# Ueber die algebraischen Curven von den Geschlechtern $p=4,\,5$ und $6$,
welche eindeutige Transformationen in sich besitze

Written by A. Wiman.
Translated by Linden Disney-Hogg,
Andrew Beckett, and Isabella Deutsch

(09/10/1895)

### Preface from the translators

This is a translation of the Wiman paper,

Wiman, Anders. “Ueber die algebraischen Curven von den Geschlechtern $p=4,\,5$
und $6$, welche eindeutigen Transformationen in sich besitzen." Bihang till
Kongl. Svenska vetenskaps-akademiens handlingar 21 (1895): 1-41.

which translates as

On the algebraic curves of genus $p=4,\,5$ and $6$, which possess unambiguous
transformations into themselves,

an influential paper written in German.
Care was taken to try and faithfully recreate the formatting of the original
article, but this was not always possible. In contrast to the original,
footnotes are numbered continuously throughout the document. Margin notes are
included to anchor the pages of the original document to this translation.
LDH performed the original transcription and translation of the paper, and
maintains the GitHub copy where errata may be reported. AB and ID contributed
equally in refining the translation and providing proofreading.
The translators would like to thank the Royal Swedish Academy of Sciences and
the Institut Mittag-Leffler for their permission to produce this English
translation. This work is licensed under a Creative Commons Attribution-
ShareAlike 4.0 International License.

## The Paper

### Introduction

The present treatise is intended to be a continuation of a work I have already
published,111Bihang till K. Vet.Akad. Handlingar, Vol. 21, Section. I. No. 1.
which, among other things, treated the unambiguous correspondences on the non-
hyperelliptic curves of genus $p=3$. For the sake of convenience, we take the
following equation from it, which was constructed using a theorem given by Mr.
ZEUTHEN:

(A) $2(p-1)=2n(p^{\prime}-1)+\sum\frac{n}{n_{i}}(n_{i}-1)\,.$

Here, on a curve of genus $p$, there should be an unambiguous correspondence
of period $n$; the summation is over the coincidences where $n$ related points
coincide in $\frac{n}{n_{i}}$ by $n_{i}$; finally, $p^{\prime}$ denotes the
genus of a curve, the points of which are in a one-to-one relationship with
the point groups of $n$ points that belong together through the
correspondence.222Refer once again to a work by Mr. Hurwitz (Math. Ann., Vol.
32, p. 290).
Since we refrain from the hyperelliptic case, which is easy to handle, we can
bring the relevant structures to the projectively invariant normal curve
$C_{2p-2}$ in the $(p-1)$-dimensional space, whose coordinates are
proportional to the ABELian integrands of the first kind. From this arises the
advantage for us that the unambiguous correspondences are represented by
collineations of the curve.333See the 5th section of my previously-mentioned
work.

###  1 p.4

††margin:  start p.4

For $p=4$ we get a normal curve of the sixth order in ordinary three-
dimensional space. This curve must be located on a surface of degree two
$F_{2}$, which, however, can also degenerate into a cone $K_{2}$.444Noted in
an essay by Mr. WEBER, Math. Ann., Vol. 13, p. 35. The generators of this
surface are formed by the trisecants of the curve.
CAYLEY has given a general theory for the singularities of a twisted
curve,555See the related presentation by SALMON-FIEDLER, Analytische Geometrie
des Raumes II, 3. Edition, p. 105., and we do not find it inappropriate to
quote some of his results for our case, since the characteristic numbers of
the invariant normal curve also designate properties of the general algebraic
structures of the relevant genus.
The $C_{6}$ mentioned has no true double points or cusps, since they can never
occur on the $\varphi$-normal curve. The only possible relevant singularities
are stationary tangents $\Theta$ and double osculating planes $\Delta$; but
those must be generators of the surface $F_{2}$, because every other degree
has at most two points with $F_{2}$, and thus with the curve, in common. We
shall find that these singularities appear in connection with the
collineations of the curve.

  * Furthermore, let $m$ be the order of the curve;

  * $r$ the order of the developable surface formed by the tangents or the class of the perspective cone (the rank of the curve);

  * $n$ the class of the curve;

  * $\alpha$ the number of stationary planes;

  * $h$ the number of apparent double points;

  * $g$ the number of lines in two osculating planes which lie in a given plane;

  * $x$ the order of the double curve;

  * $y$ the class of the double-touching developable surface.

Two systems of equations are obtained, the first by using PLÜCKER’s formulas
for an arbitrary perspective cone over the given curve. For the order, class,
double points, cusps, inflexions, double tangents one obtains the numbers $m$,
$r$, $h$, $o$, $n+\Theta$, $y$ respectively. So given ††margin:  start p.5

$m=6\,,\quad p=4$

in advance, one finds,

$r=18\,,\quad h=6\,,\quad n=36-\Theta\,,\quad y=96\,.$

So consider the cross-sectional curve of the developable surface. Here are the
PLÜCKER character numbers: $r$, $n$, $x$, $m+\Theta$, $\alpha$, $g+\Delta$.
Thus

$x=126-\Theta\,,\quad\alpha=60-2\Theta\,,\quad
g=531-\frac{65}{2}\Theta+\frac{\Theta^{2}}{2}-\Delta\,.$

The points of contact of the $\alpha$-planes have a special meaning as those
points of the structure of genus $p$ (4), in which a unique algebraic function
of lower than $(p+1)$th (5th) order becomes infinite without becoming infinite
in other places. These places have been considered by many authors and given
for their number $(p-1)p(p+1)$, so in our case 60; this is also confirmed if
only the $\Theta$-points are calculated twice.
The developable surface of order 18 and surface $F_{2}$ intersect in a curve
of order 36; this can only be calculated twice from the cuspidal curve and
consist of a number of common generators. Each generating system of $F_{2}$
delivers 12 of these, which can also be seen from the fact that each
$F_{2}$-generator has to divide the developable into 18 points, namely three
double points on the curve and twelve common generators of the other system.
Two of these 24 common straight lines move together in a stationary tangent
that may appear, which gives the upper value 12 for $\Theta$, which is
actually reached.
Meanwhile we have found 2 groups of points with projective invariant
properties, which under a collineation of the curve can only be interchanged
with each other: the 60 contact points of the stationary planes and the 24
contact points of the common generators. In addition, one could take the 24
points that cut the latter straight line from the curve. But this does not
give a new group in two actually occurring cases: if all common generators are
stationary tangents ††margin:  start p.6 or if all tangents in the above-
mentioned cut-out points yield generators of the other $F_{2}$-system.
CAYLEY also treated other singularities by means of the theory of reciprocal
surfaces.666See, for example, the presentation given by SALMON-FIELDER (Ibid
p. 660). Let us only give the following results. Let:

  * $\gamma^{\prime}$ be the number of osculating planes which also contain a tangent;

  * $t$ the number of points through which three tangents pass;

  * $t^{\prime}$ the number of planes containing three tangents.

The following formulas apply here if, of the relevant singularities, only
$\Theta$ and $\Delta$ occur:

$\displaystyle\gamma^{\prime}$
$\displaystyle=rn+12r-14n-6m-8\Theta-4\Delta\,;$ $\displaystyle t$
$\displaystyle=\frac{1}{6}\left(r^{3}-3r^{2}-58r-3r\left(n+3m+3\Theta\right)+42n+78m+78\Theta\right)\,;$
$\displaystyle t^{\prime}$
$\displaystyle=\frac{1}{6}\left(r^{3}-3r^{2}-58r-3r\left(m+3n+3\Theta\right)+42m+78n+78\Theta\right)\,,$

or in our case:

$\gamma^{\prime}=324-12\Theta-4\Delta\,;\quad t=480-12\Theta\,;\quad
t^{\prime}=120\,.$

Here we have found new invariant point groups. The $t^{\prime}$ in particular
have attracted attention in the general theory of curves, namely in the
question of the adjoint $\varphi$-curves, which each touch the base curve
simply at $p-1$ points.777See CLEBSCH-LINDEMANN, Vorlesungen über Geometrie I,
p. 847. For the number of solutions one has $2^{p-1}(2^{p}-1)$, which for
$p=4$ agrees with the one already obtained.

###  2 p.6

The collineations of $C_{6}$ into itself must also transform the $F_{2}$ into
itself. The two systems of generators can either be transformed into
themselves or exchanged with one another. A special transformation can also
occur which leaves all generators of a species individually fixed; two points
on each are fixed, which together satisfy two fixed generators of the other
kind, and the transformation has a period of three, because the points of the
curve on the fixed generators must be cyclically exchanged; the six points on
the two generators††margin:  start p.7 of the other kind mentioned remain
fixed and must have stationary tangents.
It can be seen from this that no new simple group can appear in the
transformation groups of a structure of genus $p=4$. Either the generating
systems can be exchanged, and then half of the operations yield a normal
subgroup which transforms each into itself, or, if exchange does not occur,
the group is holohedrally isomorphic with one of the known finite binary
groups, unless that every generator of a kind is transformed into itself by a
cyclic $G_{3}$, but where this $G_{3}$ must have the normal property. (The
latter also applies to the case where the surface $F_{2}$ merges into a cone,
and so the two systems coincide.)
If one uniquely assigns the ratios $x_{1}:x_{2}$ or $y_{1}:y_{2}$ to the two
sets to be generated, then it is initially evident that $C_{6}$ can be found
by a homogeneous equation $F(x_{1},x_{2},y_{1},y_{2})=0$, which is degree 3 in
both $x_{i}$ and $y_{i}$. This parameter setting is achieved by choosing
$xz=yw$ for the equation of $F_{2}$ and then using

(1) $x:y:z:w=x_{2}y_{1}:x_{1}y_{1}:x_{1}y_{2}:x_{2}y_{2}$

to determine the ratios $x_{1}:x_{2}$ and $y_{1}:y_{2}$. From the point
$x=y=z=0$ or $x_{1}=y_{1}=0$ one can now project the area and thus the curve
onto the plane $w=0$, whereby the image curve must satisfy the equation
$F(y,x,y,z)=0$.888See for example SALMON-FIELDER Ibid p. 521 or CLEBSCH-
LINDEMANN, Vorlesungen über Geometrie II 1, p. 422.
A collineation of the first kind can be replaced with two binary substitutions
as far as the area is concerned:

$\displaystyle x_{1}^{\prime}$ $\displaystyle=a_{11}x_{1}+a_{12}x_{2}\,,\quad
x_{2}^{\prime}=a_{21}x_{1}+a_{22}x_{2}\,;$ $\displaystyle y_{1}^{\prime}$
$\displaystyle=b_{11}y_{1}+b_{12}y_{2}\,,\quad
y_{2}^{\prime}=b_{21}y_{1}+b_{22}y_{2}\,.$

Two generators of each kind merge into one another, and their 4 points of
intersection remain fixed, but if one substitution is identical, all points of
the fixed generators††margin:  start p.8 of the other kind remain fixed. If
one chooses the basic elements,999It is well known that coincident basic
elements do not occur in finite groups. one gets the simpler form:

(2) $x_{1}^{\prime}=a_{1}x_{1}\,,\quad x_{2}^{\prime}=a_{2}x_{2}\,,\quad
y_{1}^{\prime}=b_{1}y_{1}\,,\quad y_{2}^{\prime}=b_{2}y_{2}\,.$

The collineation in space is now determined from (1) and (2):

(3) $x^{\prime}=a_{2}b_{1}x_{1}\,,\quad y^{\prime}=a_{1}b_{1}y\,,\quad
z^{\prime}=a_{1}b_{2}z\,,\quad w^{\prime}=a_{2}b_{2}w\,.$

In a collineation of the second kind, where the generating sets are exchanged,
one can initially start from substitutions of the following form:

$\displaystyle x_{1}^{\prime}$
$\displaystyle=\alpha_{11}y_{1}+\alpha_{12}y_{2}\,,\quad
x_{2}^{\prime}=\alpha_{21}y_{1}+\alpha_{22}y_{2}\,;$ $\displaystyle
y_{1}^{\prime}$ $\displaystyle=\beta_{11}x_{1}+\beta_{12}x_{2}\,,\quad
y_{2}^{\prime}=\beta_{21}x_{1}+\beta_{22}x_{2}\,.$

