 \chapter*{Introduction}\label{ch:0}
 \markboth{\qquad Introduction \hfill}{}

 \section{A reminder of elementary notions}\label{sec:0.1}
 \markright{\hfill A reminder of elementary notions \quad 0.1\quad}
 In this book we will usually consider an algebraic surface in ordinary
3-space and assume it is irreducible. The surface $f$ shall be defined,
in a system of Cartesian coordinates, by an equation of the form
 \[
 f(x,y,z)=0;
 \]
or else, in homogeneous coordinates
 \[
 f(x_1,x_2,x_3, x_4)=0,
 \]
where in the former case $f$ denotes a polynomial in the variables
$x,y,z$, and in the latter a homogeneous polynomial, that is, a {\em
form}.\index{form}

We assume the fundamentals of the projective theory of surfaces,
including the notions of order, class, polar surfaces, and so on, to be
well known\footnote{Cf.\ Enriques--Chisini, {\em Teoria geometrica delle
equazioni e delle funzioni algebriche}, Libro~III, Vol.~II. Bologna,
Zanichelli, 1918.}. We recall now some properties that we will use
later.

A {\em point} $O$ of the surface $f$ of order $n$ is said to be {\em
simple}\index{simple!point} if the lines passing through $O$ cut the
surface in $n-1$ points outside of $O$. The neighbourhood of a simple
point of the surface can be represented in an elementary way upon a
plane with an arbitrarily high order of approximation, that is, keeping
track of infinitesimals of orders $1,2,\dots$ and so on. To start with,
it is clear that, at the first order of approximation, the surface can
be replaced by the tangent plane at the given point. But if one wants to
consider the second order of approximation, one may assume the
Cartesian frame of reference to be chosen in a suitable fashion and
instead replace the surface by the osculating paraboloid of the same
order $z=\fie_2(x,y)$, which can be projected one-to-one onto the plane
$(x,y)$ from the point at infinity on the $z$-axis. One can proceed
similarly for the third order neighbourhood, and so on. Therefore,
given a curve on our surface with an arbitrary singularity at $O$, one
can view this singularity as being composed of multiple points and
infinitely near multiple points, exactly as on the plane\footnote{Cf.\
Enriques--Chisini, Ibidem, Libro~IV, Vol.~II.}.

Singular (or multiple) points for the surface $f$ are defined as
follows: a {\em multiple point}\index{multiple!point} $O$ of {\em order}
$r$ (for $r>1$) is such that the lines through $O$ cut the surface at
$n-r$ residual points. It is convenient to distinguish between {\em
multiple curves}\index{multiple!curves} and {\em isolated multiple
points}\index{isolated!multiple point} (without excluding consideration
of remarkable multiple points belonging to a multiple curve, which occur
as a superposition of both cases); we shall have more to say about
these distinctions later. A multiple curve of order $r$ can be regarded
in general as the intersection of $r$ surfaces or {\em branches}, which
are differentiably distinct at a generic point of the curve and
correspond to the $r$ tangent planes at the point. On the other hand,
an isolated multiple point usually has a conical aspect, in the sense
that the lines having $r+1$ intersections with the surface form a cone
of order $r$ which is generally irreducible.

But particular cases can occur. For instance, two or more branches of
the surface along a multiple curve can touch each other or even come
together in a {\em higher order branch}: the simplest example is
given by the double cuspidal curve. On the other hand, the tangent cone
at an isolated $r$-tuple point can split or even reduce to a plane
counted $r$ times: the simplest example is provided by the uniplanar
double point.

In order to master all the intricacies which can occur when dealing
with singularities of surfaces, it is convenient to define {\em
infinitely near multiple points}\index{infinitely near!points} and the
{\em (proper or infinitesimal) curves} to which they give rise, as one
does for the branches of curves passing through a point\footnote{Cf.\
Enriques--Chisini, Ibidem, Libro~IV, Vol.~IV.}. In this order of ideas,
a uniplanar point appears as a double point having three infinitely
near double points in different directions. The simplest example of an
infinitesimal double curve (line), infinitely near to an isolated
multiple point, is the  {\em tacnode}\index{tacnode}: that is, a double
point $O$ such that the plane sections through $O$ have another double
point infinitely near to $O$. But we do not want to spend more time on
this topic, for which we refer to the volume of Enriques--Chisini quoted
above.

