0$, $X_1$ and $c$ be as in section \ref{s7.2}.
Each real line passing through $c$ intersects
$X_0$ in two points and $X_1$ in one point. This defines a two-fold
covering $X_0\to X_1$. Choose on $X_0$ and $X_1$ orientations such
that with respect to them the covering is of degree $+2$. (These orientations
are defined up to simultaneous reversing, they are the complex
orientations of $\R X$.)

Let $A$ be a set that is cut on $\R X$
by other real cubic curve $Y$. Denote $X_i\cap A$ by $A_i$.
By the Bezout Theorem, $A$ may consist of at most 9 points.
Assume that $A$ {\em consists of exactly 9 points.}
One can easily see that the number of points in $A_0$ is even, and number
of points and $A_1$ is odd.

 Points of $A_i$ are cyclically ordered by their
position on $X_i$ according to the orientation.
\defLatex[length=1.9mm,width=1.6mm]{Latex[length=1.9mm,width=1.6mm]}

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[scale=.9,>=latex]
\filldraw(-.5,0)circle(1pt);
\draw[-](-1.5,-.4)--(2.0,1);
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\draw[->,very thick,blue!70!black](-0.01,-.05)--(-0.01,.07);
\draw[<-,very thick,blue!70!black](-.99,-.05)--(-.99,.07);
\draw[->,very thick,blue!70!black](1.01,-.05)--(1.01,.07);
\node at(-1.4,.2){{\color{blue!70!black}$X_0$}};
\node at(1.05,1.3){{\color{blue!70!black}$X_1$}};
\node at(-.3,-.1){$c$};
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{((\x)^3-\x)^.5});
\draw[domain=1:1.4,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\draw[domain=1:1.6,smooth=200,very thick,blue!70!black,->] plot({\x},{((\x)^3-\x)^.5});
\end{tikzpicture}
\hspace*{1cm}
\begin{tikzpicture}[scale=.9,>=latex]
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\node at(-1.3,-.2){{\color{blue!70!black}$X_0$}};
\node at(1.05,1.3){{\color{blue!70!black}$X_1$}};
\node at(2.3,.8){{\color{green!30!black}$\R Y$}};
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{((\x)^3-\x)^.5});
\draw[domain=1:1.4,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\draw[domain=1:1.6,smooth=200,very thick,blue!70!black] plot({\x},{((\x)^3-\x)^.5});
\draw[domain=-.58:.61,smooth=200,thick,green!50!black] plot({38*(\x+.5)*(\x-.5)*\x},{\x});
\filldraw(1.12,.55)circle(1pt);
\filldraw(1.08,-.425)circle(1pt);
\filldraw(1.01,-.12)circle(1pt);
\filldraw(0,0)circle(1pt);
\filldraw(-.247,.482)circle(1pt);
\filldraw(-.875,.444)circle(1pt);
\filldraw(-.985,.111)circle(1pt);
\filldraw(-.81,-.534)circle(1pt);
\filldraw(-.285,-.515)circle(1pt);
\end{tikzpicture}
\hspace*{1cm}
\begin{tikzpicture}[scale=.9,>=latex]
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\node at(-.5,-.1){{\color{blue!70!black}$X_0$}};
\node at(1.05,1.3){{\color{blue!70!black}$X_1$}};
\draw[domain=-1:0,smooth=200,very thick,blue!70!black] plot({\x},{((\x)^3-\x)^.5});
\draw[domain=1:1.4,smooth=200,very thick,blue!70!black] plot({\x},{-((\x)^3-\x)^.5});
\draw[domain=1:1.6,smooth=200,very thick,blue!70!black] plot({\x},{((\x)^3-\x)^.5});
\filldraw(1.12,.55)circle(1pt);
\draw[-Latex[length=1.9mm,width=1.6mm]](1.01,-.12) to [out=130,in=-90] (.82,.23)
to [out=90,in=210] (1.12,.55);
\filldraw(1.01,-.12)circle(1pt);
\draw[-Latex[length=1.9mm,width=1.6mm]](1.08,-.425) to [out=180,in=-90] (.88,-.31)
to [out=90,in=220] (1.01,-.12);
\filldraw(1.08,-.425)circle(1pt);
\draw[-Latex[length=1.9mm,width=1.6mm]](1.12,.55) to [out=-30,in=90] (1.5,0.05)
to [out=-90,in=30] (1.08,-.425);
\filldraw(0,0)circle(1pt);
\draw[-Latex[length=1.9mm,width=1.6mm]](0,0) to [out=40,in=-90] (.15,.3)
to [out=90,in=10] (-.247,.482);
\filldraw(-.247,.482)circle(1pt);
\draw[-Latex[length=1.9mm,width=1.6mm]](-.247,.482) to [out=110,in=0] (-.58,.8)
to [out=180,in=100](-.875,.444) ;
\filldraw(-.875,.444)circle(1pt);
\draw[-Latex[length=1.9mm,width=1.6mm]](-.875,.444) to [out=170,in=70] (-1.17,.34)
to [out=260,in=160] (-.985,.111);
\filldraw(-.985,.111)circle(1pt);
\draw[-Latex[length=1.9mm,width=1.6mm]](-.985,.111) to [out=200,in=100] (-1.22,-.34)
to [out=280,in=190] (-.81,-.534);
\filldraw(-.81,-.534)circle(1pt);
\draw[-Latex[length=1.9mm,width=1.6mm]](-.81,-.534) to [out=-90,in=180] (-.55,-.8)
to [out=0,in=-90] (-.285,-.515);
\filldraw(-.285,-.515)circle(1pt);
\draw[-Latex[length=1.9mm,width=1.6mm]](-.285,-.515) to [out=-30,in=-110] (.16,-.34)
to [out=70,in=-45] (0,0);
\end{tikzpicture}