It is easy to see that through the requirements:

$x_{1}^{\prime}:x_{2}^{\prime}=x_{1}:x_{2}\,,\quad
y_{1}^{\prime}:y_{2}^{\prime}=y_{1}:y_{2}\,,$

determine two points fixed in the collineation on the surface, provided these
conditions are not met in an infinite number of points. We choose generators
through fixed points as basic elements and then get collineations:

(4) $x_{1}^{\prime}=\alpha_{1}y_{1}\,,\quad
x_{2}^{\prime}=\alpha_{2}y_{2}\,,\quad y_{1}^{\prime}=\beta_{1}x_{1}\,,\quad
y_{2}^{\prime}=\beta_{2}x_{2}\,.$

The collineation of space now results from (1) and (4):

(5) $x^{\prime}=\alpha_{2}\beta_{1}z\,,\quad
y^{\prime}=\alpha_{1}\beta_{1}y\,,\quad
z^{\prime}=\alpha_{1}\beta_{2}x\,,\quad w^{\prime}=\alpha_{2}\beta_{2}w\,.$

If this operation is repeated, a collineation of the first kind is obtained:

(6) $x^{\prime}=\alpha_{1}\alpha_{2}\beta_{1}\beta_{2}x\,,\quad
y^{\prime}=\alpha_{1}^{2}\beta_{1}^{2}y\,,\quad
z^{\prime}=\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}z\,,\quad
w^{\prime}=\alpha_{2}^{2}\beta_{2}^{2}w\,,$

but not an arbitrary one, since all points of the straight line $y=w=0$ remain
fixed.
If we also emphasise that the equation (A) for $p=4$ only allows the prime
number solutions 2, 3, 5, then we now have the means to identify the possible
collineation groups of a structure of genus $p=4$ as quickly as possible.

###  3 p.9

††margin:  start p.9

We now want to visit these collineations. The equations of the structures in
question are written with the smallest possible number of constants, which
means that the arbitrary moduli are also given.
For a cyclic group of period 2, either the generators can be swapped or not.
In the first case one gets an infinite number of fixed points, the
intersections of two mutually exchanging generators. These form a conic
section, where the pole of the plane of the conic section with respect to the
$F_{2}$ provides the centre of perspective. The equation of the curve is
obtained by choosing two fixed points on the curve for $x_{1}=y_{1}=0$ and
$x_{2}=y_{2}=0$ and the simultaneous interchangeability of $x_{1}$, $y_{1}$
and $x_{2}$, $y_{2}$ takes into account the following:

$\displaystyle
x_{1}^{2}y_{1}^{2}(x_{1}y_{2}+x_{2}y_{1})+x_{1}y_{1}(x_{1}^{2}y_{2}^{2}+x_{2}^{2}y_{1}^{2}+ax_{1}x_{2}y_{1}y_{2})+$
(1) $\displaystyle+$ $\displaystyle
b(x_{1}^{3}y_{2}^{3}+x_{2}^{3}y_{1}^{3})+cx_{1}x_{2}y_{1}y_{2}(x_{1}y_{2}+x_{2}y_{1})+$
$\displaystyle+$ $\displaystyle
x_{2}y_{2}(d(x_{1}^{2}y_{2}^{2}+x_{2}^{2}y_{1}^{2})+ex_{1}x_{2}y_{1}y_{2})+fx_{2}^{2}y_{2}^{2}(x_{1}y_{2}+x_{2}y_{1})=0\,.$

If we go to the coordinates of space, we find (according to Eq. (5) §2, where
$\alpha_{1}=\alpha_{2}=\beta_{1}=\beta_{2}$) $x-z=0$ for the perspective plane
and $y=w=x+z=0$ for the perspective centre. The curve must be on a cone of the
3rd order, whose vertex is the centre, so that every two points on the same
generator are interchanged by the collineation. The tangents of the 6 points
in the perspective plane pass through the centre and the associated planes are
stationary. The planes of the 9 turning points of the cone are doubly
osculating, thus forming 9 $\Delta$-planes.
Now the cyclic $G_{2}$ may not effect any permutation of the generator sets.
One can bring it to the form:

$x_{1}^{\prime}=-x_{1}\,,\quad y_{1}^{\prime}=-y_{1}\,,\quad
x_{2}^{\prime}=x_{2}\,,\quad y_{2}^{\prime}=y_{2}\,.$

The equation can be written in two equivalent ways, from which we choose the
following:

(2) $\displaystyle
x_{1}^{3}y_{1}^{3}+x_{1}y_{1}x_{2}y_{2}(ax_{1}y_{1}+bx_{2}y_{2})+$
$\displaystyle
cx_{2}^{3}y_{2}^{3}+x_{1}^{3}y_{1}y_{2}^{2}+x_{1}x_{2}^{2}y_{1}^{3}+$
$\displaystyle+$ $\displaystyle
dx_{2}^{3}y_{1}^{2}y_{2}+ex_{1}^{2}x_{2}y_{2}^{3}=0\,.$ ††margin:  start p.10

Two points of the curve are transformed into themselves: $x_{1}=y_{2}=0$ and
$x_{2}=y_{1}=0$. If one takes account of the equations (2) and (3) of the
previous section, one finds that in this collineation every point of the
straight lines $x=z=0$ and $y=w=0$ remains fixed. Any straight line that
intersects these two lines must merge into itself. The connecting lines of
corresponding points on the curve thus form a ruled surface, for which those
two straight lines are guidelines, and it is easily seen that this ruled
surface is of fifth order and contains $y=w=0$ as triple $x=z=0$ as a double
directrix.
The possible location of the $C_{6}$ on a 3rd order cone or a 5th order ruled
surface thus characterises the two different types of involutory-unambiguous
correspondences on the curve. Since only 6 or 5 moduli are included in (3) and
(2) and 9 moduli are available for the general $C_{6}$, 3 or 4 conditions are
fulfilled there. The cone $K_{3}$, or the ruled surface $R_{5}$, designates a
structure of the genus $p^{\prime}=1$, or 2, to which the $C_{6}$ is
1-2-unambiguously related. No other rectilinear surfaces can simply contain
the curve if at the same time the generators are supposed to be bisecants.
In the case of equation (2) we examine the bisecant ruled surface whose
directrix is $y=w=0$, which itself contains 2 points of the curve. From each
point of this straight line there are 5 bisecants (except $y=w=0$ itself) and
in each plane through it there are 6, so that the order 11 is obtained. The
ruled surface thus consists of the mentioned $R_{5}$ and a $R_{6}$, for which
the $C_{6}$ is a double curve. The generators of this $R_{6}$, which start
from a point of the directrix, merge into each other through the
transformation $G_{2}$; but since every plane through the straight line
remains invariant, those generators must also lie in the same plane through
this straight line. One can therefore call $y=w=0$ a double tangent line for
the $R_{6}$, since the cross-sectional curves there receive tangent
nodes.101010I have dealt with this $R_{6}$ in my writing Klassifikation af
regelytorna af 6. graden p. 58 (Diss. Lund 1892) This $R_{6}$ is usually of
genus $p^{\prime}=2$, but splits into 2 $K_{3}$ when

(3) $d=e\,.$ ††margin:  start p.11

This equation says that the two points of $C_{6}$ on $y=w=0$ have the same
plane $y+dw=0$. The transformation group of the curve into itself is in this
case a Klein four-group. Two $G_{2}$ interchanging the generating sets appear:

$x_{1}^{\prime}=\pm y_{1}\,,\quad y_{1}^{\prime}=\pm x_{1}\,,\quad
x_{2}^{\prime}=y_{2}\,,\quad y_{2}^{\prime}=x_{2}\,.$

The associated perspective planes are $x\mp z=0$ and the perspective centres
$y=w=x\pm z=0$.
However, one can also obtain a Klein four-group of a different kind, which
leaves the generating groups uninterchanged altogether. We start again from
Equation (2), but at first we think of the ratios of all 8 coefficients as
completely arbitrary. Under the remaining operations of the Klein four-group,
the two generators of a family that are fixed under $G_{2}$ must be exchanged
with each other. By means of suitable normalization, the two new $G_{2}$ can
be given the following form:

$x_{1}^{\prime}=\pm x_{2}\,,\quad x_{2}^{\prime}=x_{1}\,,\quad
y_{1}^{\prime}=\pm y_{2}\,,\quad y_{2}^{\prime}=y_{1}\,.$

The equation for the curve is:

(4) $\displaystyle x_{1}^{3}y_{1}^{3}+$ $\displaystyle
x_{2}^{3}y_{2}^{3}+ax_{1}y_{1}x_{2}y_{2}(x_{1}y_{1}+x_{2}y_{2})+by_{1}y_{2}(x_{1}^{3}y_{2}+x_{2}^{3}y_{1})+$
$\displaystyle+$ $\displaystyle
cx_{1}x_{2}(x_{1}y_{2}^{3}+x_{2}y_{1}^{3})=0\,.$

A cyclic $G_{4}$ must have a $G_{2}$ as a subgroup, which leaves the
generating groups uninterchanged. A look back at equation (A) teaches us that
the two fixed points of $C_{6}$ under $G_{2}$ must also maintain their fixed
positions under $G_{4}$. In the collineations of period 4, the systems must be
exchanged; this is already evident from the fact that the necessary number of
points for the purpose of cyclic permutation are missing on the fixed
generators occurring in this case. So we start from equation (2) under the
most general assumption possible with regard to the coefficients. A
collineation of period 4, which swaps the systems and keeps the points
$x_{1}=y_{2}=0$ and $x_{2}=y_{1}=0$ fixed, can be brought to the following
form:

$x_{1}^{\prime}=y_{2}\,,\quad y_{2}^{\prime}=-ix_{1}\,,\quad
x_{2}^{\prime}=iy_{1}\,,\quad y_{1}^{\prime}=x_{2}\,.$ ††margin:  start p.12

The generator for which $x_{1}=x_{2}$ can be chosen arbitrarily. We can
therefore state that in the equation we are looking for, for example the
coefficients for $x_{1}^{3}y_{1}y_{2}^{2}$ and $x_{2}^{3}y_{1}^{2}y_{2}$
become equal. Then we get equation (4) again, only with the specialisation
that

(5) $c=-b\,.$

However, the curve determined in this way also contains the two groups of four
belonging to cases (3) and (4), so that the complete collineation group is of
the nature of a dihedral $G_{8}$.
One now easily proves that the 5 groups discussed are the only possible ones
without collineations of higher prime number period.