 \section{Rational transformations}\label{sec:0.2}
 \markright{\hfill Rational transformations \quad 0.2\quad}
 A rational transformation of the space to which the surface $f$
belongs is defined by letting the new coordinates $X,Y,Z$ be rational
functions of the coordinates $x,y,z$:
 \begin{equation}
 X=\frac{\fie_1(x,y,z)}{\fie_0(x,y,z)},\quad
Y=\frac{\fie_2(x,y,z)}{\fie_0(x,y,z)},\quad
 Z=\frac{\fie_3(x,y,z)}{\fie_0(x,y,z)},
 \label{eq0.2.1}
 \end{equation}
where the $\fie$ are polynomials, which we can assume to be of the same
order. Through this transformation, which we assume to be
nondegenerate, to every generic point of the space $(xyz)$ there
corresponds a point of the space $(XYZ)$. Conversely, to a point
$(XYZ)$ corresponds in general a certain finite number $n$ $(\ge1)$ of
points $(xyz)$, which can be obtained by solving the equations
(\ref{eq0.2.1}), where the $XYZ$ are regarded as given, and the $xyz$
as unknown. For $n>1$ the groups of points thus obtained form, in the
space $(xyz)$, an involution $I_n$ of order $n$, that is, a series of
groups of $n$ points such that every point belongs in general to one
and only one group.

Now, in general, the transformation given by (\ref{eq0.2.1}) takes the
$\infty^2$ points of the surface $f(x,y,z)=0$ into $\infty^2$ distinct
points, describing a surface $F(X,Y,Z)=0$, the transform of $f$. To a
generic point of $f$ shall therefore correspond a single point of $F$,
and conversely, to a generic point of $F$ shall usually correspond a
single point of $f$, the transformation turning out to be one-to-one
invertible for the points of the surface $F$.\footnote{Hence the
distinction between a {\em birational transformation} relating two
surfaces and a {\em Cremona transformation}, that extend to the space
containing them.} This invertibility, and the particular circumstances
under which it fails to hold, require some explanation. We saw indeed
that the transformation (\ref{eq0.2.1}), regarded as a map from the
space $(xyz)$ to the space $(XYZ)$, is not in general one-to-one
invertible, but, in fact, carries the points $(XYZ)$ to the groups of
$n$ points of an involution $I_n$ in the space $(xyz)$. But if $f$ is
chosen in a general way inside this space, it will not belong to the
involution $I_n$, and thus a point of $f$ shall have $n-1$ conjugate
points in $I_n$, lying outside the surface. Therefore to a generic
point of $F$ shall correspond, via (\ref{eq0.2.1}), one point of $f$ and
$n-1$ points outside of $f$, which shall describe the conjugate surface
of $f$ with respect to $I_n$.

Only in the case when $f$ belongs to the involution $I_n$, or to an
involution partially contained in it, will the surface in which $F$ is
transformed reduce to a multiple surface, in $[1,n]$ or $[1,m]$
correspondence with $f$ (with $m<n$).

Now, considering the general case, in which $f$ and $F$ are in a
one-to-one correspondence, we should note that the transformation given
by (\ref{eq0.2.1}) is not in general everywhere defined without
exceptions\footnote{One should bear in mind analogous considerations
concerning transformations of curves: cf.\ Enriques--Chisini, Ibidem,
Libro~V, Vol.~III.}. Indeed, the formulas expressing the transformation
become indeterminate at all those points $(xyz)$ for which all the
$\fie$ vanish simultaneously (that is, for the base points of the
transforming system, which is determined by the surfaces $\fie_0=0,
\fie_1=0, \fie_2=0, \fie_3=0$). An exception of this sort can take
place first of all at a simple point of $f$; then such a point is in
general transformed into a line or into a curve of $F$, and this curve,
which is the transform of a simple point, is called an {\em exceptional
curve}\index{exceptional!curve} of the surface $F$.