\end{center}
\label{f1}
\end{figure}

A continuous change of $Y$ forces $A$ to move along $\R X$. We consider only
those changes of $Y$ during which $\R Y$ stays transversal to $\R X$. Assume
that $A$ moved for a while and came back to its original position. This
gives rise to a monodromy permutation $A\to A$, which splits to permutations
$A_0\to A_0$ and $A_1\to A_1$. The cyclic order in $A_i$ defined by the
complex orientation of $X_i$ does not change. So, the permutations
$A_i\to A_i$ are cyclic.   Ay\c{s}eg\"ul \"Ozt\"urkalan \cite{Ay} proved
that

{\it If none of
$A_i$ is empty, then the permutations on $A_0$ and $A_1$ are closely
related to each other:
the pair of permutations are realized if and only if they are the same
powers of the cyclic
permutations moving each point of $A_i$ to the next one along the complex
orientation of $X$.

If $A_0=\varnothing$, then only the identity permutation
of $A_1$ is realizable.}

These restrictions on the permutations follow from Theorem
\ref{Th1} (a) and (b) of this paper and the relation between the monodromy
and tracing described in section \ref{s3.3} above.

\subsection{Separating morphisms}\label{s7.4}
In the example of section \ref{s7.2} the projection $X\to P^1$ from
point $c$ is such
that the preimage of $\R P^1$ is $\R X$. Morphisms of real algebraic curves
with this property are called {\em separating.\/} The name was motivated by
the fact that existence of a separating morphism $X\to P^1$ implies that $X$
is of type I, i.e., $\R X$ separates halves of $\C X$ from each other.

Originally separating morphisms were used to prove that the curve under
consideration is of type I. For example, if $X$ is a real plane projective
curve of degree $d$, which has a nest of depth $[\frac{d}2]$, then
projection from any real point which surrounded by ovals of the nest is a
separating morphism. Hence $X$ is of type I.

Gabard \cite{Gab06} proved that existence of a separating morphism is
necessary and sufficient condition for a curve being of type I.

Notice that if $f:X\to P^1$ is a separating morphism, then the family
$\{f^*(y)\mid y\in\R P^1\}$ consists of
purely real simple divisors. For any $D\in\{f^*(y)\mid y\in\R P^1\}$, the
image of $\Re\tr:\pi_1(\Def(D))\to H_1(\R X)$ is generated by the class
of a complex orientation of $X$.

\begin{rem}\label{rem4.2.1}
In all examples that I am aware about, if $D$ is a simple purely real
divisor on a real non-singular projective curve $X$ such that the
$\Re\tr:\pi_1(\Def(D))\to H_1(\R X)$ is not trivial, there exists
a separating morphism $f:X\to P^1$ such that $D=f^*(y)$ for some
$y\in\R P^1$. Is this always true?
\end{rem}

\subsection{Looking at the dual curve}\label{s7.5}
In the example of section \ref{s7.2}, the divisors linear equivalent to
$D$ are cut on $X$ by lines. Lines form the projective plane
$\check P^2$ dual to $P^2$.
Thus there is a bijection between $\Leq(D)$ and
$\check P^2$.

 Non-simple divisors in $\Leq(D)$ correspond to
points of the curve $\check X$ dual to $X$. Simple divisor in $\Leq(D)$
correspond to points of its complement $\C\check P^2\sminus\C\check X$.
The set $\Def(D)$ is a connected
component of the complement of  $\R\check X$ in the
$\R\Leq(D)$.  Let us look at the curve $\R\check X$.

\begin{center}
\includegraphics[scale=.23]{M-cubic.pdf}\hspace*{2cm}
\includegraphics[scale=.2]{dual2M-cubic.pdf}\\
M-cubic \hspace*{4cm} the dual curve
\end{center}
$\R\check X$ consists of two connected components. The component $\check
X_1$ dual to the one-sided component $X_1$ has three cusps (which
correspond to inflection points on $X_1$). The component $\check X_0$ dual
to the oval $X_0$ is an oval enclosing $\check X_1$.

Each connected component of  $\R\check P^2\sminus\R\check X$ is $\Def(E)$
for a divisor $E$ corresponding to any of its points.
The non-orientable component of $\R\check P^2\sminus\R\check X$ is $\Def(D)$. It contains
lines (in particular, the infinity-line) which correspond to pencils of line
passing through points encircled by $X_0$. The fundamental group of this
component is $\Z$. Points of this connected
component of $\R\check P^2\sminus\R\check X$ correspond to
purely real divisors each of which consists of two points on $X_0$ and a
point on $X_1$.

The component enclosed in $\check X_1$ consists
of purely real divisors, which are collinear triples of points on $X_1$.
The fundamental group of this component is trivial.

The third component is bounded by both  $\check X_0$ and $\check X_1$.
It consists of divisors with a single real point and two complex conjugate
points. The fundamental group of this component is $\Z$. The image of
real tracing homomorphism is generated by $2[X_1]$.

This example illustrates part (a) of Theorem \ref{Th2}. Indeed, in this
example a real divisor on an
M-cubic curve cut on it by a real line intersecting the one-sided component
in a single point and missing the oval, and the image of $\Re\tr$ is
generated by $2[X_1]$.

\subsection{A generalization of dual curve}\label{s7.6} Let $X$ be a
non-singular real projective curve and $D$ a simple real divisor on $X$.
The variety $\Leq(D)$ of divisors linear equivalent to $D$ is a real
projective space. Its complex points are divisors linear equivalent to $D$.
Non-simple divisors form a real algebraic hypersurface of $\Leq(D)$.
In section \ref{s7.5} this hypersurface was identified with curve
$\check X$ dual to $X$.