###  4 p.12

We now add the cyclic groups of period 3. With such a group, either each
generator of a group can merge into itself or not. In the latter case, the
collineation can be created in the following form:

$x_{1}^{\prime}=j^{2}x_{1}\,,\quad x_{2}^{\prime}=x_{2}\,,\quad
y_{1}^{\prime}=jy_{1}\,,\quad
y_{2}^{\prime}=y_{2}\,,\quad\left(j=e^{\frac{2\pi i}{3}}\right)$

or in the coordinates of space (according to (2) and (3) §2):

$x^{\prime}=jx\,,\quad y^{\prime}=y\,,\quad z^{\prime}=j^{2}z\,,\quad
w^{\prime}=w\,.$

The equation of a $C_{6}$ that merges into itself through this transformation
can be written with only 3 moduli:

(6)
$x_{1}^{3}y_{1}^{3}+x_{2}^{3}y_{2}^{3}+a(x_{1}^{3}y_{2}^{3}+x_{2}^{3}y_{1}^{3})+bx_{1}^{2}y_{1}^{2}x_{2}y_{2}+cx_{1}y_{1}x_{2}^{2}y_{2}^{2}=0\,,$

whereby, of course, one disregards structures with double points and thus
lower genus. But the curve (6) also has three perspective $G_{2}$ into itself:

$x_{1}^{\prime}=y_{1}(jy_{1},j^{2}y_{1})\,,\quad y_{1}^{\prime}=x_{1}\,,\quad
x_{2}^{\prime}=y_{2}\,,\quad y_{2}^{\prime}=x_{2}(jx_{2},j^{2}x_{2})\,.$

or (according to (4) and (5) §2):

$x^{\prime}=z(j^{2}z,jz)\,,\quad y^{\prime}=y\,,\quad
z^{\prime}=x(jx,j^{2}x)\,,\quad w^{\prime}=w\,.$

The relevant perspective planes are:

$x-z=0\,,\quad x-j^{2}z=0\,,\quad x-jz=0\,,$ ††margin:  start p.13

and the perspective centres are obtained as the intersections of the straight
lines $y=w=0$ with $x+z=0$, $x+j^{2}z=0$ and $x+jz=0$.
The transformation group of the curve (6) is thus a dihedral group $G_{6}$.
Six related points lie in a plane through $y=w=0$. The straight lines
connecting the corresponding points at the $G_{2}$ form 3 $K_{3}$, the
vertices of which are the above-mentioned perspective centres. These $K_{3}$
are cyclically exchanged by the $G_{3}$. One can now easily deduce that a
plane through $y=w=0$ must either touch all or none of $K_{3}$. There are 6
planes of that kind, which must also touch the curve three times; the points
of contact belong to stationary planes, and their tangents each pass through
one of the three vertices.
We now think of the $G_{3}$ as a subgroup of a higher group that does not swap
systems. This group cannot be a cyclic $G_{6}$, since the non-fixed points
permute each other cyclically to six on such a fixed generator, and not an
octahedral group, because the cyclic $G_{4}$ required for this do not occur
according to the previous section. The possible cases to be discussed here are
thus dihedral $G_{6}$ and the tetrahedral group.
In the case of the other collineations of a linear dihedral group, those
elements which remain fixed in the cyclic subgroup must be exchanged. In order
for this exchange to be possible, a condition must be fulfilled in equation
(6), for which we can set

(7) $b=c\,.$

The complete collineation group of the curve is then of the type of a dihedral
$G_{12}$. As subgroups we mention the dihedral $G_{6}$, which has the general
curve (6), the dihedral $G_{6}$, which does not commute the generators, whose
three collineations of period 2 are the following:

$x_{1}^{\prime}=x_{2}(jx_{2},j^{2}x_{2})\,,\quad x_{2}^{\prime}=x_{1}\,,\quad
y_{1}^{\prime}=y_{2}(j^{2}y_{2},jy_{2})\,,\quad y_{2}^{\prime}=y_{1}\,,$

and finally the cyclic $G_{6}$, which is formed by repeating the operation:

$x_{1}^{\prime}=jy_{2}\,,\quad y_{2}^{\prime}=x_{1}\,,\quad
x_{2}^{\prime}=y_{1}\,,\quad y_{1}^{\prime}=j^{2}x_{2}\,.$ ††margin:  start
p.14

We now assume that the subgroup which does not interchange the systems is a
tetrahedral group. As a normal subgroup, we have a Klein four-group here. We
are therefore looking for the further conditions which the coefficients of
equation (4) must satisfy in this case. The fixed elements in the individual
operations of the Klein four-group are to be cyclically exchanged in the
collineations of period 3 of the tetrahedron group. This collineation of
period 3 can now easily be represented in the following way:

$x_{1}^{\prime}=\pm i(x_{1}\pm x_{2})\,,\quad x_{2}^{\prime}=x_{1}\pm
x_{2}\,,\quad y_{1}^{\prime}=\pm i(y_{1}\pm y_{2})\,,\quad
y_{2}^{\prime}=y_{1}\pm y_{2}\,,$

and thus obtains for the equation of the curve merging into itself:

(8)
$(x_{1}y_{1}+x_{2}y_{2})^{3}+b(x_{1}y_{1}-x_{2}y_{2})(x_{1}y_{2}-x_{2}y_{1})(x_{1}y_{2}+x_{2}y_{2})=0\,.$

The complete collineation group of curve (8) in itself turns out to be
holohedrally isomorphic with an octahedral group. The plane
$x_{1}y_{1}+x_{2}y_{2}=0$ or $y+w=0$ always merges into itself. The pole of
this plane in relation to the area $F_{2}$ must therefore remain fixed for all
24 collineations. The curve lies on 6 cones $K_{3}$ whose vertices are in the
invariant plane.

###  5 p.14

We now assume that each generator of the $y$ family merges into itself under a
$G_{3}$. The $G_{3}$ can be generated by the collineation:

$x_{1}^{\prime}=jx_{1}\,,\quad x_{2}^{\prime}=x_{2}\,,\quad
y_{1}^{\prime}=y_{1}\,,\quad y_{2}^{\prime}=y_{2}\,,$

and the equation of an associated curve is:

(9) $x_{1}^{3}f_{3}(y_{1},y_{2})+x_{2}^{3}\varphi_{3}(y_{1},y_{2})=0\,.$

This curve has 3 independent moduli.
We now consider the case that $f_{3}$ and $\varphi_{3}$ contain a common
transformation group within them. The same must of course also hold for their
HESSian covariants, and we first assume that these do not coincide. There are
two possibilities here: either the points of one covariant can be base points
of a $G_{2}$, ††margin:  start p.15 which swaps those of the other, or the
points of both covariants are swapped by a $G_{2}$, whose base points must be
supplied by its simultaneous quadratic covariant; but that case turns out to
be impossible, because in order for the three points $f_{3}=0$ (resp.
$\varphi_{3}=0$) to merge into each other through a $G_{2}$, one of them must
remain fixed and thus provide a base point.
But one also comes to higher groups if $f_{3}=0$ and $\varphi_{3}=0$ can be
exchanged with each other. Because the 6 points $f_{3}\varphi_{3}=0$ have to
be ordered cyclically into one or more series, the period of a collineation
can only be 6 or 2. By the same way of reasoning for the HESSian covariants,
one obtains 2 as the only possible period. The equation of the curve can be
found in the following form:

(10) $\displaystyle x_{1}^{3}(y_{1}-y_{2})(y_{1}-ay_{2})(y_{1}-by_{2})+$
$\displaystyle+$ $\displaystyle
x_{2}^{3}(y_{1}+y_{2})(y_{1}+ay_{2})(y_{1}+by_{2})=0\,.$

The transformation group is a dihedral $G_{6}$ whose collineations of period 2
are the following:

$x_{1}^{\prime}=x_{2}(jx_{2},j^{2}x_{2})\,,\quad
x_{2}^{\prime}=x_{1}(j^{2}x_{1},jx_{1})\,,\quad y_{1}^{\prime}=y_{1}\,,\quad
y_{2}^{\prime}=-y_{2}\,.$

Since the points of the HESSian covariants can be paired in two ways, one has
the possibility to consider that $f_{3}=0$ and $\varphi_{3}=0$ are also
swapped in two ways. But then $f_{3}=0$ and $\varphi_{3}=0$ must each contain
a transformation different from the identity, which can be obtained by
combining two permutations. The equation of the curve can be brought into the
following form:

(11)
$x_{1}^{3}y_{1}(y_{1}^{2}+ay_{2}^{2})+x_{2}^{3}y_{2}(ay_{1}^{2}+y_{2}^{2})=0\,.$

The collineation group of this curve is a dihedral $G_{12}$. A generating
operation of the cyclic subgroup $G_{6}$ is found in:

$x_{1}^{\prime}=-jx_{1}\,,\quad x_{2}^{\prime}=x_{2}\,,\quad
y_{1}^{\prime}=-y_{1}\,,\quad y_{2}^{\prime}=y_{2}\,.$

The 6 remaining collineations are the following:

$x_{1}^{\prime}=\pm x_{2}(\pm jx_{2},\pm j^{2}x_{2})\,,\quad
x_{2}^{\prime}=x_{1}\,,\quad y_{1}^{\prime}=\pm y_{2}\,,\quad
y_{2}^{\prime}=y_{1}\,.$ ††margin:  start p.16

We now examine the case where the HESSian covariants of $f_{3}$ and
$\varphi_{3}$ coincide. The equation of the curve can easily be brought into
the form:

(12)
$x_{1}^{3}y_{1}^{3}+x_{1}^{3}y_{2}^{3}+x_{2}^{3}y_{1}^{3}+a^{3}x_{2}^{3}y_{2}^{3}=0\,.$

This curve has 12 stationary tangents whose tangents are cut out in threes
each by the generators $x_{1}=0$, $x_{2}=0$, $y_{1}=0$ and $y_{2}=0$. Here one
has a $G_{3}$ for each system, in which each generator remains fixed:

$\displaystyle x_{1}^{\prime}$ $\displaystyle=jx_{1}\,,\quad
x_{2}^{\prime}=x_{2}\,,\quad y_{1}^{\prime}=\phantom{j}y_{1}\,,\quad
y_{2}^{\prime}=y_{2}\,;$ $\displaystyle x_{1}^{\prime}$
$\displaystyle=\phantom{j}x_{1}\,,\quad x_{2}^{\prime}=x_{2}\,,\quad
y_{1}^{\prime}=jy_{1}\,,\quad y_{2}^{\prime}=y_{2}\,.$

By combining these $G_{3}$ one gets a $G_{9}$, which consists of 4 $G_{3}$ and
is normal in the overall group. If one adds a collineation that exchanges the
generators $x_{1}=0$ and $x_{2}=0$ or $y_{1}=0$ and $y_{2}=0$:

$x_{1}^{\prime}=ax_{2}\,,\quad x_{2}^{\prime}=x_{1}\,,\quad
y_{1}^{\prime}=y_{2}\,,\quad y_{2}^{\prime}=\frac{1}{a}y_{1}\,,$

and combines this with $G_{6}$, one gets 9 collineations from the period 2. So
one has a $G_{18}$, which converts every generator system into itself and is
obviously holohedrally isomorphic with the collineation group of a general
planar $C_{3}$. There are also 18 collineations that swap systems. One gets
these by combining the $G_{18}$ with any of them, like:

$x_{1}^{\prime}=y_{1}\,,\quad y_{1}^{\prime}=x_{1}\,,\quad
x_{2}^{\prime}=y_{2}\,,\quad y_{2}^{\prime}=x_{2}\,.$

Of these 18 new collineations, 12 have period 6 and 6 have period 2.
With the mentioned 36 collineations of the curve (12), 2 straight lines remain
fixed: the straight lines connecting the points $x_{1}=y_{1}=0$ and
$x_{2}=y_{2}=0$ or $x_{1}=y_{2}=0$ and $x_{2}=y_{1}=0$, or (Eq. (1) §2)
$x=z=0$ and $y=w=0$. The transformation groups of these straight lines are
dihedral $G_{6}$ and thus obtained with $G_{36}$ from the combination of these
two $G_{6}$, and by combining subgroups of $G_{6}$ subgroups of $G_{36}$
arise. The vertices of three $K_{3}$ containing the curve lie on each of the
two fixed straight lines.††margin:  start p.17 The mentioned dihedral $G_{6}$
arise by summarising all possible permutations of these vertices. The $G_{36}$
is therefore to be regarded as a subgroup of the group of all possible
permutations of 6 objects, namely the above-mentioned $G_{18}$ that does not
permute the systems is obtained as the subgroup of $G_{36}$, in which
permutations occur in an even number.
The question arises whether the curve (12) can contain further collineations.
The generator $x_{1}=0$ can only be exchanged with the other three generators
containing $\Theta$-points: $x_{2}=0$, $y_{1}=0$, $y_{2}=0$. So if $x_{1}=0$
merges into itself under $n$ collineations, then the order of the group is
$4n$. The number $n$ is obtained by combining $G_{3}$, which leaves every
point of the imagined generator unchanged, and the common transformation group
of $f_{3}$ and $\varphi_{3}$ in itself. However, the latter group is usually a
$G_{3}$ if the HESSian covariants coincide, but a dihedral group of order 6 if
$f_{3}$ and $\varphi_{3}$ are mutual third-degree covariants. In this case,
where $a^{3}=-1$ and thus the equation of the curve:

(13)
$x_{1}^{3}y_{1}^{3}+x_{1}^{3}y_{2}^{3}+x_{2}^{3}y_{1}^{3}-x_{2}^{3}y_{2}^{3}=0\,,$