Alternatively, it may happen that a base point of the transforming
system $\fie$ occurs at a singular point of $f$. And if there is a
multiple curve of $f$ on which the $\fie$ all vanish simultaneously,
then in general this curve will be transformed into a simple curve: to
every $r$-tuple point of the curve will correspond $r$ distinct points,
related to the $r$ branches of $f$ through the point. If instead a base
point of the $\fie$ occurs at an isolated multiple point $O$ of $f$,
this is in general transformed into a curve whose points correspond to
the points infinitely near to $O$.

These statements fail when multiple points and curves infinitely near
to the singular points under consideration intervene. In such cases the
overall effect of the transformation is to simplify the singularity,
transforming infinitely near multiple points of the first order into
proper multiple points, and more generally, reducing from $r$ to $r-1$
the order of infinitesimal neighbourhood of the points under
consideration -- at least, under the assumption that the surfaces $\fie$
passing through a point $O$ do not have any fixed tangent in common.

Nevertheless, the transformation which achieves a simplification of the
singularities of a surface does in general introduce new singularities.
In fact, there will in general be $\infty^1$ pairs of points of $f$ at
which the $X,Y,Z$ take the same values, so that these points will
correspond to the points of a double curve of $F$. This follows
essentially from an easy count of constants. For $X,Y,Z$ to take the
same values at points $(x,y,z)$ and $(x',y',z')$, the following five
equations in six unknowns have to be satisfied:
 \[
 \renewcommand{\arraystretch}{1.2}
 \begin{array}{l}
 \fie_1(x,y,z)\fie_0(x',y',z')-\fie_1(x',y',z')\fie_0(x,y,z)=0, \\
 \fie_2(x,y,z)\fie_0(x',y',z')-\fie_2(x',y',z')\fie_0(x,y,z)=0, \\
 \fie_3(x,y,z)\fie_0(x',y',z')-\fie_3(x',y',z')\fie_0(x,y,z)=0, \\
 \quad f(x,y,z)=0 \hfill  f(x',y',z')=0. \quad
 \end{array}
 \]
In general, these admit $\infty^1$ solutions $(x,y,z,x',y',z')$, in
addition to the $\infty^2$ trivial solutions\footnote{The first three
equations define three hypersurfaces of the space $S_6$, which have in
common the $S_3$ given by ($x=x',y=y',z=z'$) and, outside this, a
certain 3-dimensional variety $V$; this $V$ is intersected by the
cylindrical varieties $f(x,y,z)=0$ and $f(x',y',z')=0$ along a curve.}
 \[
x=x',\quad y=y',\quad z=z'.
 \]

In particular cases, one may have $\infty^1$ groups of more than two
points of $f$ corresponding to a multiple curve of $F$. But there may
also be curves of $f$ on which the $XYZ$ are constant, corresponding to
isolated multiple points of $F$. And as the singularities of $f$ get
more complicated, and as other particular circumstances occur, we
cannot exclude the genesis of higher singularities for $F$.

 \section{Surfaces in hyperspaces $S_r$ and resolution of
singularities}\label{sec:0.3}
 \markright{\hfill Surfaces in $S_r$ and resolution of singularities
\quad 0.3 \quad}
 The transformation of ordinary space considered above can be
generalized, thus defining a transformation carrying the surface $f$
into a projective space $S_r$ of higher dimension:
 \[
 X_1=\frac{\fie_1(x,y)}{\fie_0(x,y)}, \quad
 X_2=\frac{\fie_2(x,y)}{\fie_0(x,y)},\dots, \quad
 X_r=\frac{\fie_r(x,y)}{\fie_0(x,y)}.
 \]
The surface $f(x,y,z)=0$ will be transformed into a surface $F$ of $S_r$
defined by the above formulas, but may also be defined in other ways as
a partial intersection of varieties of the hyperspace $S_r$.
\footnote{Cf.\ Enriques--Chisini, Ibidem, Libro~I, \S23. }

Most of what we said for a transformation of $S_3$ can be repeated for
this transformation; it usually turns out to be one-to-one for the
points of the two surfaces (but not for the points of the spaces
containing them). An ordinary multiple curve of $f$ that is a base
curve for the transforming system may, in general, be resolved, and
becomes the locus of the $r$ simple points corresponding to the
$r$-tuple points of $f$ that make up the curve. The same can be said
for isolated multiple points that map to curves, and so on.