In general, when $\dim \Leq(D)>2$, we still have a real projective space
$\R\Leq(D)$ and a real projective variety $\GS\Leq(D)$ of non-simple
divisors linear equivalent to $D$. The complement
$\R\Leq(D)\sminus\R\GS\Leq(D)$ is an open set, its connected components
are the spaces of deformation $\Def(E)$ for real simple divisors $E$
linear equivalent to $D$. Of course, a knowledge on geometry of $\GS\Leq(D)$
may be quite valuable for understanding of $\Def(E)$.

redpajama_set_name: RedPajamaArXiv


text:
Q: how to program if score equals 50 or below minus 2 points? Im trying to get it right just can't seem to get the right line.
if (score is 50 or below) then (-2 from score)

how would I do this correctly - programatically!?

A: Just use a simple if statement...
if(score <= 50)
    score = score - 2;

Quick google search provides some useful docs

redpajama_set_name: RedPajamaStackExchange


Read More Projected stats for the rest of the season as calculated by Steamer, which uses past performance and aging trends to develop a future projection for players.It also uses pitch tracking data to help forecast pitchers. Pitched for the Chatham A's in the Cape Cod League, where he was the recipient of the 2007 BFC Whitehouse Award as the league's top pitcher (6-1, 2.92era)...is, but it's something some of us may have endured due to the sheer curiosity of it…or giving in to the coaxing friend(s)…or you were dragged out of your home, kicking and screaming. To some it's an awful fact, and maybe some people have found success stories through a night of it.I don't judge, and there was interest at one point joining a few (now former) coworkers.Projected stats for the rest of the season as calculated by Steamer, which uses past performance and aging trends to develop a future projection for players.It also uses pitch tracking data to help forecast pitchers.
To show off the new app's potential, Hinge has put together a list of the 40 most eligible people in New York City.Read More Is married.....wife's name is Tina....couple has a daughter named Mia• Has a puggle named Kona and an English bulldog named Harlee... Follow him on Twitter (@Tommy Milone_33) .....enjoys playing golf.Went 3-5 with a 5.71 ERA in 19 games (12 starts) with Minnesota... Went 4-0 with a 1.66 ERA in 7 starts at Triple-A Rochester from 5/10-6/14...And the dating app Hinge, which has gone through some changes of its own, wants to help.


text:
Dan Ashkenazi
German Talmudist and exegete; flourished in the second half of the thirteenth century. Dan, who was one of the most prominent Talmudists of Germany and the teacher of Mordecai ben Hillel, emigrated to Spain toward the end of the thirteenth century, probably in consequence of the cruel persecutions to which the Jews of Germany were subjected at that time, when many were driven to seek asylum in other countries.
In Spain, where he was called "Ashkenazi" (German), he met the foremost rabbinical authorities, who thought highly of him. Dan, however, was so imprudent as to give a letter of recommendation to a youth who pretended to be a prophet (compare Abraham Of Avila); and when the latter turned it to account, Solomon Adret cast scorn upon the German rabbi in his circular letter on the pseudoprophet (Adret's Responsa, No. 548). Even before this occurrence the relations between these two men do not seem to have been very friendly, since Dan declared at Saragossa that, from the strict point of view of the Halakah, there could be no objection to the slaughtering of animals by Christians, as the reason given in the Talmud for forbidding the slaughtering of animals by pagans did not apply to Christians; for the pagan regarded the slaughtering as a sacrifice to his idols, while the same could not be said of the Christians (ib. No. 529; but Dan's reasons are not clearly stated).
Dan, who was a person of much individuality, was misunderstood by many, and acts were ascribed to him which he certainly did not commit (ib. No. 530). Adret's five responsa (Nos. 1229-1233) show that Dan was not a man to be overlooked; Adret's successor as rabbi of Barcelona, Nissim b. Reuben, also recognizes Dan's scholarship (Responsa, No. 32; ed. Rome, p. 72). Yom-Ṭob b. Abraham of Seville (RIṬBA) calls Dan "our teacher" (ib.), although this did not prevent him from writing a pamphlet against Dan regarding their dispute over an important halakic question (RIṬBA to Yibum, 109).
Dan was also very independent as an exegete; the fragments of his exegesis that have been preserved in manuscript, and also in the works of Baḥya ben Asher and in the collection "Hadrat Zeḳenim" (Leghorn, 1840), are highly interesting on account of their rationalism, which was not to be expected from one who had allowed himself to be misled by a false prophet. For instance, he interpreted , in Exodus 23:20, as "messenger," not "angel," and supposed it to refer to Joshua. It is curious to note that in the collection of responsa "Besamim Rosh," which Saul Berlin published as Asher b. Jehiel's work, the statement is made that Dan wrote his Tefillin in Aramaic (No. 24).
Fuenn, Keneset Yisrael;
S. Eppenstein, in Rev. Et. Juives, 37:318;
D. Kaufmann, R. Dan Aschkenasi Exégète, in Rev. Et. Juives, 36:287-291;
Perles, R. Salomo b. Abraham, pp. 9, 63;
Kohn, Mordechai b. Hillel, pp. 37, 38.
Singer, Isidore, Ph.D, Projector and Managing Editor. Entry for 'Dan Ashkenazi'. 1901 The Jewish Encyclopedia. https://www.studylight.org/encyclopedias/eng/tje/d/dan-ashkenazi.html. 1901.
Dan-Jaan