36 new collineations appear. One gets this by combining the $G_{36}$ with any
new one, e.g

$x_{1}^{\prime}=x_{2}\,,\quad x_{2}^{\prime}=x_{1}\,,\quad
y_{1}^{\prime}=y_{1}\,,\quad y_{2}^{\prime}=-y_{2}\,.$

Of the 36 collineations thus added, those which interchange the systems are of
period 4; of the rest, 12 have period 6 and 6 have period 2. Those lines fixed
under $G_{36}$ are swapped under the 36 new collineations, and one can
consider the $G_{72}$ as a group of permutations of 6 objects, where about the
three first and three last things always form the same 2 commuting groups. The
$G_{72}$ has three normal subgroups of order 36: the collineation group of the
general curve (12), the subgroup that does not interchange the systems, and
finally a group of even permutations of the 6 $K_{3}$ vertices. The latter is
holohedrally isomorphic with the collineation group of the plane harmonic
$C_{3}$. As mentioned, ††margin:  start p.18 the curve (12) lies on 6 $K_{3}$,
the vertices of which are occupied three times each on the two distinguished
straight lines. The same curve also lies on 9 $R_{5}$, whose triple directrix
connects two cone vertices. The special curve (13) also lies on 6 other
$R_{5}$ whose triple directrices each connect two $\Theta$-points; these lines
are double tangent straight lines of 6 $R_{6}$ for which (13) is double curve.
In general, 72 points of the curve (13) permute in the $G_{72}$ as a closed
system. But the 12 $\Theta$-points only permute among themselves, and the same
applies to the system of the 36 $\alpha$-points, and finally also to 18 points
in the 9 straight lines connecting two $K_{3}$ vertices, which form the points
of contact of 9 double-osculating planes. Each of these points remains fixed
under a $G_{6}$, or $G_{2}$ or $G_{4}$.

###  6 p.18

It remains to consider the case where the curve is transformed into itself by
a $G_{5}$. With an odd number of periods, of course, the generating systems
cannot be interchanged. Since the points of the curve on the 4 fixed
generators cannot be cyclically permuted to 5 each, they must be occupied in
the 4 fixed points of intersection. However, this can only be achieved by
simply touching the curve in each of the 4 points. It is now easy to see
through experiments that the equation of a curve of the type considered here
with a collineation of period 5 can be brought into the following form:

(14)
$x_{1}^{3}y_{1}^{2}y_{2}+x_{1}^{2}x_{2}y_{2}^{3}+x_{1}x_{2}y_{2}^{3}+x_{1}x_{2}^{2}y_{1}^{3}+a^{5}x_{2}^{3}y_{1}y_{2}^{2}=0\,,$

where if $\varepsilon^{5}=1$, the collineations are represented as follows:

$x_{1}^{\prime}=\varepsilon x_{1}\,,\quad
x_{2}^{\prime}=\varepsilon^{4}x_{2}\,,\quad
y_{1}^{\prime}=\varepsilon^{2}y_{1}\,,\quad
y_{2}^{\prime}=\varepsilon^{3}y_{2}\,.$

However, the curve (14) also has 5 collineations of period 2:

$x_{1}^{\prime}=a^{2}\varepsilon^{4}x_{2}\,,\quad
x_{2}^{\prime}=\frac{\varepsilon}{a}x_{1}\,,y_{1}^{\prime}=\varepsilon^{3}y_{2}\,,\quad
y_{2}^{\prime}=\frac{\varepsilon^{2}}{a}y_{1}\,,$

so that the complete group of its collineations is a dihedral $G_{10}$.
††margin:  start p.19 One easily finds that no collineation of period 10 can
transform the curve (14) into itself; because either all 4 fixed points of
$G_{5}$ would also remain fixed here, which contradicts equation (A), or 2 of
them would be swapped with each other, which would be geometrically
inconsistent. Collineations with a higher number of periods are also
impossible for the same reasons. One only has to consider the case in which a
generator is transformed into itself by an icosahedron group. Because the
$G_{3}$ occurring here must be of the type dealt with in the 4th section, and
the systems were always exchanged there, the complete group must be of order
120. With such a curve with 120 transformations in itself, the 24 tangents,
which are also generators of $F_{2}$, form 6 quadrilaterals corresponding to
the 6 $G_{5}$ within the icosahedron group, whose corners lie in the points of
contact.
If an exchange of the systems for the curve (14) is to be possible, the
quadrilateral on which it is based must be converted into itself if one does
not want to assume several such quadrilaterals, and thus also several $G_{5}$
and one $G_{120}$. One easily finds the condition $a^{5}=-1$; then there are 5
cyclic $G_{4}$ whose generating operations are the following:

$x_{1}^{\prime}=\varepsilon^{2}y_{1}\,,\quad
y_{1}^{\prime}=-\varepsilon^{4}x_{2}\,,\quad
x_{2}^{\prime}=\varepsilon^{3}y_{2}\,,\quad y_{2}^{\prime}=\varepsilon
x_{1}\,,$

where $\varepsilon$ is a fifth root of unity.
Meanwhile Mr. GORDAN recognised111111See P. GORDAN, über die Auflösung der
Gleichungen vom fünften Grade (Math. Ann. Vol. 13) or F. KLEIN, Vorlesungen
über das Ikosaeder, p. 194ff. that the curve obtained in this way:

(15)
$x_{1}^{3}y_{1}^{2}y_{2}+x_{1}^{2}x_{2}y_{2}^{3}+x_{1}x_{2}^{2}y_{1}^{3}-x_{2}^{3}y_{1}y_{2}^{2}=0$

allows 120 collineations. Mr. KLEIN calls this curve Bring’s curve, for the
following reason. If one introduces pentahedron coordinates $z_{1}$, $z_{2}$,
$z_{3}$, $z_{4}$, $z_{5}$ with the identical relation

$\sum z_{i}=0\,,$ ††margin:  start p.20

one obtains the same as an intersection curve of two surfaces, the main
surface and the diagonal surface:

(15a) $\sum z_{i}^{2}=0\,,\quad\sum z_{i}^{3}=0\,.$

The 120 collineations of the curve result from the permutations of the
$z_{i}$, and the 60 even permutations leave the generating systems unchanged.
The nature of the individual collineations is briefly indicated here. By
cyclic permutation of all $z_{i}$ one gets 24 collineations of period 5.
Twenty collineations of the type:

$z_{1}^{\prime}=z_{2}\,,\quad z_{2}^{\prime}=z_{3}\,,\quad
z_{3}^{\prime}=z_{1}\,,\quad z_{4}^{\prime}=z_{5}\,,\quad
z_{5}^{\prime}=z_{4}\,,$

have the period 6. The resulting 10 cyclic $G_{6}$ have subgroups of period 3,
which yield 20 new collineations, E.g.:

$z_{1}^{\prime}=z_{2}\,,\quad z_{2}^{\prime}=z_{3}\,,\quad
z_{3}^{\prime}=z_{1}\,,\quad z_{4}^{\prime}=z_{4}\,,\quad
z_{5}^{\prime}=z_{5}\,;$

In addition, there are the subgroups of period 2:

$z_{1}^{\prime}=z_{2}\,,\quad z_{2}^{\prime}=z_{2}\,,\quad
z_{3}^{\prime}=z_{3}\,,\quad z_{4}^{\prime}=z_{5}\,,\quad
z_{5}^{\prime}=z_{4}\,.$

These 10 collineations are perspective; the perspective plane in the chosen
example is $z_{4}-z_{3}=0$ and the perspective centre
$z_{1}=z_{2}=z_{3}=z_{4}+z_{5}$. The 60 points of contact of the stationary
planes are in the perspective planes. The 10 straight lines $z_{i}=z_{k}=0$
each go through three perspective centres; then the straight line
$z_{1}=z_{2}=0$ goes through the centres assigned to the planes $z_{3}=z_{4}$,
$z_{4}=z_{5}$ and $z_{5}=z_{3}$ and stays with the permutations of $z_{3}$,
$z_{4}$, $z_{5}$ resulting dihedral $G_{6}$. Thirty collineations have period
4; in these, one pentahedron plane remains fixed and the 4 others are
cyclically interchanged. By repeating one of these, 15 collineations of period
2 »with axes» arise. As an example we choose:

$z_{1}^{\prime}=z_{2}\,,\quad z_{2}^{\prime}=z_{1}\,,\quad
z_{3}^{\prime}=z_{4}\,,\quad z_{4}^{\prime}=z_{3}\,,\quad
z_{5}^{\prime}=z_{5}\,.$ ††margin:  start p.21

The one axis $z_{1}+z_{2}=z_{3}+z_{4}=z_{5}=0$ goes through 2 of the
aforementioned perspective centres: $z_{1}+z_{2}=z_{3}=z_{4}=z_{5}=0$ and
$z_{3}+z_{4}=z_{1}=z_{2}=z_{5}=0$, and in fact each of the 15 connecting lines
is such an axis. According to the discussion in Equation (2) of section 3,
each such axis passes through the tangents of a doubly osculating plane.
The higher subgroups within the $G_{120}$ are: the icosahedron groups, which
leave the systems uninterchanged, 5 octahedron groups, in each of which one
pentahedron plane remains fixed, 6 $G_{20}$ of the kind previously discussed,
10 dihedral $G_{12}$, in which the pentahedron planes permute in 2 systems of
2 and 3, and finally dihedral groups of the order 10, 8, 6 and 4, which,
however, form only subgroups of the subgroups discussed.
Corresponding to the 10 perspective $G_{2}$ and the 15 $G_{2}$ »with axes»,
BRING’s curve lies at 10 $K_{3}$ and 15 $R_{5}$. Each turning plane of this
$K_{3}$ osculates the curve twice. However, 15 turning points, the mentioned
axes, belong to 2 $K_{3}$. Sixty others are obtained, so that the total number
of planes doubly osculating the curve is 75. Of those planes $\gamma^{\prime}$
which touch the curve at one point and osculate at another, 300 are absorbed
(according to a formula at the end of the 1st section) by the 75
$\Delta$-planes; the remaining 24 osculate in the points whose tangents are
generators of $F_{2}$, because the other generator through such a point also
contains a tangent of the curve. One could easily draw out several such
remarkable properties of BRING’s curve.
The 60 $\alpha$-points, the 30 points of tangency of the system of 15
$\Delta$-planes and the 24 points in which the tangents are generators of the
$F_{2}$ form the only systems of less than 120 points that are permuted in a
closed manner under $G_{120}$ .

###  7 p.21

Now that we have found all possible collineation groups of a structure of
genus $p=4$ whose normal curve of the 6th order lies on a genuine surface of
the 2nd degree, the results obtained are summarised here. $M$ denotes the
number of independent moduli of the entities concerned. ††margin:  start p.22

  1. a.

$M=6$.

    1. 1)

$G_{2}$, which swaps the generating systems of the surface $F_{2}$. Equation
(3).

  2. b.

$M=5$.

    1. 2)

$G_{2}$ without swapping the systems. Eq. (2).

  3. c.

$M=4$.

    1. 3)

Klein four-group with swapping of the systems. Eq. (3).

  4. d.

$M=3$.

    1. 4)

Klein four-group without swapping of the systems. Eq. (4).

    2. 5)

Dihedral $G_{6}$ with exchange of systems. Eg. (6).

    3. 6)

$G_{3}$. Eq. (9).

  5. e.

$M=2$.

    1. 7)

Dihedral $G_{8}$ with exchange of systems. Eq. (5).

    2. 8)

Dihedral $G_{12}$ with exchange of systems. Eq. (7).

    3. 9)

Dihedral $G_{6}$ without swapping of the systems. Eq. (10).

  6. f.

$M=1$.

    1. 10)

Octahedral group. Eq. (8).

    2. 11)

Dihedral $G_{6}$ without swapping of the systems. Eq. (11).

    3. 12)

Dihedral group of order 10 without interchanging the systems. Eq. (14).

    4. 13)

Group of Order 36. Eq. (12).

  7. g.