But here, at least when $r\ge5$, no new singular points for $F$ are
produced in general. Indeed when $r\ge5$, there do not exist, in
general, pairs of points of $f$ where the $X_1,X_2,\dots,X_r$ take the
same values, whereas for $r=4$ one has, in general, a finite number of
such pairs, giving rise to the same number of double points for $F$.

One can guess in this way that it will always be possible to resolve
the singularities of a surface $f$ and to transform it into another
surface of $S_r$ completely free of singularities, that is, having only
simple points. But this reduction, which turns out to be easy for
surfaces only having ordinary multiple curves and points (multiple
curves with distinct branches and isolated multiple points with
completely general osculating cone), requires a more detailed
examination when $f$ possesses arbitrarily complicated infinitely near
singularities. However the analysis required for this purpose can be
decomposed into two parts: a first analysis in the differential
geometric sense, where the main issue is to resolve the singularities
defined in a neighbourhood of a point $O$ belonging to a curve in
ordinary space, and where it is allowed to use birational or Cremona
transformations of the entire space $S_3$; and secondly a critical
analysis of the new singularities which can be produced by the
transformation itself. But this last step, which involves delicate
difficulties when dealing with the reduction of singularities through
birational transformations of $S_3$\footnote{A problem solved by
O.~Chisini, in {\em Mem.\ Acc.\ Bologna}, 1921.}, can be avoided via a
simple remark of B.~Levi. In fact, if the surface $f(x,y,z)=0$ is
transformed into another $F(X,Y,Z)=0$, the surface of 6-dimensional
space, which is the locus of the points $(x,y,z,X,Y,Z)$, will only have
singularities corresponding to the common singularities of $f$ and $F$.

We do not propose to dwell here on an explanation of the reduction of
surface singularities, as first rigorously obtained by B.~Levi. We
shall limit ourselves to stating the main result that can be obtained,
referring for a proof to the most important Memoirs which contain and
develop it in different manners\footnote{B.~Levi, Risoluzione delle
singolarit\`a delle superficie algebriche, {\it Atti Acc.\ Torino}, 1897
(cf.\ {\it Ann.\ di Mat.}, 1897).

O. Chisini, La risoluzione delle singolarit\'a delle superficie
mediante trasformazioni birazionali dello spazio. {\it Mem.\ Acc.\
Bologna}, 1921.

G. Albanese, Trasformazione birazionale di una superficie algebrica
qualunque in un'altra priva di punti multipli, {\it Rendic. Circ.\ Mat.\
Palermo}, 1924.

R. J. Walker, Reduction of singularities of an algebraic surface, {\it
Ann.\ of Math.}, 1935.

O. Zariski, The reduction of the singularities of an algebraic
surface, {\it Ann.\ of Math.}, 1939. cf.\ {\it Ann.\ of Math.}, 1941.}.

 \begin{proc}
 A surface $f$ admitting arbitrary singularities can always be
birationally transformed into another one, free of singularities, and
belonging to a $5$-dimensional space.
 \end{proc}

 \section{Surfaces with normal singularities}
 \label{sec:0.4} \index{surface!with normal singularities}
 \markright{\hfill Surfaces with normal singularities \quad 0.4\quad}
 Given a surface with arbitrary singularities, we begin by transforming
it into a surface $F$ of $S_5$ completely free of multiple points. If
we project $F$ successively from two points to an ordinary space, we
shall obtain in the latter space an image $\Phi$ of $F$. We deal with
the question of what the singularities of $\Phi$ will be when the
projection of $F$ is made from two generic points. We want to show that
$\Phi$ shall have only a nodal double curve (with generally distinct
tangent planes), endowed with triple points which are also triple for
the surface.

It is clear first of all that this is the type of {\em normal
singularities}\index{normal!singularities}, which will in general
appear on a surface of ordinary space, obtained as a projection of a
surface free from singular points in a higher dimensional space; this
comes out of a simple computation of constants. Observe in fact that
the chords of a surface of the space $S_r$ are always $\infty^4$, hence
for $r=5$ there is a finite number of chords passing through a point,
whereas for $r=4$ there are $\infty^1$ such chords. Hence, by
projecting the surface $F$ from a generic point of $S_5$, the projected
surface $F'$ shall acquire a finite number of double points,
corresponding to the chords of $F$ passing through the centre. By then
projecting $F'$ from a generic point of $S_4$, the projected surface
acquires an (apparent) double curve to which the double points of $F'$
will map. In other words, through two points of $S_5$, or through the
line joining them, there passes an $\infty^1$ series of planes meeting
$F$ at two points and a finite number of planes meeting it at three
points. Hence the projection of $F$ performed from those centres shall
indeed possess a nodal double curve endowed with triple points, triple
both for the double curve and for the surface.