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text:
Public Statement of Stephen Pietzak,
Before the Board of Directors of the
Allegheny Regional Asset District
CLP - "Old" Allegheny Regional Library's Best Site Is Where It Is Located Now --- ARAD FUNDING
I'm Steve Pietzak, 403 Mountain Ave., on the South Side Slopes—and was main Reference Librarian at Allegheny Regional Library for 13 years.
The "Old" Allegheny Regional Library is on the way to ANY location on the North Side. The Old Library has LOCATION, LOCATION, LOCATION, which the new proposed Library site on Federal Street doesn't have.
The "old" Allegheny Regional Library is the " 1st PUBLICLY SUPPORTED ANDREW CARNEGIE LIBRARY IN THE U.S." Any major city would relish the "Cradle of Andrew Carnegie Public Libraries," and preserve, save and enhance this Old Library as a "Historical Time Marker."
IT MAKES GOOD BUSINESS SENSE AND HISTORICAL SENSE TO MAINTAIN AND ENHANCE THE OLD ALLEGHENY LIBRARY INTO A SHOWCASE FOR ALLEGHENY COUNTY, the CITY OF PITTSBURGH, and LIBRARIES.
The LIBRARY'S PRIORITY of BOOK, MAGAZINE AND NEWSPAPER COLLECTIONS HAS BEEN COMPROMISED.
The Carnegie Library will be the Number #1 Source for Information, if the library doesn't DISCARD and SELL its resources. If the Library MAINTAINS its book and magazine collection, the Library will be the #1 Information Source; but these Collections are in danger of being "lost" and may be seriously damaged. The Library is on a PERILOUS COURSE of Disaster, if the Administration constantly tries to eliminate its source of riches, its OLDER COLLECTION of Books and Magazines—its CORE COLLECTION—this is the "Mental Food" of People!
Many Librarians and staff have worried over the last 7 years over the myopicness of the Carnegie Library of Pittsburgh Directors and their top staff. These Directors have been concerned more about Coffee Shops and branch library smallness. Library Collections have shrunk significantly, especially the Science and Technology, Classics, Literature and Music Collections. Over the last 7 years, massive amounts of Books and Bound Magazines have been discarded. Dumpsters have been placed in back of several libraries and books have been discarded wholesale. One customer at Allegheny Library said that she has a better history collection than the library. She told me that she got the books from the dumpster. Another person I talked to from the East Liberty Carnegie Library said they had a dumpster a year or two ago and were throwing massive amounts of books.
On one weekend when I worked at the main Library, I counted 95 boxes of books that were discarded.
The recent two Library Directors seemed to forget the importance of the older collection of books and magazines, "the mental food," both to the Library and to Allegheny County.
Many Librarians and library staff fear for their jobs—and would throw the library collection away if it meant they maintained their job! One branch manager told me " I'm not going to lose my position over this stupid library branch, I'll do what they tell me!"
In the past, the Library Administration withheld over $1 million dollars from the Library Operating Budget from ARAD funds for a Library Capital Campaign----this has crippled many Library Services, has crippled Library Collections Development and crippled Library Facilities and Hours of Service. 2
The Library Administration was giving more ARAD Money to Marketing Staff and Projects, with little money going to Public Library Services Areas of Staffing, Collections Development and Facilities Development.
The Library is on a PERILOUS COURSE of Disaster, if it doesn't maintain its Good Collections and upgrade all of its Public Service Facilities; and doesn't go out and seek funding from Foundations and Private Philanthropy in addition to ARAD funds.
Stephen Pietzak
403 Mountain
epietzak@hotmail.com 10/2/2006
Appendix to Public Statement --
How do you measure the BOOKS and MAGAZINES, the "MENTAL FOOD, of the Library?
A. August Wilson credits his local Carnegie Library for molding his mind---he read the Roman and Greek classics and tragedies—which is reflected in his many plays! How does the library measure this?
B. As a newspaper reporter, Theodore Dreiser read the Roman and Greek Classics at the Carnegie Library Allegheny Library. He wrote the American Tragedy, which is on the Top 100 Novels of all Time Lists. How does the library measure this?
Andrew Carnegie, as a young man, read books at the Colonel Anderson Library on the ore mines of the Great Lakes, on telegraphy, on coal fields and coal mining, on iron and steel manufacturing industries—which helped to form his great steel empire! How does the library measure this?
D. When good Scientific and Patent Collections are discarded and cannibalized--How does the library measure the loss?
E. When Classic Literature Books are discarded wholesale --How does the library measure the loss?
F. When a person reads a self-help book from the library and applies it to his/her life –how does the library measure this?