$M=0$.

    1. 14)

Group of Order 72. Eq. (13).

    2. 15)

Group of Order 120. Eq. (15).

###  8 p.22

We now consider the case in which the normal curve of a structure of genus
$p=4$ sits on a cone of the 2nd degree $K_{2}$. This special case has
attracted the attention of Messrs. NOETHER and SCHOTTKY in the literature. As
a strange property of the curve, it has been recognised that the same curve
(in the typical form of a $C_{9}$ with 8 triple points) can be a coincidence
curve with an unambiguous transformation of the plane in itself. One also has
in itself a more general class of unambiguous transformations of the plane,
that »with 8 points », which is characterised by an unambiguous correspondence
among the points ††margin:  start p.23 of such a $C_{9}$. For this reason Mr.
S. KANTOR, in his investigations of the isolated types of unambiguous periodic
transformations of the plane, looked for the cyclic transformations of the
above structures with $p=4$ in themselves;131313Acta Mathematica, Vol. 19, p.
164. but not the complete groups, because that was not necessary for the
problem to be solved. It is to this more general task that we shall concern
ourselves here. We want to use a method that differs somewhat from KANTOR’s.
If the curve is projected onto a plane from a point of the cone, the image
curve is a $C_{6}$ with 2 infinitely close triple points, which come from the
branches located on the generator of the cone through the starting
point.141414Cf. CLEBSCH-LINDEMANN, Vorlesungen II, p. 431. So the equation of
the image curve can be brought to the following form:

$z^{3}y^{3}+z^{2}y^{2}f_{2}(x,y)+zyf_{4}(x,y)+f_{6}(x,y)=0\,,$

where $y=x=0$ forms a triple point with $y=0$ as the common tangent of the 3
branches. The generators of the cone are mapped by the pencil of rays
$x=\lambda y$, and the points of the curve occupied on one generator are
represented by the equation:

$z^{3}+z^{2}f_{2}(\lambda,1)+zf_{4}(\lambda,1)+f_{6}(\lambda,1)=0=\varphi_{3}(z)\,,$

where $z=\infty$ is the vertex of the cone or the common point of all referred
to as generators. The polars of the pole $z=\infty$ for $\varphi_{3}(z)$ are
of course invariant structures. Here we consider the linear polar
$3z+f_{2}(\lambda,1)=0$ whose location satisfies the conic $3zy+f_{2}(x,y)=0$.
But this conic section is the image of a cross-section of the cone. From now
on we choose the centre of projection on this cross-section; the conic section
then turns into a straight line, which we can denote by $z=0$, and for the
general form of the equation of the image curve we get:

(I) $z^{3}y^{3}+zyf_{4}(x,y)+f_{6}(x,y)=0\,.$

The collineation group of the normal curve occupied on the cone must be of
such a nature that both a point††margin:  start p.24 , the vertex of the cone,
remains fixed as well as a plane, the location of the linear polars mentioned.
The question arises whether, with a collineation belonging here, each
generator can remain fixed; the 3 points of the curve must be cyclically
permuted on a generator, so that the only possibility is a $G_{3}$. So we can
state the theorem:
The collineation group of a normal curve of genus $p=4$ located on a cone
$K_{2}$ is either a linear group or a combination of a linear group with a
$G_{3}$.
If we now start from the projected curve (I), we have to consider a group of
quadratic transformations instead of a collineation group. However, the
distinguished straight line $z=0$ must always merge into itself, and 2
fundamental points must also lie in the infinitely neighboring triple points.
The equation of a related transformation can be brought into the form:

(a)
$x^{\prime}:y^{\prime}:z^{\prime}=(a_{1}x+b_{1}y)(a_{2}x+b_{2}y):(a_{2}x+b_{2}y)^{2}:zy\,.$

However, it can be seen here that the substitution:

$x^{\prime}:y^{\prime}=(a_{1}x+b_{1}y):(a_{2}x+b_{2}y)$

must convert the functions $f_{6}$ and $f_{4}$ into itself. One has therefore
first to look for the common transformation group of these two functions.
The generators of the cone, which are fixed for a single spatial collineation,
can be chosen for $x=0$ and $y=0$. The 4 fixed points in the collineation are
the vertex of the cone and 3 points in the aforementioned fixed plane, 2 of
which lie on those fixed generators, and the third is the pole of their
connecting line with respect to the cross-section of the cone. But there are
cases where all the points of a line joining two of these points, or of a
plane through three of them, remain fixed. The planar transformation (a) turns
into a collineation here:

($a^{\prime}$) $x^{\prime}:y^{\prime}:z^{\prime}=\alpha x:\beta y:\gamma z\,.$

This collineation transforms $f_{6}(x^{\prime},y^{\prime})$ into $k\cdot
f_{6}(x,y)$ and $f_{4}(x^{\prime},y^{\prime})$ into $l\cdot f_{4}(x,y)$. The
individual terms of (I)††margin:  start p.25 are to be multiplied by the same
number $k$. From this we get the conditions:

(b) $\beta^{3}\gamma^{3}=\beta\gamma l=k\,;\quad\gamma=\frac{k}{\beta
l}\,;\quad k^{2}=l^{3}\,.$

Projecting the curve from the fixed point on the generator $x=0$, the equation
of the image curve is:

$z^{3}x^{3}+zxf_{4}(x,y)+f_{6}(x,y)=0\,,$

and the collineation is transformed into the following plane

($a_{1}^{\prime}$) $x^{\prime}:y^{\prime}:z=\alpha x:\beta
y:\gamma_{1}z\,;\quad\gamma_{1}=\frac{k}{\alpha
l}=\frac{\beta}{\alpha}\gamma\,.$

If one sets $\alpha=\beta$ in ($a^{\prime}$), each generator is transformed
into itself; one gets $k=\alpha^{6}$, $l=\alpha^{4}$ and $\gamma=\alpha$. The
collineation is thus an identity. However, other conditions arise when $f_{4}$
vanishes identically, which, however, is initially excluded.
If $\beta=\gamma$, each point of the generator $x=0$ is transformed into
itself. The same applies to the generator $y=0$ if $\alpha=\gamma_{1}$. If
these relations are valid at the same time, one gets $\alpha^{2}=\beta^{2}$ or
$\alpha=-\beta$. The spatial collineation is then a harmonic homology whose
plane of perspective goes through the solid generators.
If one takes $\alpha=\gamma$, then $\beta=\gamma_{1}$. The straight line with
nothing but fixed points in the spatial collineation is here the polar line of
the plane through the fixed generators.
By means of the above indications, the properties of the spatial collineations
can be deduced by considerations in the plane.

###  9 p.25

First, we assume that $f_{4}$ does not vanish identically. The collineation
group of the normal curve itself is obtained from the joint group of $f_{4}$
and $f_{6}$, which satisfies condition (b) $k^{2}=l^{2}$. The curve is touched
by the 12 generators, $27f_{6}^{2}+4f_{4}^{3}=0$, of the cone. So that the
curve does not have a double point, no 2 of those 12 straight lines may
coincide; unless $f_{6}$ and $f_{4}$ share common factors that include
stationary tangents; but $f_{6}$ must not contain a double or multiple factor,
which also goes into $f_{4}$. Here is a brief list of the different types.
††margin:  start p.26

  1. 1)

Perspective collineation of period 2:
$x^{\prime}:y^{\prime}:z^{\prime}=-x:y:z$.

(1)
$z^{3}y^{3}+zy(ax^{4}+bx^{2}y^{2}+cy^{4})+dx^{6}+ex^{4}y^{2}+fx^{2}y^{4}+gy^{6}=0\,.$

  2. 2)

Collineation »with axes» of period 2 :
$x^{\prime}:y^{\prime}:z^{\prime}=x:-y:z$.

(2)
$z^{3}y^{2}+z(ax^{4}+bx^{2}y^{2}+cy^{4})+x(dx^{4}+ex^{2}y^{2}+fy^{4})=0\,.$

Here the order of the image curve is reduced because the centre of projection
is a point on the curve. The axes are the straight lines connecting two fixed
points of the curve in the invariant plane and their polar line.

  3. 3)

Klein four-group, consisting of 2 operations of type 1) and one of type 2).
One gets the equation:

(3)
$z^{3}y^{2}+z[a(x^{4}+y^{4})+bx^{2}y^{2}]+x[c(x^{4}+y^{4})+dx^{2}y^{2}]=0\,.$

Here $f_{6}$ and $f_{4}$ merge into themselves when $x$ and $y$ are swapped.

  4. 4)

Klein four-group, which consists of nothing but operations of the type 2).
Since 2 points of $f_{6}=0$ remain fixed for each substitution, $f_{6}$ must
be the JAKOBIan covariant of $f_{4}$.

(4) $z^{3}y^{2}+z[a(x^{4}+y^{4})+bx^{2}y^{2}]+x(x^{4}-y^{4})=0\,.$

  5. 5)

Dihedral $G_{8}$, which contains as subgroups a Klein four-group of each of
the two previous species.

(5) $z^{3}y^{2}+azx^{2}y^{2}+x(x^{4}+y^{4})=0\,.$

The distinguished cyclic subgroup derives from the repetition of the
operation, $x^{\prime}y^{\prime}:z^{\prime}=x:iy:-z$.

The 5 species thus obtained fully correspond in sequence to the 5 given in the
3rd section. The following species, however, is a genuinely new one

  1. 6)

cyclic $G_{4}$ containing a $G_{2}$ of type 1) as a subgroup.

(6) $z^{3}y^{2}+z(x^{4}+ay^{4})+y(bx^{4}+cy^{4})=0\,.$

$x^{\prime}:y^{\prime}:z^{\prime}=ix:y:z$.††margin:  start p.27
The generator $x=0$ remains fixed at each point. In addition, one point of the
curve moves into the projection centre on $y=0$, which causes the order of the
image curve to decrease. The case follows from (1) if there $b=d=f=0$.

  2. 7)

Dihedral $G_{6}$.

(7) $z^{3}y^{3}+azy^{3}x^{2}+x^{6}+bx^{3}y^{3}+y^{6}=0\,.$

The generating operation of the cyclic $G_{3}$ is the following:
$x^{\prime}:y^{\prime}:z^{\prime}=x:jy:z$. With this $G_{3}$ every point of
the polar line remains fixed. The 3 $G_{2}$ are of type 1) and are obtained by
swapping $x$ and $y$. If one has $b=0$ in 7), the transformation group is a

  3. 8)

Dihedral $G_{12}$.

(8) $z^{3}y^{3}+azy^{3}x^{2}+x^{6}+y^{6}=0\,.$

The generating operation of the cyclic subgroup $G_{6}$ is:
$x^{\prime}:y^{\prime}:z^{\prime}=-x:jy:z$.

Cases 7) and 8) correspond exactly to cases (6) and (7) of section 4.

  1. 9)

Cyclic $G_{3}$ where the points of a generator remain fixed.

(9) $z^{3}y^{3}+zy^{2}(ax^{3}+by^{3})+x^{6}+cx^{3}y^{3}+dy^{6}=0\,.$

$x^{\prime}:y^{\prime}:z^{\prime}=jx:y:z$.

The points of $x=0$ remain fixed here. If $f_{6}$ is also the Jacobian
covariant of $f_{4}$, the transformation group consists of 4 such $G_{3}$ and
a Klein four-group of the kind 4), which together form a

  1. 10)

tetrahedral group.

(10) $z^{3}y^{3}+azy^{2}(x^{3}+y^{3})+x^{6}+20x^{3}y^{3}-8y^{6}=0\,.$

One also gets the equation under the form 4) if only $b=2ai\sqrt{3}$, by which
condition the equianharmonic invariant of $f_{4}$ vanishes. A special case of
both (1) and (9) is the following.

  1. 11)

Cyclic $G_{6}$.

(11) $z^{3}y^{3}+azy^{5}+x^{6}+by^{6}=0\,.$

$x^{\prime}:y^{\prime}:z^{\prime}=-jx:y:z$.
††margin:  start p.28 Also with this $G_{6}$ all points of the generator $x=0$
remain fixed. If one has $b=0$ here, the group is a

  2. 12)

cyclic $G_{12}$.

