\input ensmath
%\francais  %% If the paper were in French, we would remove the %-sign here!
            %% Try it to see the difference!
\input macutil
%
\let\phi=\varphi
\def\Vd{V_{\delta}} 
 
\titrebref{\maj{Rational curves}}
 
\titre{\maj{Some interesting rational curves\\ on K3 surfaces}}
 
\auteurbref{D. Coray and S. Others}
 
\auteur{Daniel \sn{Coray} and Several \sn{Others}}
 
\abstract{This is a mock paper to illustrate some features of the macros
of {\sl L'Enseignement Math\'ematique}\note{For more details on how to
use this set of macros, the reader is referred to the specific instructions,
which are at present only available in French.}.}

\sec{}{Introduction}
 
The following assertion was made by B. Segre:
 
\proclaim{($\SS$)\noperiod}{The general quartic surface $F$ contains a
finite number $c_h>0$ of rational curves of degree $4h$  
$(\text{\it for } h=1,2,3,\dots)$.}
 
It is interesting to compare with a similar statement proved
by Mumford and by Bogomolov (see [M-M]):
 
\theorem{(Bogomolov \&\ Mumford)}{Every algebraic K3 surface over~$\CC$
contains a singular rational curve and a pencil of singular elliptic
curves.}
 
\parag{Scholia}
What happens in reality is somewhat surprising. We shall try to explain
here why this is so.
 
\proclaim{Scholium~1}{\smallpenalty\rm
It is known that the Chow variety of space curves of degree~$m$ has an 
irreducible component $\Rr_m$, of dimension~$4m$.}
 
\sec{1.}{About finiteness}
 
Segre's heuristic starting point was that a complete linear system of 
arithmetic genus~$g$ on a smooth quartic is also of dimension~$g$. 
 
\lemma{1.1}{
No K3 surface carries any one-dimensional algebraic system of 
irreducible rational curves.}
 
\proof{
We denote by $p\: \Jj\to X$ the first
projection and observe that $p$ is dominant, unless the family is 
constant.\qed}
 
\remark{}{
Lemma 1.1 does not imply that a given K3 surface cannot contain
infinitely many smooth rational curves.}
 
\subsec{2.}{About existence}
 
Let $\YYY\subset \Vd\times F$ be the universal Cartier divisor of
the flat family. 
 
\remarks{
1)\enspace 
The schemes parametrizing all curves of a given geometric genus
have been well examined.
 
2)\enspace 
We can think of a smooth cubic surface as being $\PP^2$ with six points
blown up.}
 
\definition{2.1}{A {\it quartic\/} in $\PP^3$ is a smooth surface of 
degree~$4$.}
 
In the present paper we are especially interested in the case where
$\lambda=3$. 
 
\EMdemo{Proof of Theorem~4.1}{
As explained at the beginning of \S3, we can work over an open
set of reduced and irreducible curves.\qed}
 
\corollary{$4.2'$\noperiod}{
The smooth quartics in $\PP^3$ carrying curves of degree~$8$ form 
a constructible set in~$\Ll_K$.\qed}
 
\begreflab{References}{1.4cm}
 
\ref Ca {\smc Cayley, A.} On a new analytical representation of curves
in space. {\it Quart. J. of Pure and App. Math. 3\/} (1860), 225--236
({\it Collected Papers, vol.~4}, 446--455).
 
\ref De {\smc Deligne, P.} Le th\'eor\`eme de Noether. In:
{\it SGA~7~II}, Lecture Notes in Mathematics no. 340,
expos\'e 19, pp.~328--340. Springer (Berlin--Heidelberg--New York), 1973.
 
\ref Ha {\smc Hartshorne, R.} {\it Algebraic Geometry.} Springer
(New York), 1977.
 
\ref M-M {\smc Mori, S. {\rm and} S. Mukai.} The uniruledness of the
moduli space of curves of genus~$11$. In: {\it Lecture Notes in Maths
1016\/} (1983), 334--353.
 
\endref
 
\adresse
Daniel Coray \\
Section de Math\'ematiques
2--4, rue du Li\`evre
CH--1211 Gen\`eve 24
Switzerland
{\it e-mail:\/} coray\@ibm.unige.ch
\endadresse
\smallskip 
\adresse
Several Others \\
Universit\'e de Gen\`eve
C.P. 240
CH--1211 Gen\`eve 24
Switzerland
{\it e-mail:\/} Several.Others\@math.unige.ch
\endadresse
 
\bye
 