text:
David Tolan - Music Producer
BPI Awarded Music Producer based @ Hope Mill Recording Studios, Manchester UK
Music Producer, Engineer and Musician
Creative, innovative music production delivered by David at Hope Mill Recording Studios – a first rate recording studio in the North West. A critically acclaimed Music Producer and Musician with a vast wealth of music production, recording and mixing experience.
Credits include :- Johnny Marr, The Charlatans, Tori Amos, Denise Johnson, Primal Scream, Tears for fears and many more..
David delivers first rate music production at his own recording studio in Manchester. He aims to bring out the very best in every artist he works with. Whether building songs from a simple idea or augmenting and refining a band or artist songs, he will get the best out of you!
He has over 25 years experience and has acheived critical acclaim for his production skills over the years.
A highly experienced recording, mixing and mastering engineer – David learned his craft from an early age and has worked in some of the best studios in the UK.
He has a great ear for what sounds good and is not afraid to try new ideas. He has an in depth knowledge of recording, mixing and mastering techniques, and aims to get the best sonic quality on your recording sessions.
David started his career playing drums and has gone on to record and perform with some great artists over the years. Alongside drums, he has developed his skills with bass, guitar, keys and BV's too.
His knowledge and skills as a musician gives him a great understanding of what it's like to be on 'the other side of glass' which is beneficial in understanding a musicians needs and helping them feel at home in the studio.
What to expect…
Working With David
Inspiring and creative Music Production!
Before embarking on any work, David likes to get to know more about the artist and what records that have inspired them. What music they love? and what they're listening to at present. He likes to build up a picture of what the artist is about and what they are aiming for which helps create more seamless and symbiotic productions.
A CLEAR PLAN
All productions will be set out with a clear plan and budget, which is tailored for an artists specific needs, whether it be a single, e.p or album.
ENTHUASIASM & DRIVE
There'll be lots of enthusiasm and drive on your sessions, time to experiment with ideas and some fun along the way too!
A GREAT STUDIO
Sessions will take place at Hope Mill Recording Studios in Manchester – a great studio for Pre Production, Recording and Mixing.
GREAT SOUNDING RESULTS!
David always aims for sonic excellence and delivers first rate productions, as his previous track record shows.
Discograghy
Here's a selection of Artists David's worked with
Don't take our word for it – here's what David's clients say:
I have known Dave Tolan for a long time, in fact back to my days managing The Joy. This guy wins awards-which means Dave sets standards on excellence that most can only aspire to. You can most definitely learn from this guy. A major thing that sets him apart from others is his drive to take his successes further. You can most certainly learn from that attitude! I whole hardheartedly support him and can highly recommend him as a first rate producer.
Brian Shepherd
Head of A&R EMI records 1978-1980
Managing Director Phonogram records 1980-1985
Managing Director A&M Records 1985-1990
Dave produces passionately with every artist he works with, his aim is to get the best out of every individual so that they perform above the ability they currently have.
More importantly for me, he really knows how to capture a great vocal and make it come alive in his recordings
. His production knowledge and skill means that the finished track is always of the highest standard"
Primal Scream, Electronic,
A Certain Ratio, Johnny Marr, Gay Dad, Hypnotone, Micheal Hutchinson
Dave always makes fantastic sounding records. As a producer his determination and dedication runs from pre production right through to the finished Master.
As a Musician he has helped me out on my own productions over the years, adding great drums to songs. His knowledge as a producer means his playing is perfect for the production and he always brings something special to the table.
Jim Spencer
Johnny Marr, Charlatans, The Cribs, New Order, The Vaccines
Keep up to date with David's latest news
Please feel free to contact me using the form below
Hello, please use the form below in order to get in touch!


text:
Gaza militants fire 12 rockets, Israel strikes Hamas targets
by: JOSEPH KRAUSS, Associated Press
Posted: Aug 20, 2020 / 11:46 PM PDT / Updated: Aug 21, 2020 / 12:36 PM PDT
Palestinian residents walk in the Shati refugee camp in west Gaza City as the coastal enclave gets maximum four hours of electricity a day, Wednesday, Aug. 19, 2020. (AP Photo/Hatem Moussa)
JERUSALEM (AP) — Palestinian militants fired 12 rockets at Israel from the Gaza Strip overnight, nine of which were intercepted, and Israel responded with three airstrikes on targets linked to the territory's militant Hamas rulers, the Israeli military said early Friday.
Friday evening, the military added that the militants fired at least one more projectile that was also intercepted.
It was the most serious exchange of fire along the Gaza frontier in months, but there were no reports of casualties. Police said buildings and vehicles in Israel were damaged, and that bomb-disposal units had been dispatched to pick up shrapnel and rocket parts.
In recent weeks, groups affiliated with Hamas have launched incendiary balloons into Israel, igniting farmland in a bid to pressure Israel to ease the blockade it imposed on Gaza when the Islamic militants seized power in 2007. The rocket fire marks a significant escalation.
Israel and Hamas have fought three wars and several smaller battles over the last 13 years. Neither side is believed to be seeking war, but any casualties could ignite a wider conflict.
After a meeting with the military chiefs to assess the situation, Israeli Defense Minister Benny Gantz threatened Hamas with a serious blow. He said the army "is prepared, is protecting and will continue to protect the people of the south, and will attack the attackers in turn, inflicting serious damage."
"We will not stand by while Hamas is out of control," Israeli President Reuven Rivlin said in a statement. The military "will respond with force and determination, sortie after sortie, and will continue even if it requires patience and time."
Israel has closed Gaza's only commercial crossing, causing the territory's sole power plant to shut down for lack of fuel and limiting the territory's 2 million residents to around four hours of electricity a day. Israel has also banned fishing in Gaza's coastal waters, measures it says are in response to the incendiary balloons.
Egyptian mediators were in Gaza earlier this week to try and shore up an informal truce but left without announcing any progress. Israel has allowed the Gulf nation of Qatar to provide hundreds of millions of dollars in aid to Gaza in recent years to keep the economy from collapsing and preserve calm.
"We will not allow the enemy to continue the unjust siege on our people, who have the right to express their rejection to this siege by all means," Hamas and other Palestinian armed groups said in a joint statement.
Describing the incendiary balloons as "peaceful tools" of resistance, the factions said they "responded and will respond" to any Israeli airstrikes on militant sites.
The latest exchange began late Thursday when militants fired two rockets that landed near the security fence. A few hours later, a volley of three rockets was intercepted by Israeli missile defenses. Another seven rockets were fired early Friday, six of which were shot down.
The Israeli military said it carried out airstrikes targeting Hamas military infrastructure, including a compound used to manufacture rocket ammunition, in response.
It was the most serious cross-border exchange since February, when the smaller Islamic Jihad militant group fired around 80 rockets into Israel after one of its fighters was killed near the border while allegedly planting explosives. Israel struck dozens of targets across the territory.
Associated Press writer Fares Akram in Gaza City, Gaza Strip contributed.


text:
The Kleist Prize is an annual German literature prize. The prize was first awarded in 1912, on the occasion of the hundredth anniversary of the death of Heinrich von Kleist. The Kleist Prize was the most important literary award of the Weimar Republic, but was discontinued in 1933.

In 1985 the prize was awarded for the first time in over fifty years. Between 1994 and 2000 it was awarded biennially. A monetary sum of €20,000 accompanies the award.