(12) $z^{3}y^{3}+zy^{5}+x^{6}=0\,.$

$x^{\prime}:y^{\prime}:z^{\prime}:=-jx:iy:-iz$.

  3. 13)

Cyclic $G_{5}$.

(13) $z^{3}y^{2}+azy^{4}+x^{5}+by^{5}=0\,.$

$x^{\prime}:y^{\prime}:z^{\prime}=\varepsilon x:y:z$.
The points of $x=0$ remain fixed. If $b=0$, the group is a

  4. 14)

cyclic $G_{10}$. $x^{\prime}:y^{\prime}:z^{\prime}=\varepsilon x:-y:z$.

(14) $z^{3}y^{2}+zy^{4}+x^{5}=0\,.$

The subgroup $G_{2}$ here is of the type 2).

In the remaining cases, $f_{4}$ vanishes identically. The normal curve always
merges into itself through a perspective $G_{3}$, which leaves each generator
unchanged. In general,

  1. 15)

this $G_{3}$, $x^{\prime}:y^{\prime}:z^{\prime}=x:y:jz$, is also the complete
group of the curve

(15) $z^{3}y^{3}=f_{6}(x,y)\,.$

But if $f_{6}$ has a linear transformation group in itself of order $\mu$, one
obtains the group of the curve by combining this $G_{\mu}$ with $G_{3}$, from
which the order $3\mu$ results. Within this $G_{3\mu}$ the perspective $G_{3}$
is normal.
The possible groups of $f_{6}$ in itself have already been compiled in our
earlier discussion.151515Bihang till K. Sv. Vet.-Akad. Handl. Vol 21, Section.
I, No. 1 We found there in the 4th section that the same are the following:
$G_{2}$, Klein four-group, Dihedral $G_{6}$, $G_{5}$, Dihedral $G_{12}$ and
octahedral group. We can therefore immediately write down the following normal
equations:

(16) $\displaystyle z^{3}y^{3}$
$\displaystyle=x^{6}+ax^{4}y^{2}+bx^{2}y^{4}+y^{6}\,;$ (17) $\displaystyle
z^{3}y^{2}$ $\displaystyle=x(x^{4}+ax^{2}y^{2}+y^{4})\,;$ (18) $\displaystyle
z^{3}y^{3}$ $\displaystyle=x^{6}+ax^{3}y^{3}+y^{6}\,;$ (19) $\displaystyle
z^{3}y^{2}$ $\displaystyle=x^{5}+y^{5}\,;$ (20) $\displaystyle z^{3}y^{3}$
$\displaystyle=x^{6}+y^{6}\,;$ (21) $\displaystyle z^{3}y^{2}$
$\displaystyle=x(x^{4}+y^{4})\,.$ ††margin:  start p.29

The group of (16) is a cyclic
$G_{6}:x^{\prime}:y^{\prime}:z^{\prime}=-x:y:jz$, and that of (19) is a cyclic
$G_{15}:x^{\prime}:y^{\prime}:z^{\prime}=\varepsilon x:y:jz$. The group of
(17) is of order 12 and consists of 3 cyclic $G_{6}$, all of which have the
perspective $G_{3}$ as a subgroup. Of these $G_{6}$, 2 are of the same nature
as that of Curve (16); but the third,
$x^{\prime}:y^{\prime}:z^{\prime}=x:-y:jz$, has a $G_{2}$ »with axes» as a
subgroup. We do not dwell on the groups of curves (18) and (20) of the
respective orders 18 and 36. The group of order 72 of curve (21) has 3 cyclic
$G_{12}$, which are assigned to the cyclic $G_{4}$ of an octahedron group. The
one $G_{12}$ is transformed into the plane collineation,
$x^{\prime}:y^{\prime}:z^{\prime}=x:iy:-jz$.
Regarding the cyclic groups, our results differ in a few points from those
given by Mr. KANTOR.161616Ibid, p. 166. Mr. KANTOR does not mention the
$G_{2}$ »with axes» (our case 2) and the cyclic $G_{6}$ belonging to the curve
(17), which has a perspective $G_{3}$ and a $G_{2}$ »with axes» as subgroups.
We have found 2 types of cyclic $G_{12}$ here: one belongs to the curve (12)
and contains as subgroups a perspective $G_{2}$ and a $G_{3}$ with a generator
fixed at each point, the other to the curve (21) and contains a perspective
$G_{3}$ and a $G_{2}$ »with axes». Mr. KANTOR considers these two cases
(designated 12 and 16 in his account) to be equivalent, which is not
necessarily true. On the other hand, KANTOR’s forms 6 and 15, 9 and 10 are
equivalent to each other (and to our cases 6 and 9), but it should be noted
that KANTOR’s forms 6, 15 and 10 are not given with the full number of terms.

###  10 p.29

We now turn to the treatment of structures of genus $p=5$. The plane curve of
the lowest order, into which a general structure of this genus can be
unambiguously transformed, is known to be a $C_{6}$ with 5 double points; but
if the structure has a special linear series $g_{3}^{1}$, it can be
transformed into a $C_{5}$ with a double point.171717See CLEBSCH-LINDEMANN,
Vorlesungen über Geometrie, I, p. 709.††margin:  start p.30 In the latter
case, the pencil of rays intersects the $g_{3}^{1}$ through the double point,
and the adjoint $\varphi$-curves are conics through the same point. If one now
goes to the $\varphi$-normal-curve in 4-dimensional space, one finds that it
must lie on a third-order ruled surface, which is the image of our original
plane, with the rays from the double point being transformed into generators
and the double point itself in be transformed into a simple straight line of
the ruled surface. Each generator is thus a trisecant and the directrix a
bisecant of the normal curve. From this it can be seen that the ruled surface
or the plane of the $C_{5}$ plays an distinguished role. Consequently, the
unambiguous transformations of the $C_{5}$ must be birational in themselves,
because no quadratic or higher collineations are possible. The latter is
concluded from the fact that the system of adjoint conics must also be
transformed into itself in such a birational transformation; but within this
system there is an distinguished degenerating subset, and the associated
conics each consist of a straight line through the double point and an
arbitrary straight line; the system of straight lines must therefore be
converted into itself, which only occurs with a collineation. The possible
collision groups of $C_{5}$ must be of a very trivial nature. One descends
from the total group to the normal subgroup, in which the two branches in the
double point remain uninterchanged, and from this to a (cyclic) normal
subgroup, in which each straight line merges into itself through the double
point. However, these normal groups may consist of the total group or just
identity.
According to a theorem by Mr. WEBER, an algebraic structure with any $p$ is
characterised by $\frac{1}{2}(p-2)(p-3)$ linearly independent quadratic
relations between the $\varphi$-functions.181818Math. Ann., Vol. 13. However,
these relations are not sufficient to define the structure if a $g_{3}^{1}$
exists.191919Based on a theorem of L. KRAUS, Math. Ann., Vol. 16. It should be
noted here that the relations $\Phi^{(2)}=0$ are not sufficient to define the
structure even in a case without $g_{3}^{1}$, namely for the plane $C_{5}$
without double points. So††margin:  start p.31 for $p=5$ we have three
quadratic relations, $\Phi_{1}^{(2)}=0$, $\Phi_{2}^{(2)}=0$,
$\Phi_{3}^{(2)}=0$, the normal curve $C_{8}$ can generally be considered as
its intersection curve;202020Cf. a treatise by Mr. NOETHER, Math. Ann., Vol.
26. but the $\Phi^{(2)}=0$ in the case of the $g_{3}^{1}$ have a common
surface, namely the previously mentioned ruled surface of the third order.

###  11 p.31

From now on we stick to the general case. The normal curve is determined in
the 4-dimensional space by the three quadratic relations:

(1) $F_{1}(x_{1},x_{2},x_{3},x_{4},x_{5})=0\,,\quad F_{2}=0\,,\quad
F_{3}=0\,.$

We take any linear combination of $F_{i}$:

(2) $\lambda_{1}F_{1}+\lambda_{2}F_{2}+\lambda_{3}F_{3}=0$

and look for its discriminant:

(3) $\Delta_{5}(\lambda_{1},\lambda_{2},\lambda_{3})=0\,,$

which can be written in the known way in the form of a determinant with 5 rows
and 5 columns. The nature of the covariant curve $\Delta_{5}=0$ in the
$\lambda$ plane is of fundamental importance for our normal curve. Equation
(3) determines those combinations of $\lambda_{i}$ for which the left term of
(2) is a function of only 4 variables, say $x_{1}^{\prime}$, $x_{2}^{\prime}$,
$x_{3}^{\prime}$, $x_{4}^{\prime}$. The location of the vertices
$x_{1}^{\prime}=x_{2}^{\prime}=x_{3}^{\prime}=x_{4}^{\prime}=0$ clearly
corresponds to the curve $\Delta_{5}$. However, this location is determined by
the determinant group

(4) $\begin{vmatrix}F_{1}^{1}&F_{1}^{2}&F_{1}^{3}&F_{1}^{4}&F_{1}^{5}\\\
F_{2}^{1}&F_{2}^{2}&F_{2}^{3}&F_{2}^{4}&F_{2}^{5}\\\
F_{3}^{1}&F_{3}^{2}&F_{3}^{3}&F_{3}^{4}&F_{3}^{5}\end{vmatrix}=0\,,$

where $F_{i}^{k}=\frac{\partial F_{i}}{\partial x_{k}}$. The system (4)
represents a curve of order 10 in the 4-dimensional space,212121See SALMON-
FIEDLER, Algebra der linearen Transformationen, 2. Edition (1877), p. 368.
which we will denote by $D_{10}$. The normal curve cannot have a double point
because then its genus is reduced.††margin:  start p.32 However, as is easy to
find, both equations (1) and (4) should hold for a double point. The necessary
condition follows that the curve $D_{10}$ must not separate the normal curve.
Thus, no double points of $D_{10}$ and $\Delta_{5}$ can come from such
intersections. But the curve $\Delta_{5}$ can have double points, namely in
those points of the $\lambda$-plane, for which the left term of (2) as a
function of only 3 variables, say $x_{1}^{\prime}$, $x_{2}^{\prime}$,
$x_{3}^{\prime}$, can be written; such a double point corresponds to a
straight line on $D_{10}$, $x_{1}^{\prime}=x_{2}^{\prime}=x_{3}^{\prime}=0$.
The relations $F_{i}=0$ between 3 variables correspond to just as many systems
of infinitely many ABELian root forms. L. KRAUS now wanted to show that there
are at most 3 such systems;222222Ibid. but his proof is incorrect, and we
shall find that as many as 10 such systems can occur. In this case,
$\Delta_{5}$ has 10 double points, which is only possible if it consists of 5
lines, and $D_{10}$ consists of 10 lines, for which each 4 intersect at 5
points, these lines and multiple points of $D_{10}$ correspond to double
points and straight lines of $\Delta_{5}$. But we shall come back to this case
later.
According to BRILL-NOETHER’s law of reciprocity, the special linear series
$g_{4}^{1}$ of the structures of genus $p=5$ are assigned to each other in
pairs. The curve $D_{10}$ has a simple connection with the system of
$g_{4}^{1}$. If one projects the normal curve from a point of $D_{10}$, one
obtains a curve $C_{8}$ as its image in 3-dimensional space, which extends to
a surface of the 2nd degree
$F(x_{1}^{\prime},x_{2}^{\prime},x_{3}^{\prime},x_{4}^{\prime})=0$, where
$F=0$ gives the corresponding relation with only 4 variables. The generating
systems of this surface cut out 2 reciprocal $g_{4}^{1}$ to each other. But if
that relation were between only 3 antecedents, the projected curve lies on a
second-degree cone, so that the two series $g_{4}^{1}$ coincide, and the only
$g_{4}^{1}$ is its own reciprocal; the infinitely many associated ABELian root
forms are assigned to the tangent planes of the cone. Thus the system of
$g_{4}^{1}$ is 1-2-unambiguously related to the curve $D_{10}$, as we later
want to illustrate geometrically in a different way.††margin:  start p.33 With
a collineation of the normal curve in itself, the system of relations
$\lambda_{1}F_{1}+\lambda_{2}F_{2}+\lambda_{3}F_{3}=0$ must of course also be
converted into itself. The $\lambda_{i}$ thus experience linear substitutions,
in which the covariant curve $\Delta_{5}$ is necessarily transformed into
itself. This is not contradicted by the fact that a collineation of the
$\Delta_{5}$ is not always the result of a collineation of the normal curve.
On the other hand, the normal curve can undergo unambiguous transformations in
itself in which each point of $\Delta_{5}$ remains fixed, and therefore each
relation (2) is retained unchanged. With such a collineation, the curve
$D_{10}$ must also merge point by point, if one disregards straight lines that
correspond to double points of $\Delta_{5}$; the remaining curve $D_{10}$ can
consequently not be an actual 4-dimensional curve, but can either be composed
of a point and a 3-dimensional curve or of a straight line and a flat curve,
depending on a point in the collineation and the points of a space or the
points of a straight line and a plane remain fixed. As a necessary condition
for this one immediately obtains that each relation (1) $F_{i}=0$ can be
written either in the form