One might nevertheless wonder whether the projection from two generic
points of $S_5$ of a particular surface of $S_5$, even if free from
multiple points, might acquire higher singularities, that is, a multiple
curve instead of a double curve, or multiple points of multiplicity
greater than 3 for the double curve. In order to exclude the first
doubt, it is enough to remark that the $\infty^4$ chords of a surface of
any space of dimension greater than 3 cannot be trisecants, since
otherwise the chords of a twisted curve, hyperplane section of the
surface, would also be trisecants, which is impossible\footnote{Cf.\
Enriques--Chisini, Ibidem, Libro~III, Chapter~IV, \S43, Vol.~II,
page~289.}. It follows from this that no trisecants of $F$ can pass
through a generic point of $S_5$, and that, when $F$ is projected from a
generic point to a surface $F'$ of $S_4$, only finitely many trisecants
can pass through a generic point of $S_4$.

It remains to prove that, if the centres of projection in $S_5$ are
chosen in a generic way, it is impossible for the trisecant planes
through the centres to meet the surface $F$ in more than three points
and therefore that the double curve of the projected surface $\Phi$
possesses points of higher multiplicity rather than triple
points\footnote{Cf.\ Enriques, Sulle singolarit\`a che nascono per
proiezione di una superficie o variet\`a algebrica, in {\em Scritti
Matematici offerti a L.~Berzolari}, Pavia, 1936, page~351.}. For this,
it is enough to exclude the possibility that all planes subject to the
condition of meeting the surface at three points would, as a
consequence, meet it at four or more points. In fact, if every
trisecant plane of $F$ were quadrisecant, then the same would happen
for every twisted curve, hyperplane section of $F$ (which cannot belong
to some $S_3$). Therefore the projection of this curve from one of its
points to $S_3$ should also give a curve every chord of which is a
trisecant, which, as we already recalled, is impossible.

In such a way one can rigorously prove a fundamental theorem already
discovered by Noether (1888):
 \begin{proc}
 A surface with arbitrary singularities can always be transformed into
another surface of ordinary space having only normal singularities,
that is, a {\em nodal double curve} possessing in general a finite
number of triple points that are also triple points for the surface.
 \end{proc}
The double curve can also possess a finite number of double points
({\em crossings}), arising from the chords of the surface of $S_4$
being projected to $S_3$ that are double for the cone of projecting
chords.

It is also useful to recall that, on the above {\em nodal curve}, there
will in general be a finite number of {\em pinch points}\index{pinch
point} (punti cuspidali) at which the two linear branches of the
surface come together in a second order branch, and which turn out to
be simple base points for the curves obtained as variable intersections
with the polar surfaces.

In what follows, as we develop a theory of surfaces concerned with the
properties that are invariant under birational transformations, we may
always consider either a model surface free from singularities of
5-dimensional space, or a surface of ordinary space endowed with the
normal singularities, as in the preceding statement. In this last case
a biplanar point of the double curve should be viewed as the
superposition of two simple points whose neighbourhoods correspond to
the two branches, as in the invariant theory of curves a node may be
considered as the superposition of two points belonging to the two
branches.

 \section{Note}\label{sec:0.5}
 \markright{\hfill Note \quad 0.5\quad}
 What we have said about the transformation from a surface to another
surface with normal singularities in ordinary space can be made more
precise in the following sense: we may suppose that {\em the transform
of any surface is endowed with an irreducible double curve}, which, as
we said above, possesses a certain number of triple points, triple both
for the surface and for its nodal curve.