Winners
Listings of the Kleist Prize winners are maintained by the Kleist-Archiv Sembdner. and Kleist Gesellschaft

1912–1932

1912 Hermann Burte and Reinhard Sorge
1913 Hermann Essig and Oskar Loerke
1914 Fritz von Unruh and Hermann Essig
1915 Robert Michel and Arnold Zweig
1916 Agnes Miegel and Heinrich Lersch
1917 Walter Hasenclever
1918 Leonhard Frank and Paul Zech
1919 Anton Dietzenschmidt and Kurt Heynicke
1920 Hans Henny Jahnn
1921 Paul Gurk
1922 Bertolt Brecht
1923 Wilhelm Lehmann and Robert Musil
1924 Ernst Barlach
1925 Carl Zuckmayer
1926 Alexander Lernet-Holenia and Alfred Neumann (Rahel Sanzara did not accept her prize), Honorable Mention: Martin Kessel
1927 Gerhard Menzel and Hans Meisel
1928 Anna Seghers
1929 Alfred Brust and Eduard Reinacher
1930 Reinhard Goering
1931 Ödön von Horvath and Erik Reger
1932 Richard Billinger and Else Lasker-Schüler

1933–1984
Discontinued

1985–present

1985 Alexander Kluge for Die Macht der Gefühle
1986 Diana Kempf for Der Wanderer
1987 Thomas Brasch for Robert, ich, Fastnacht und die anderen
1988 Ulrich Horstmann
1989 Ernst Augustin for Der amerikanische Traum
1990 Heiner Müller for Hamletmaschine
1991 Gaston Salvatore for Lektionen der Finsternis
1992 Monika Maron for Stille Zeile Sechs
1993 Ernst Jandl for life works
1994 Herta Müller for Herztier
1996 Hans Joachim Schädlich for Der Kuckuck und die Nachtigall
1998 Dirk von Petersdorff for Wie es weitergeht & Zeitlösung
2000 Barbara Honigmann for Alles, alles Liebe!
2001 Judith Hermann for Sommerhaus, später
2002 Martin Mosebach for Eine lange Nacht, Der Nebenfürst & Das Grab der Pulcinellen
2003 Albert Ostermaier for Vatersprache
2004 Emine Sevgi Özdamar for Seltsame Sterne starren zur Erde. Wedding – Pankow 1976/77
2005 Gert Jonke for Seltsame Sache & Die versunkene Kathedrale
2006 Daniel Kehlmann for Ich und Kaminski & Die Vermessung der Welt
2007 Wilhelm Genazino for Mittelmäßiges Heimweh
2008 Max Goldt for life works
2009 Arnold Stadler (destined by Péter Esterházy) for Einmal auf der Welt. Und dann so
2010 Ferdinand von Schirach for Verbrechen
2011 Sibylle Lewitscharoff for Blumenberg
2012 Navid Kermani for life works
2013 Katja Lange-Müller for life works
2015 Monika Rinck
2016 Yoko Tawada
2017 Ralf Rothmann
2018 Christoph Ransmayr
2019 Ilma Rakusa
2020 Clemens J. Setz
2021 not awarded
2022 Esther Kinsky

See also
 German literature
 List of literary awards
 List of years in literature
 List of years in poetry