(a) $a_{i}x_{1}^{2}+f_{i}(x_{2},x_{3},x_{4},x_{5})=0\quad(i=1,2,3)$

or in the form

(b) $\varphi_{i}(x_{1},x_{2})+\psi_{i}(x_{3},x_{4},x_{5})=0\,.$

In the first case one has the collineation:

($\alpha$) $x_{1}^{\prime}=-x_{1}\,,\quad
x_{i}^{\prime}=x_{i}\quad(i=2,3,4,5)$

with 8 fixed points of the curve in space $x_{1}=0$; in the latter case the
collineation:

($\beta$) $x_{1}^{\prime}=-x_{1}\,,\quad x_{2}^{\prime}=-x_{2}\,,\quad
x_{i}^{\prime}=x_{i}\quad(i=3,4,5)$

without fixed points of the curve. The straight lines connecting corresponding
points in cases (a) and (b) thus form (according to equation (A) introduction)
a ruled surface of genus $p^{\prime}=1$ or $p^{\prime}=3$. However, symmetric
transformations with the assigned genus $p^{\prime}=2$ can also occur, but
without transforming every point of the curve $\Delta_{5}$ into itself. In
case (a) the discriminant of equation (2) breaks up into a linear and a
biquadratic ††margin:  start p.34 factor and in case (b) into a quadratic and
a cubic factor, so that the curve $\Delta_{5}$ consists of a straight line and
a $C_{4}$, respectively, a conic section and a $C_{3}$. But these curves can
also break up, so that the following 7 typical cases are obtained.

  1. 1)

$\Delta_{5}$ is an actual $C_{3}$.

  2. 2)

$\Delta_{5}$ breaks up into a straight line and a $C_{4}$. The intersection
points of the straight lines with $C_{4}$ correspond to 4 systems of
infinitely many ABELian root forms. But here, as in the previous case, other
such systems can appear, corresponding to the possible double points of
$C_{4}$ or $C_{5}$.

  3. 3)

$\Delta_{5}$ breaks up into a conic section and a $C_{3}$. The 6 points of
intersection of these curves correspond to 6 systems of ABELian root forms.

  4. 4)

$\Delta_{5}$ breaks up into 2 straight lines and a $C_{3}$. Any relation
$F_{i}=0$ has the form

(c) $a_{i}x_{1}^{2}+b_{i}x_{2}^{2}+\psi_{i}(x_{3},x_{4},x_{5})=0\,.$

Here one has 2 collineations of type ($\alpha$) and one of type ($\beta$).
Seven systems of ABELian root forms always occur.

  5. 5)

$\Delta_{5}$ breaks up into a straight line and 2 conics.

(d) $F_{i}=a_{i}x_{1}^{2}+\varphi_{i}(x_{2},x_{3})+\psi_{i}(x_{4},x_{5})=0\,.$

A collineation of type ($\alpha$) and 2 of type ($\beta$). Eight systems of
ABELian root forms.

  6. 6)

$\Delta_{5}$ breaks up into 3 lines and a conic section.

(e)
$F_{i}=a_{i}x_{1}^{2}+b_{i}x_{2}^{2}+c_{i}x_{3}^{2}+\psi_{i}(x_{4},x_{5})=0\,.$

There are 3 collineations of type ($\alpha$) and 4 of type ($\beta$). Nine
systems of ABELian root forms.

  7. 7)

$\Delta_{5}$ breaks up into 5 straight lines.

(f)
$F_{i}=a_{i}x_{1}^{2}+b_{i}x_{2}^{2}+c_{i}x_{3}^{3}+d_{i}x_{4}^{4}+e_{i}x_{5}^{2}=0\,.$

In this case one has 5 collineations of type ($\alpha$) and 10 of type
($\beta$). From the 3 independent relations $F_{i}=0$ one can eliminate any
pair $x_{k}^{2}$ and $x_{i}^{2}$ and thus obtain 10 relations between 3
variables, which correspond to just as many systems of Abel root forms.

††margin:  start p.35

The question can now be raised as to how one can arrive at the structure in
4-dimensional space given $\Delta_{5}$. First it should be emphasised that
both a general $C_{5}$ and a structure of the genus $p=5$ have twelve moduli
that are indestructible by unambiguous transformation, so that a finite number
of solutions can be expected. It turned out

$\Delta_{5}(\lambda)=\sum\pm u_{11}u_{22}u_{33}u_{44}u_{55}$

in determinant form, where the $u_{ik}=u_{kl}$ are linear functions of the
$\lambda$. The unambiguous transformation connecting the curves $\Delta_{5}$
and $D_{10}$ has the form

$x_{1}:x_{2}:x_{3}:x_{4}:x_{5}=U_{i1}:U_{i2}:U_{i3}:U_{i4}:U_{i5}$

or

$x_{i}x_{k}=\rho U_{ik}\,,$

where $U_{ik}$ is a subdeterminant from $\Delta_{5}$ to $u_{ik}$. From these
equations one can produce the $\lambda_{1}^{4}$,
$\lambda_{1}^{3}\lambda_{2},\,\dots$ as a linear function of the $x_{1}^{2}$,
$x_{1}x_{2},\,\dots$. According to this, the intersections of $\Delta_{5}$
with all fourth-order curves correspond to the intersections of $D_{10}$ with
the manifolds represented by the quadratic relations; in particular, the
intersections of $D_{10}$ with the doubly counted 3-dimensional spaces
correspond to a quadruple infinite system of fourth-order tangent curves at
$\Delta_{5}$. This system results from changing the determinant $\Delta_{5}$.
So one would have to look for that class of touch [contact. ID: tangent?, LDH:
Maybe, discuss further with AB] systems by which $\Delta_{5}$ can be written
in the form of a symmetric determinant.232323Cf. HESSE’s investigations on the
contact conic sections of a $C_{3}$ and on the systems of the first kind of
$C_{3}$ in contact with a $C_{4}$, CRELLE’s Journal, Vol. 36,49. The
formations in the 4-dimensional space originating from different contact
systems are generally not collinearly related, because for this it would be
necessary for the systems to also be related in this way. It should also be
noted that a collineation of the $\Delta_{5}$ must also convert the underlying
contact system into itself, so that a corresponding collineation exists for
the normal curve with $p=5$. The contact systems in question must be adjoint
††margin:  start p.35 and pass through the double points of $\Delta_{5}$,
which is evident from the fact that the double points of the $\Delta_{5}$
correspond to straight lines of $D_{10}$.

###  12 p.36

A $C_{7}$ in the usual 3-dimensional space and a flat $C_{6}$ can always, if
their genus $p=5$, come out as projections of the normal curve, because the
$\infty^{2}$ planes resp. $\infty^{2}$ straight lines cut out special linear
series in the two cases mentioned. Now the question arises how, given a plane
$C_{6}$ with 5 double points, one can determine the two points of the curve
which are to be regarded as successive projection centres. These points,
together with an arbitrary pair united in a double point, should form a group
$G_{4}$, which belongs to a special linear series $g_{4}^{1}$. However, this
condition is only satisfied by those points which are cut out by a conic
section drawn through the 5 double points.
The $C_{7}$ projected from the various points of the normal curve each reveal
the peculiarities of the system of special linear series $g_{4}^{1}$ through
their trisecant ruled surface. The 3 points which, together with the
projection centre, form a group $G_{4}$ of such a special linear series must
be projected in a straight line. Hence the generators of the ruled surface and
the series $g_{4}^{1}$ correspond unambiguously. The point groups of the
reciprocal $g_{4}^{1}$ are cut from the planes through the trisecant
associated to the given $g_{4}^{1}$, and in particular 2 generators
corresponding to reciprocal $g_{4}^{1}$ intersect each other, their common
plane going through the image point of the projection centre. The locus of the
intersections of two such reciprocal generators is the projected image of the
curve $D_{10}$. If a $g_{4}^{1}$ and consequently the corresponding trisecant
is reciprocal to itself, the trisecant must be double torsal such that the
torsal plane contains the tangents of the 3 points lying on the torsal of the
$C_{7}$. The $C_{7}$ is the fivefold curve and the $D_{10}$ double curve of
the trisecant end ruled surface. A generator thus intersects 12 others in the
3 points of $C_{7}$ and one in the point of $D_{10}$, and the degree of the
ruled surface is consequently 15. For the genus $P$ of a cross-section one
obtains according to PLUCKER’s formulas:

$P=11-\delta\,,$ ††margin:  start p.37

where $\delta$ means the number of the above-mentioned double torsals with the
same plane. Depending on the nature of $\Delta_{5}$ and $D_{10}$ one has
(corresponding to the typical cases of the previous section) the following 7
cases.

  1. 1)

$\Delta_{5}$ is an actual $C_{5}$, which can have a maximum of 6 double
points. The genus of $R_{15}$ is usually 11, but can be lowered to 5 by
$\delta$.

  2. 2)

$\Delta_{5}=C_{4}+C_{1}$. $R_{15}=R_{12}+K_{3}$. The genus of $R_{12}$
alternates between 7 and 4, with the reduction being effected by double points
of $C_{4}$. $K_{3}$ is of genus 1, and the cusp contains that point of $C_{7}$
which corresponds to the centre of projection in the transformation ($\alpha$)
of the previous section. The generators of the cone connect the points
corresponding to this transformation. $K_{3}$ and $R_{12}$ intersect in
$C_{7}$, counted quadruple, touching each other along the 4 double torsals,
which correspond to the intersections of the constituent parts of
$\Delta_{5}$. Also in the following cases, in which $R_{15}$ breaks up, the
sectional curves only consist of $C_{7}$ and double torsals.

  3. 3)

$\Delta_{5}=C_{3}+C_{2}$. $R_{15}=R_{9}+R_{6}$. $R_{9}$ has genus 4 or
(through a double point of $C_{3}$) 3. $R_{6}$ is always of genus
$P=2$.242424Regarding this $R_{6}$ see my work, Klassifikation af regelytorna
af 6. graden p. 57 (Lund 1892. Diss.)

  4. 4)

$\Delta_{5}=C_{3}+2C_{1}$. $R_{15}=R_{9}+2K_{3}$.

  5. 5)

$\Delta_{5}=2C_{2}+C_{1}$. $R_{15}=2R_{6}+K_{3}$.

  6. 6)

$\Delta_{5}=C_{2}+3C_{1}$. $R_{15}=R_{6}+3K_{3}$.

  7. 7)

$\Delta_{5}=5C_{1}$. $R_{15}=5K_{3}$. The $C_{7}$ lies here on $5K_{3}$, each
pair of which touches each other along the connecting line of the cusps.