We present the proof of this property,\footnote{Cf.\ A.~Franchetta,
Sulla curva doppia della proiezione di una superficie dello $S_4$ da un
punto generico su uno $S_3$. {\it Rendic.\ R. Acc.\ d'Italia}, 1941, see
also {\it Rendic.\ Acc.\ Lincei}, s.~VIII, Vol.~II, (3), 1947.} assuming
known the principles of the theory of linear systems, which we treat in
the first sections of this book.

For this purpose, it is convenient to start from a surface free from
singularities in the space $S_5$, or, even better, from a surface $F$
which is a generic projection of the former in $S_4$, and is therefore
endowed only with {\em ordinary} improper double points,\index{improper
double points} each of which may be viewed as an apparent singularity,
representing a pair of {\em distinct} superimposed points. It is
convenient to analyze the singularities of a surface $\Phi$ of ordinary
space, which is a projection of $F$ from a generic point $O$ of $S_4$:
in which case it could happen that the double curve of $\Phi$ always
turns out to be reducible as the centre of projection $O$ varies.

This examination leads to establishing that: if a surface $F$, endowed
with improper ordinary double points in $S_4$, is projected from a
generic point to a surface $\Phi$ of $S_3$, whose double curve is
reducible, then $\Phi$ is a projection of the Veronese surface (the
surface of the fourth order in $S_5$, represented on the plane by the
$\infty^5$ system of all conics).

To obtain this result it is convenient first to state the following:

 \begin{Lemma} There do not exist surfaces of $S_4$ possessing
$\infty^3$ pairs of tangent planes incident along a line, in such a way
that the lines joining their points of contact fill up the whole space.
 \end{Lemma}

Let $P$ be a generic point of $F$ and $\pi$ the tangent plane at $P$.
By assumption there are $\infty^1$ tangent planes to $F$ that meet
$\pi$ along a line; their points of contact fill up a curve, possibly
reducible, of which we consider a component $C$. Let $Q$ be one of its
points and $Q'$ the point infinitely near to $Q$ on $C$. The tangent
planes to $F$ at $Q$ and $Q'$ each cut $\pi$ in a line; hence each of
them lies together with $\pi$ in some $S_3$. But the tangent plane at
$Q'$ contains $Q$, hence the two above planes determine, together with
$\pi$, a unique $S_3$, which is tangent to $F$ at all points of $C$. As
$P$ varies, the curve $C$ describes an $\infty^1$ system (at least);
and if, as we are assuming, the surface $F$ does not belong to some
$S_3$, each of these curves $C$, which must lie in both the $S_3$
determined by it and in the $S_3$ determined by the infinitely near
curve, is contained in a plane (that could, in particular, reduce to
a line); the planes of the two infinitely near curves $C$ lie in an
$S_3$. We conclude that a surface for which our assumption holds is
either ruled or contains $\infty^1$ plane curves whose planes form a
developable variety. In the former case, the lines joining the points
of contact of two planes incident along lines are the generators of the
ruling; in the latter, they are the lines of the planes of the
developable. In neither case do they fill up $S_4$.

With this setting we assume we have a surface $F$ in $S_4$, endowed
only with finitely many improper ordinary double points, whose
projection $\Phi$ from a generic point $O$ possesses a reducible double
curve. We claim first that this curve must be connected, that is,
formed of two or more parts having some common point. Indeed, the
apparent double curves
 \[
 C=\sum K
 \]
corresponding to the sets of those parts $K$ form a rational (or at
least unirational) system whose elements correspond to the points $O$ of
the space $S_4$, hence the $C$ belong to a linear system of curves on
$F$.\,\footnote{This theorem has been established for rational series of
groups of points over a curve and extended to varieties by Enriques.
cf.\ Enriques--Chisini, Ibidem, Libro~V, Chapter~I, \S9, Vol.~III,
page~78 (cf.\ page~485).} If this system is irreducible, then the
reducible curves $C$ which belong to it must be connected\footnote{The
principle of degeneration, cf.\ Enriques--Chisini, Ibidem, Libro~V,
Chapter~III, \S36, Vol.~III, page~405.}; if instead all the $C$ in the
above linear system are reducible, then they must be composed with the
curves $K$ of a pencil, which do not have variable intersections.

But this case cannot occur, because two curves $K$ containing two pairs
of points such as $AA_1$ and $AA_2$ certainly have (in $A$) a common
point. Note that they cannot coincide with one and the same curve
since, varying $A_2$, this curve would contain all the points of the
surface.