References

External links
 Heinrich von Kleist Gesellschaft

Heinrich von Kleist
German literary awards

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text:
The Golf Cart Is Here for Good
It brought the game to the elderly and infirm, and at Palm Springs it has started a whole new way of life
Alfred Wright
I can't be absolutely sure about the date, but I think I saw a golf cart for the first time somewhere around 1952 at the Burlingame Country Club near San Francisco. At the time it seemed to be worthy of immediate applause. It made 18 holes of golf possible for a gallant and impetuous friend of mine who had left most of one of his legs on a beach in the Marianas nearly 10 years before. This fellow had been a brilliant athlete in school and college, and it was heart-warming to discover that he could once again use his coordination and grace on the golf course. There were qualifications to my enthusiasm, however. On our way around the course I had a few short rides with Ferdy in his new cart, and the way he drove it down gullies and across bridges, I'd just as lief have been with him on that beach at Saipan.
Not too long after that we all learned how the cart was making it possible for President Eisenhower to resume his golfing after his first heart attack. That was another merit badge for carting, bringing home what a boon this invention would be to those who could still play the game but, for one reason or another, couldn't walk the full 18 holes. I also remember how much criticism we heard at the time on the subject of the President's golfing, and not all of it came from the political type who just spouts to get his name in the paper. I think a lot of people felt, as I did, that 18 holes of golf was too much for a man who had experienced a heart disturbance—that is, if he were going to have any energy left over for his job. But, as I say, I knew nothing about golf carts, and though I knew that it must be easier to ride around the course than to walk it, it seemed to me that 18 holes of golf was still a pretty rough grind for a semi-invalid.
Those of us who must live and play our golf along the northern reaches of the eastern seaboard are accustomed to a somewhat Spartan approach to the game, part of which involves walking around the course on one's own two pins. At one club where I play there is an elderly gentleman—I should judge him to be pushing 80—who trots through 18 holes carrying his own clubs summer and winter, even when the ground is frozen and the course otherwise abandoned, for reasons that have nothing to do with thrift. At another club I visit now and then there is an extremely wealthy lady in her middle 70s who does more or less likewise. In the face of such examples, no self-respecting golfer would dare ride around in a cart as long as he was sound of limb and body.
So it was with some surprise that I discovered golf carts to be as common as citrus fruit when I visited Florida a season or two ago. Younger golfers in their 30s and 40s rode them without apology and even used them—not without protest—during informal tournaments. A set of brothers with no good reason not to walk other than an aversion to unnecessary exercise had even installed walkie-talkie radio sets in their carts so they could communicate across the course with one another when they weren't playing in the same match. You might say that at that point the golf cart had arrived at an advanced state of decadence almost before it had reached maturity.
It is no doubt appropriate that, since the golf cart has been slowly coming into its own—and even creeping inexorably into the hardier climates of the North—it should attain full flower in Palm Springs, that burgeoning California desert resort area that refers to itself as The Winter Golf Capital of the World. With 11 courses serving communities whose populations total only 30,000 permanent residents, the desert enterprisers have truly done more than any other spot in the country to bring what was once a rich man's game to Everyman. When Everyman gets into the act he usually manages to streamline an activity and adapt it to the pace of the times while ignoring the surly mutterings of the traditionalists. "The oldtimers may think they can avoid it," says Jimmy Hines, one of the veteran pros, "but the cart is here to stay. It's got to stay. Just think what it's done for golf."
Hines today is the vice-president and part owner of a country club that may be to golf what gunpowder was to warfare. At Eldorado, in Palm Desert—and before that, down the road at Thunderbird—Hines and his friends in the West have built a golf world that is so completely mechanized that the resort might more aptly refer to itself as The Golf Cart Capital of the World. As of last week the 11 courses were operating 729 golf carts among them, and it is a rare golfer indeed who will set out on one of those courses on foot. In fact, if he does he is likely to bring a few frowns and protests for jamming the traffic.
WHITHER THE CART?
This saturation of golf by the golf cart raises an inevitable question: What is the cart doing to golf? Since it obviously is no longer simply a Samaritan to the aged and the infirm, the cart must obviously serve some important functional role. Does it, for instance, speed up the game? Does it make golf physically less demanding and thereby improve any given golfer's ability? Does it make golf more fun? You can get a yes or no to any of these questions, but everyone around Palm Springs, whether approving of carts or not, seems agreed that they represent progress. Claude Harmon, who began his new duties as the head pro at Thunderbird in October, expresses the philosophy of the golf cart protagonists with this question: "Would you walk to the market to do your shopping when you could get in your car and drive there?"
As far as the best memories around Palm Springs can reconstruct it, the first golf cart made its appearance there about nine years ago, coinciding with the beginning of the golfing boom in the area. It was a walloping four-wheeled, six-passenger vehicle which had been brought West by a prosperous oilman from Houston named D. B. McDaniels. Its two-cycle gasoline engine chugged and wheezed and spat noxious fumes and shattered the nerves of virtually every golfer on the Thunderbird course except its owner. However, Palm Springs is a youthful, aggressive sort of place, patronized by energetic and successful business people from all across the country who refuse to turn their backs to the future, and many of them were quick to see prospects in this monster. The pros in the golf shop were soon in touch with a Long Beach firm that produced electric carts for invalids, and before long a quiet, electric-motored golf cart was in the making. The outcome of this empirical period was a two-passenger cart called the Autoette, and except for refinements, it varied little from the majority of carts in operation today. The first Autoettes were put into use at Thunderbird during the 1951—52 winter, and that could be called the Year One in the era of the cart.
It should be noted that Palm Springs has a problem—if it can be called a problem—that is endemic among expensive resorts. A high percentage of its best customers have made their nest eggs and are now taking it easy. If these citizens can be mounted, their patronage offers a wonderful source of revenue for country clubs such as those at Palm Springs, where the golf courses have to be raised on an arid desert that previously supported nothing thirstier than sagebrush and now require upward of $100,000 a year for items of maintenance such as 500,000 to a million gallons of water a day.
The elderly people of Palm Springs were understandably delighted with the carts, and it wasn't long before their enthusiasm was shared by many of their able-bodied juniors who went around explaining that that way they could play more golf in a day. Nowadays, with the cart as common as it is in Palm Springs, it is taken as a matter of course that a player will ride one and no explanations are needed. Claude Harmon put the case for the cart as well as anyone when he said, "The most important element in a person's life today is time. Every hour you spend on golf you must take away from something else. It may take as much as four and a half to five hours to play a full round of golf on a crowded course. If everyone uses carts you can shave this by an hour and a quarter or an hour and a half. You can even play 36 holes in one day if you want."