###  13 p.37

In order to determine the complete collineation group of a given normal curve,
one first has to consider the covariant curve $\Delta_{5}$. One classifies
them in one of the 7 cases of the 11th section and then look for their
collineation group. From this group one has to remove the subgroup to which
collineations of the normal curve also belong. A given linear substitution in
the $\lambda_{i}$ corresponds to the transposed substitution in the $F_{i}$
due to the equation $\lambda_{1}F_{1}+\lambda_{2}F_{2}+\lambda_{3}F_{3}=0$.
Consequently, one can search for the typical periodic collineations in the
4-dimensional space from the outset, through which a linear transformation is
effected under ††margin:  start p.38 3 quadratic relations whose intersection
curve has no double point. If we transfer the result of such an investigation
to the $\lambda$-plane, it follows that only the collineations from period 2
and the non-perspective $G_{3}$, $G_{4}$ and $G_{5}$ need to be considered. In
4-dimensional space there are also cyclic $G_{6}$ and $G_{8}$; however, the
associated subgroup $G_{2}$ leaves each quadratic relation unchanged.
The finite planar and simple collineation groups are (apart from the cyclic
ones with a prime number period) the icosahedron group, the $G_{168}$ known
from the theory of a special $C_{4}$ and a $G_{360}$.252525This $G_{360}$ was
discovered by Mr. VALENTINE (Kong. Dansk. Vid. Selsk. Skrift. (6) V, 1889),
but otherwise seems to be unknown. The first two are known not to have an
invariant form of 5th order; the same must be true for $G_{360}$ because it
contains icosahedral groups. If one then also takes the results of section 10
regarding the structures with a $g_{3}^{1}$ into account, we have the
following theorem:
All groups occurring within the non-hyperelliptic structures of genus $p=5$
can be broken up into a series of purely cyclic groups.
If $\Delta_{5}$ now has a group of order $\mu$, to which collineations of the
normal curve also belong, then one has the following numbers for the order of
the total group of the latter, depending on the typical case of $\Delta_{5}$:
$\mu,\,2\mu,\,2\mu,\,4\mu,\,4\mu,\,8\mu,\,16\mu$.
Finally, those curves are listed here in their normal equations whose moduli
are completely determined by the condition of admitting certain collineation
groups.

  1. 1)

Group of Order 192.

$\displaystyle F_{1}$ $\displaystyle=x_{1}^{2}+x_{4}^{2}+x_{5}^{2}=0\,,$
$\displaystyle F_{2}$ $\displaystyle=x_{2}^{2}+x_{4}^{2}-x_{5}^{2}=0\,,$
$\displaystyle F_{3}$ $\displaystyle=x_{3}^{2}+x_{4}x_{5}=0\,.$

One has (with the identity) 8 collineations by sign changes from the $x_{i}$,
where each $F_{i}$ is unchanged. Then one has to notice that the form
$x_{4}x_{3}\cdot(x_{4}^{2}+x_{5}^{2})\cdot(x_{4}^{2}-x_{5}^{2})$ is an
octahedron form broken up into its 3 distinct quadratic factors††margin:
start p.39 . One uses the group of this binary form with suitable permutations
and substitutions of $x_{1},\,x_{2},\,x_{3}$ and then by combining it with
$G_{8}$ the complete $G_{192}$ is obtained.262626In the theory of elliptic
modular functions, this $G_{192}$ belongs to the eighth order main congruence
group. See KLEIN-FRICKE, Modulfunctionen I, p. 338ff. Eliminating $x_{5}$
gives the equations:

$x_{1}^{2}+x_{2}^{2}+2x_{4}^{2}=0\,,\quad
x_{4}^{2}(x_{2}^{2}+x_{4}^{2})-x_{3}^{4}=0\,.$

The curve $\Delta_{5}$ consists of a conic section and 3 straight lines, which
form a polar triangle in relation to the conic section. In the 3 following
cases, $\Delta_{5}$ consists of 5 straight lines, which are also in a special
position.

  2. 2)

Group of order 64.

$\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}$
$\displaystyle=0\,,$ $\displaystyle x_{1}^{2}+ix_{2}^{2}-x_{3}^{2}-ix_{4}^{2}$
$\displaystyle=0\,,\quad(i^{2}=-1.)$ $\displaystyle
x_{1}^{2}-x_{2}^{2}+x_{3}^{3}-x_{4}^{2}$ $\displaystyle=0\,.$

The $G_{64}$ is created by combining the sign changes with the cyclic
permutation of $x_{1},,\ x_{2},\,x_{3},\,x_{4}.$

  3. 3)

Group of order 96.

$\displaystyle x_{1}^{2}+x_{4}^{2}+x_{5}^{2}$ $\displaystyle=0\,,$
$\displaystyle x_{2}^{2}+jx_{4}^{2}+j^{2}x_{5}^{2}$
$\displaystyle=0\,,\quad(j^{3}=1.)$ $\displaystyle
x_{3}^{2}+j^{2}x_{4}^{2}+jx_{5}^{2}$ $\displaystyle=0\,.$

The $G_{96}$ is obtained by combining the sign changes and swapping
$x_{1},\,x_{2}$ and $x_{3}$ on the one hand, and $x_{4}$ and $x_{5}$ on the
other.

  4. 4)

Group of order 160.

$\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}$
$\displaystyle=0\,,$ $\displaystyle x_{1}^{2}+\varepsilon
x_{2}^{2}+\varepsilon^{2}x_{3}^{2}+\varepsilon^{3}x_{4}^{2}+\varepsilon^{4}x_{5}^{2}$
$\displaystyle=0\,,\quad(\varepsilon^{5}=1.)$
$\displaystyle\varepsilon^{4}x_{1}^{2}+\varepsilon^{3}x_{2}^{2}+\varepsilon^{2}x_{3}^{2}+\varepsilon
x_{4}^{2}+x_{5}^{2}$ $\displaystyle=0\,.$ ††margin:  start p.40

Here the allowed permutations of the $x_{i}$ form a Dihedral $G_{10}$. Namely,
the array $(x_{1}\,x_{2}\ x_{3}\,x_{4}\,x_{5})$ can both be cyclically shifted
into itself and reversed into the array $(x_{5}\,x_{4}\,x_{3}\,x_{2}\,x_{1})$.
The $G_{160}$ is created by combining this $G_{10}$ with the sign changes.

###  14 p.40

Finally, a few hints about the formations of genus $p=6$ shall be permitted.
Apart from the hyperelliptic case, there are 3 essentially different types,
which we want to treat here in series.

  1. 1)

The curve has a special linear series $g_{5}^{2}$, i.e. it can be transformed
into a plane $C_{5}$. With an unambiguous transformation of $C_{5}$ in itself,
the point groups $G_{5}$ of $g_{5}^{2}$ must form a closed invariant system.
Such a transformation must consequently be a collineation, and according to a
theorem in the previous section composed entirely of cyclic groups.

  2. 2)

The curve has a special linear series $g_{3}^{1}$. Such a curve is easily
unambiguously converted into a $C_{6}$ with a triple and a double point. In
the case of a group of unambiguous transformations of the curve in itself, the
point groups $G_{3}$ of the special linear series must be permuted in a closed
manner, which of course can only be effected by one of the finite linear
groups; but not by an icosahedron group, because the number of $G_{3}$
containing 2 coincident points is 16, and no 16 points in this group form a
closed system. In addition, the transformation group can still be composed of
a cyclic group of period 3, which cyclically exchanges the points of each
$G_{3}$.

  3. 3)

In the general case, the curve has 5 $g_{4}^{1}$ and can be unambiguously
converted into a $C_{6}$ with 4 double points, whereby the point groups of the
$g_{4}^{1}$ are cut out by the pencil of rays by a double point each and the
conic pencil by all 4 $\delta$. With an unambiguous transformation of the
curve in itself, these $g_{4}^{1}$ can either be permuted in themselves or in
some way. The former is not possible if there are no 3 double points in a
straight line. Each permutation of the $g_{4}^{1}$ can be effected by
collineations and successive quadratic transformations. One also gets a case
where all 120 permutations of the 5 $g_{4}^{1}$ transform the curve into
itself:††margin:  start p.41

$2\sum(x^{4}yz+y^{3}z^{3})-2\sum x^{4}y^{2}+\sum
x^{3}y^{2}z-6x^{2}y^{2}z^{2}=0\,.$

The double points are here in the coordinate corners and in the point $x=y=z$.

Finally, as a result of our investigations, we can cite the following general
propositions, which Mr. DYCK has already given for the genera
$p=0,\,1,\,2,\,3$.272727Über reguläre Riemann’sche Flächen, Math. Ann., Vol.
17, p. 474
The only simple and finite groups of unambiguous transformations of a curve in
itself within the genera $p=0,\dots,6$ are the cyclic groups of prime order,
the icosahedron group and a $G_{168}$ (at $p=3$). The composite groups within
our genera, with 3 exceptions, can be broken up into a sequence of merely
cyclic groups. The exceptions are: BRING’s curve ($p=4$) and the just
mentioned curve with $p=6$, whose groups are holohedrally isomorphic with the
group of 120 permutations of 5 things, as well as a hyperelliptic
curve282828The equation of this curve is $y^{2}=x(x^{10}+11x^{5}-1)$. In the
theory of elliptic modular functions, it supplies the modules belonging to a
normal congruence subgroup of the 10th order. See KLEIN-FRICKE,
Modulfunctionen I, p. 651 of genus $p=5$, whose group is also of order 120;
namely, within these groups, an icosahedron group is normal.


To understand better the structure of $\mathfrak{M}_{EB}\left(L,\tau\right)$ or more generally the moduli space of solutions to Kähler-Yang-Mills equations [2, 3], it is inevitable to discuss the “degeneration” of the metric when the Higgs field varies, which represents a variation of complex structure [15] on the $SU(2)$ bundle over $\mathbf{P}^1\times \mathbf{P}^1$.  In this perspective, we should mention a general existence result about “multiple strings” Einstein-Bogomol'nyi metrics on $\mathbf{C}$ <cit.> since it might be related to the “degeneration” problem in the full moduli space $\mathfrak{M}_{EB}\left(L, \tau\right)$ when we allow $\bm\phi$ to vary along the sequence and volume goes to $+\infty$ simultaneously. In terms of the equation (<ref>), this existence result can be phrased as follows: for any holomorphic function $\bm\phi$ on $\mathbf{C}$ (which is a holomorphic section of the trivial line bundle on $\mathbf{C}$) vanishing at $p_1, \cdots, p_d$ with multiplicities $n_1, \cdots, n_d$ respectively,  finite energy solutions exist under the condition on the total string number

\[
N':= n_1+\cdots + n_d<\frac{1}{\alpha\tau}.
\]

It is also showed that the underlying Riemannian metrics on $\mathbf{C}$ of those solutions are complete if and only if

\[
N':=n_1+\cdots + n_d\leq \frac{1}{2\alpha\tau}.
\]

It follows from the quasi-isometry relation <cit.> of the obtained metric with standard model metric on complement of a compact set in $\mathbf{C}$ that the metric is asymptotically conical in case $N'<\frac{1}{2\alpha\tau}$, and asymptotically cylindrical in case $N'=\frac{1}{2\alpha \tau}$. Taking into consideration of all the results obtained in this article, we can make the following reasonable conjecture:

Let $\left(\omega_i, h_i, \bm\phi_i\right)\in \mathfrak{M}_{EB}\left(L,\tau\right)$ be a sequence of Einstein-Bogomol'nyi metrics with $\bm\phi_i$ all being polystable, and $\bm\phi_i\to \bm\phi$ is also polystable.

* If $\Vol_{\omega_i}\to V\in \left( \underline V, +\infty\right)$, then there exists some sequence $\sigma_i\in PSL(2,\mathbf{C})$ such that $\sigma_i^*\left( \omega_i, h_i, \bm\phi_i\right)$ converges to some $\left(\omega, h, \bm\phi\right)$;

* If $\Vol_{\omega_i} \to +\infty$, then any nonflat Cheeger-Gromov limit of the sequence can be realized as one of the “multiple strings" solutions of Yang.

This study was funded by the start up grant supported by ShanghaiTech University under the No. 2018F0303-000-03. The author would like to thank Professor Mingliang Cai and Oscar Garcia-Prada for some valuable discussions. He would also like to express the gratitude to Mario Garcia-Fernandez, Vamsi Pingali and Song Sun for their interests in the work.

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