We are led therefore to examine whether and how the surface $\Phi$, a
projection of $F$ from a generic point $O$, may have a double curve
composed of two or more parts connected between them.

A connection point of such parts can be either:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item a simple crossing, that is, a double point that is common to two
simple branches, which shall therefore be a tacnode for the surface, or

 \item a triple point, through which pass three branches of the double
curve, two of which may a priori also belong to the same irreducible
component.
 \end{enumerate}

\paragraph{(1)} A common point $\Pbar$ of two simple branches of the
double curve $\Cbar$ of $F$ will in general be a projection of two
points $P$ and $P'$ which are collinear with the centre $O$ of $S_4$.
One sees immediately that the tangent planes to $F$ at $P$ and $P'$
belong to the same hyperplane whose trace is the tangent plane to
$\Phi$ at $\Pbar$, and thus they are incident along a line. But since
the double point $\Pbar$ is assumed to exist for every position of the
centre $O$, the surface $F$ should possess $\infty^3$ pairs of incident
tangent planes, and the lines joining the contact points of these pairs
of planes should fill up the whole space; by the above lemma this is
absurd.

A special examination is required in the case where the two points $P$
and $P'$ are infinitely near, thus specializing to points which a
tangent line through $O$ has in common with $F$. In this case the point
$\Pbar$, a crossing point for the two branches of the double curve of
$\Phi$, shall again be a tacnode for $\Phi$; hence every plane through
$\Pbar$ shall cut $\Phi$ along a curve still having a double point
infinitely near to $\Pbar$. Hence it follows that the hyperplane joining
$O$ with the tacnodal plane of $\Phi$ must cut $F$ along a curve which
has a fourfold intersection with every plane through the tangent line
$PP'$, and thus it has $P$ and $P'$ as double points. Note that a
separate examination is needed in the particular cases where the
effective multiplicity of $P$ becomes bigger than the virtual
multiplicity 2, which however decreases the multiplicity of $P'$. The
assumption under consideration implies that, as $O$ moves in the space
$S_4$, there is always a tangent through $O$, call it $p$, meeting $F$
at two infinitely near points, that would be double points for the
section with the tangent hyperplane at $P$. Therefore, each of the
$\infty^3$ tangent hyperplanes of $F$ should touch the surface at two
infinitely near points $P$ and $P'$. But it is easy to see that this
property cannot hold for a surface $F$ of $S_4$. In fact, a generic
projection of such a surface in $S_3$ would be such that each of its
tangent planes would also be tangent to it at an infinitely near point,
hence it would turn out to be a developable surface. In which case the
surface in $S_4$ of which it is the projection would also be a
developable surface. However a developable surface in $S_4$ admits only
$\infty^1$ and not $\infty^3$ hyperplanes touching it at two infinitely
near points.

The conclusion we draw from the above discussion is the impossibility
of the above assumption, namely that the projection of our surface $F$
from a generic point $O$ of $S_4$ is a surface $\Phi$ whose double
curve is composed of two components meeting in a point $\Pbar$ which is
the image of two infinitely near points $P$ and $P'$ of $F$. But one
should really examine the particular case which we previously
mentioned: that instead of a tangent hyperplane at $P$ whose
intersection with $F$ has two double points $P$ and $P'$, we encounter
a tangent hyperplane whose section has a triple point at $P$ and a
simple point at $P'$, or just a quadruple point at $P$. Of the two
cases we are confronted with, the first does not give rise to any
exception to the previous argument, since there will still be
$\infty^3$ tangent hyperplanes to $F$ that would also (virtually) touch
it at an infinitely near point. It is instead necessary to dwell for a
moment on the second exception, according to which we would a priori
have only $\infty^2$, rather than $\infty^3$, hyperplanes cutting $F$
along curves endowed with a quadruple point. As a matter of fact, this
exception cannot occur because, if we have a surface of $S_4$ such that
for each of its points one has a tangent hyperplane cutting it along a
curve endowed with a quadruple point, then that surface cannot belong
to $S_4$, but is entirely contained in a space $S_3$. In fact, its
hyperplane sections are curves such that the osculating plane at any
point has a fourfold contact with the curve itself, hence they are
plane curves.