By as early as 1955, the cart population of Palm Springs was in the hundreds. Local residents still note with pride that there were 206 carts following the Ryder Cup matches at Thunderbird that year, and hardly a soul who was there will ever forget the awesome sight of this motorized audience swarming down a fairway like one of Hitler's Panzer divisions invading Poland. So far, no one has seriously disputed the claim that this was the world record for the largest number of golf carts ever operated on one course at one time, but it must be remembered that golf carting is so new that its statistics are kept rather haphazardly.
Inasmuch as everyone in Palm Springs insists that the golf cart is here to stay, traditionalists and those who are still devoted to the ideal of physical fitness might just as well stop fretting and look at the bright side of the picture. For instance, golf carts make money. Although privately owned carts account for nearly half of those in the Palm Springs area, most courses are not partial to them, and some, like the new Indian Wells Club and the new public course, would like to ban them completely. For the fact of the matter is that each golf cart in the rental fleet at a course is capable of earning as much as $700 during a five-to six-month season. It works like this. The going rate for a cart rental for 18 holes is $7 or $7.50. A course that knows its prospective trade and thus is not overstocked with carts can figure to rent each of them at least 150 times a year for a gross of better than $1,000 per cart. A good golf cart nowadays costs about $1,200 new and has a life expectancy of five years if properly maintained During that time it will need only one new set of the six-volt, 170-ampere batteries that power its one-to 1½-horsepower motor (the batteries must be recharged after every round of golf at a cost of about 56 for juice). A good cartman is capable of maintaining and servicing about 50 carts, and his wages should run to about $400 a month—or roughly 30¢ a day per cart. Adding it all up, a cart will gross around $5,000 during its lifetime, and it will cost about $1,500 to purchase and maintain it during that period. Hence the profit of $700 a year. Anyone can quickly see what this means to a club like Tamarisk, whose 61 rental carts represent the biggest rental fleet in Palm Springs.
57 VARIETIES
However, it is only natural in the world of cartifacts that plenty of people should want to own their own. Man's pride in his carriage, which is almost as old as the wheel itself, has a pronounced manifestation in the golf cart—and particularly at Thunderbird, where there are 165 privately owned carts housed in the Thunder-bird Buggy Stable, a wood and cinder-block shack out behind the clubhouse kitchen. There you can see enough cart varieties to satisfy a Heinz. Two partners in a Plymouth agency have fashioned special rear ends on their carts to resemble the hind end of a Plymouth Fury—tail fins, spare-tire housing and all. Many private-cart owners favor gaggy nicknames like Lagniappe or Jambon, which they paint on the front or rear of the machine. One owner, getting right to the heart of the matter, has painted on his cart: "Let's Play Faster!!!" Bandleader Phil Harris and his favorite golfing partner, Milt Hicks, had a built-in bar and ice chest, to say nothing of a radio, in the cart they owned jointly just prior to their present one; also, a bottle of bourbon was painted on the side where Harris liked to sit and a bottle of gin on Hicks's side. Everyone who knows them got the point. Most private carts, however, simply settle for some pedestrian identification like the owner's name or nickname or initials.
The brand names of the carts are legion—Autoette, Turf Rider, Golf-mobile, Golf Pony, Marketeer and Taylorcar, to name a few—but their variations are not conspicuous. Most of them seat two people side by side and have an open compartment behind the seat for carrying two golf bags. They generally have one floor pedal for a single-speed throttle, another for the brake, and a hand lever for switching from forward to reverse. Just about all of them have a tiller for steering the single front wheel and are geared to travel about 8 mph. Above and beyond the basic cart it is possible to get extras worth hundreds of dollars, such as a radio, a top, a collapsible top, a top with a white-tasseled fringe, a cigarette lighter, headlights, chromium trimmings, a windshield to fend off bugs and the cold evening air, and terry-cloth seat covers that won't overheat in the sun.
Naturally, the private-cart owner cannot be sure he will always be playing on the same golf course, particularly at Palm Springs, where it is not unusual for one person to belong to several golf clubs within a few miles of one another. So some cart owners have cart trailers they can attach to the rear of their automobiles. Then there is the status-happy type who will maintain a cart at each of his clubs; and perhaps another one at home to take him back and forth to the golf house if, like so many of the new Palm Springs resorters, his house is built somewhere on the club premises. A man named McCulloch, who once manufactured a very fine singleseater cart but abandoned the project when he found he couldn't sell it for less than $800, actually keeps five of these little carts at one of his clubs and four at another for the convenience of his guests.
The single most famous cart now in existence belongs to President Eisenhower. It is a Turf Rider IV, a $1,385 fiber-glass job which the experts almost unanimously rate as the Cadillac of golf carts. All the Palm Springs golf clubs combined to give this cart to Eisenhower while he was vacationing among them last October. It has a cigarette lighter and a radio and a fringed top, and eventually inscribed in dignified lettering on the front are the words "Our President." Ike liked it so much he took it home to Gettysburg with him.
A golfer who finds himself playing in a cart for the first time may discover that for all its comfort it is a mixed blessing. For one thing, a cart lends itself to irrelevant social chatter, and before he knows it the golfer has arrived at his ball and is about to hit a shot to which he hasn't given sufficient advance thought; in other words, his concentration is likely to suffer grievously. Another problem is the meandering cart driver who will dump you off some distance from your ball and then head for his own while you try to decide whether to hit your shot with the wrong club or delay the whole foursome by testily demanding that he bring your clubs back. Nor are injuries unknown to cart riders. Ankles have been instantly snapped when a careless rider has let his spikes trail outside the cart, and rattled drivers have been known to plunge into ravines and come up badly bloodied. Many clubs now demand that carters sign a form releasing the club from all liability, and in at least one case where these forms have been broadly and carelessly drawn up, the carters out playing golf could be held legally liable for almost any catastrophe that happened to anyone within five miles of the clubhouse.
Despite the tremendous success of golf carts on the desert courses of Palm Springs, it would be premature to say that they are ready to take on full-time duty everywhere. Many courses, particularly those in areas of heavy rainfall, would be seriously scarred by a fleet of carts scampering along their fairways every day. But such problems are not insoluble. Right now the cart industry is working on a broad, low-pressure tire with a tread of 12 inches or so—a principle similar to that used by the swamp buggy—and presumably this cart will be able to go anywhere, including through sand traps and across greens without causing the slightest damage. Nobody, however, has yet come up with an idea for a foolproof driver.
Nevertheless, heed the words of Claude Harmon: "The golf cart is a terrific thing for golf. It's brought the game to people who could never play otherwise. It means more people can play more golf in a shorter time. It means the average fellow who isn't in the best of shape can play 18 holes and not come back so exhausted he's no more good to anybody that day. Most people get too much exercise playing 18 holes of golf anyway. I tell you the cart is here for good, and a darn good thing it is, too. Believe me, I never want to walk again when I can ride."
GLAMOUR GAG is a Palm Springs trademark, so it is used here to dramatize some happy assets of the golf cart capital.
BING CROSBY HELPED POPULARIZE CARTS
JIM HAGERTY HITCHES A RIDE WITH IKE
GROUCHO MARX LETS A FRIEND DO THE DRIVING
CART OF GENERAL BRADLEY (THIRD FROM LEFT) HAS FIVE STARS
CROONER DEAN MARTIN RESTS EASILY WHILE MILES BRADY PLAYS SHOT
DANNY THOMAS SPORTS HEADLIGHT ON CART FOR LATE DRIVES
IKE'S NEW CART, A FIBER-GLASS JOB, WAS PRESENTED BY ADMIRING PALM SPRINGERS