\paragraph{(2)} A point $\Pbar$ which is a triple point for the double
curve $C$ of $\Phi$ and which is a point connecting two irreducible
components of it may be either:
 \begin{enumerate}
 \renewcommand{\labelenumi}{\alph{enumi})}
 \item a simple point common to three irreducible components of $\Cbar$;
or
 \item a point which is simple for an irreducible component of $\Cbar$
and is double for another component.
 \end{enumerate}

 \subparagraph{a)} The point $\Pbar$ is the projection of three points
of $F$, say $P_1,P_2,P_3$; and the components $\Kbar_1,\Kbar_2,\Kbar_3$
of $C$ for which $\Pbar$ is a triple point are the projection of three
curves, respectively $K_1,K_2,K_3$, for which we may assume that the
first passes through the points $P_2$ and $P_3$; the second through
$P_1$ and $P_3$, the third through $P_1$ and $P_2$.

We recall that the surface $\Phi$ obtained as the projection of $F$
must have a singularity similar to that at $\Pbar$, no matter how the
centre $O$ of the projection varies in $S_4$; in particular, we let it
move along the line $P_1P_2P_3$. Now, corresponding to such a variation
in $O$, it can either happen that the apparent double curves
$K_1,K_2,K_3$ on $F$ remain fixed, or that they vary with $O$. If one
of them, say $K_1$, remains fixed, we deduce that it is a conic. In
fact, for $\Kbar_1$ to project always to the same curve, we must have
that every line joining a point of $\Kbar_1$ with a point of the line
$P_2P_3$ meets $K_1$ at another point, and thus $K_1$ is contained in a
plane through the line $P_2P_3$. A plane curve that projects, as
this one does, to a double curve (and not to a curve of higher
multiplicity) must be a conic. Thus the surface $F$, possessing
$\infty^2$ conics, must be a projection of the Veronese surface.

We should now examine the hypothesis that the curve $K_1$ varies as $O$
varies on the line $P_2P_3$. In such a case, for a suitable position of
$O$, $K_1$ will also pass through the point $P_1$ of this line, and
correspondingly we shall have a surface $\Phi$ endowed with a
particular triple point $\Pbar$, where two branches, say those
corresponding to the neighbourhoods of $P_1,P_2$, are tangent to each
other. But this implies that the tangent planes to $F$ at $P_1$ and
$P_2$ meet along a line. Then under our assumptions, $F$ will have
$\infty^3$ pairs of incident tangent planes in such a way that the
lines joining the contact points of these pair of planes fill up the
space $S_4$; this is excluded by the above lemma.

 \subparagraph{b)} We examine now the hypothesis that the triple point
$\Pbar$ of $\Phi$ is simple for one component $\Kbar_1$ and double for a
component $\Kbar_2$ of the double curve. $\Kbar_1$ and $\Kbar_2$ are
projections of two apparent double curves $K_1$ and $K_2$ of $F$, and we
may assume that the former passes simply through $P_2$ and $P_3$, and
the latter passes simply through $P_2$ and $P_3$ and doubly through
$P_1$. We now let the centre of projection $O$ vary on the line
$P_1P_2P_3$; it may happen either that the curve $K_1$ stays fixed or
that it varies with $O$. If it stays fixed, we conclude as before that
it is a plane curve, and thus a conic, whence it follows that $\Phi$ is
a projection of the Veronese surface. If instead $K_1$ varies with $O$,
for a suitable position of $O$, it will also pass through $P_1$. The
singularity that $\Phi$ presents at $\Pbar$ is now a triple point in
which at least two branches touch each other, and therefore the tangent
planes to $F$ at two such points as $P_1$ and $P_2$ will turn out to
meet along a line. The surface $F$ possesses $\infty^3$ pairs of
incident tangent planes, and we fall back to the case we already
excluded.

{From} all the preceding discussion we conclude that
 \begin{proc}
 the projection to the space $S_3$ of a surface of $S_5$ free from
singularities has in general an irreducible double curve; the only
exception being the case of the Veronese surface which in fact is
projected onto the Steiner surface, having three double lines passing
through a triple point.
 \end{proc}
