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\issueinfo{3}{3}{January}{1997}
\dateposted{April 8, 1997}
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\PII{S 1079-6762(97)00019-X}
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\theoremstyle{remark}
\newtheorem{gen}[definition]{Generalizations}
\newtheorem{ques}[definition]{What is missing?}
\newtheorem{notation}[definition]{Notation}
\newtheorem{remarks}[definition]{Remarks}
\newtheorem{procedure}[definition]{A procedure to find semi-stable models}
\newtheorem{termination}[definition]{Termination of the procedure}
\newtheorem{requirements}[definition]{The requirements for the theory}

\newtheorem{nota}[prop]{Notation}

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\begin{document}

\title[Polynomials with integral coefficients]{Polynomials with integral 
coefficients,\\
equivalent to a given polynomial}
\author{J\'{a}nos Koll\'{a}r}
\address{University of Utah, Salt Lake City, UT 84112}
\email{kollar@{}math.utah.edu}
\subjclass{Primary 11G35, 14G25, 14D10; Secondary
11C08, 11E12, 11E76, 11R29, 14D25, 14J70}
\keywords{Polynomials, hypersurfaces, geometric invariant theory, class
numbers, quadratic forms}
%\issueinfo{3}{1}{}{1997}
%\copyrightinfo{1997}{American Mathematical Society}
\commby{Robert Lazarsfeld}
\date{January 30, 1997}
\begin{abstract}Let $f(x_{0},\dots ,x_{n})$ be a homogeneous polynomial with rational
coefficients. The aim of this paper is to  find a polynomial with
integral coefficients $F(x_{0},\dots ,x_{n})$ which is ``equivalent" to $f$
and as ``simple" as possible. The principal ingredient of the proof 
is to connect this question with the geometric invariant theory of
polynomials. Applications to binary forms, class numbers, quadratic
forms and to families of cubic surfaces are given at the end.
\end{abstract}
\maketitle


%\setcounter{section}{1}
\section{Introduction}\label{sec:1}
Let $f(x_{0},\dots ,x_{n})$ be a homogeneous polynomial with rational
coefficients. The aim of this paper is to  find a 
polynomial with integral
coefficients $F(x_{0},\dots ,x_{n})$ which is ``equivalent" to $f$ and
as ``simple" as possible.

There are two frequently considered notions of equivalence for polynomials.
We say that two polynomials $f,g$ are {\em affine equivalent} if there is an
invertible matrix $M\in GL(n+1,\q )$  such
that
$f(\mathbf{x})= g(M\mathbf{x})$.
  $f$ and $g$ are  called {\em projectively equivalent} if there is an
invertible matrix $M\in GL(n+1,\q )$ and a nonzero constant $c\in \q ^{*}$ such
that
$f(\mathbf{x})=c\cdot g(M\mathbf{x})$. 
Both of these concepts are  equivalence relations which preserve the degree.

It is much less clear how to define which polynomials are ``simple". In fact,
the main aim of the present paper is to develop such a definition.

Many special cases of this question have been considered in the past. 

\subsection{} \label{sub:1.1}
The study of quadratic number fields $K=\q (\sqrt {D})$ is intimately
connected with the study of binary quadratic forms $Q=ax^{2}+bxy+cy^{2}$. Let
$d_{K}$ denote the discriminant of $K$. A quadratic form $Q$ is  called
fundamental for $K$ if $b^{2}-4ac=d_{K}$. The study of equivalence classes of
fundamental quadratic forms is  equivalent  to the study of the ideal class
group of $K$; see, for instance, \cite[VII.2]{Frohlich-Taylor91}. 

\subsection{} \label{sub:1.2}
In the theory of elliptic curves it is frequently useful to look at a
Weierstrass equation 
\begin{equation*}y^{2}z+a_{1}xyz+a_{3}yz^{2}=x^{3}+a_{2}x^{2}z+a_{4}xz^{2}+a_{6}z^{3},\qtq {where $a_{i}\in \z $.}
\end{equation*}
   Such an equation is called minimal if its discriminant is the smallest
possible; see \cite[VIII.8]{Silverman86} for details. (If 
$a_{1}=a_{2}=a_{3}=0$, then
the discriminant is the usual expression
$4a_{4}^{3}+27a_{6}^{2}$, up to a normalizing factor. The general form is quite
complicated but it can be reduced to this one by suitable substitutions.)

\subsection{}\label{sub:1.3}
In the fifties Demyanov, Lewis, and Springer studied the solvability of
equations $f=0$ in the $p$-adic field $\q _{p}$ (Artin's conjecture). Their 
method was to find an equivalent equation $F=0$ with 
coefficients in the $p$-adic
integers
$\z _{p}$ such that the reduction of $F$ modulo $p$ is relatively simple. Their
main concern was to ensure that $F$ modulo $p$ is irreducible, even over the
algebraic closure of the finite field $\f _{p}$. See \cite{Lewis89} for an
overview and references.

\subsection{}\label{sub:1.4}
Recently, Corti \cite{Corti96} studied families of Del Pezzo surfaces over
smooth curves. His aim was to use birational transformations to obtain  a
family whose total space has very mild singularities. For cubic  surfaces
over Dedekind domains
he proposed a program to obtain such a model. He  proved that the program works
over  curves in characteristic zero, but the positive characteristic
and the arithmetic cases were left open.

The intention of this paper is to  propose a  unified
generalization of  the above special cases. 

\setcounter{definition}{4}
\begin{definition}\label{defn1.5} 
Let $f(\mathbf{x})\in \q [x_{0},\dots ,x_{n}]$ be a homogeneous polynomial. For any $M\in GL(n+1,\q )$ and
$c\in \q ^{*}$ consider the new polynomial obtained by the coordinate change 
$F(\mathbf{x}):=c\cdot f(M\mathbf{x})$. It may happen that 
$F(\mathbf{x})$ has integer coefficients, in which case I call $F$ 
an {\em integral model} of $f$. If $F(\mathbf{x})$ is an integral model 
of $f$ and $A\in GL(n+1,\z )$, then $\pm F(A\mathbf{x})$ is 
also an integral model. Such integral
models will be considered {\em integrally equivalent}.

The ``simplest" integral models will be called {\em semi-stable over $\z $}.
(The choice of terminology should become clear later.)
Before giving the precise definition  (see (\ref{defn3.3})),   I list the
main properties that such a concept should possess.
\end{definition}

\setcounter{subsection}{5}
\subsection{The requirements for the theory}\label{rem1.6} 
Semi-stable models  over $\z $
should satisfy the following   properties:

(1.6.1) Being semi-stable over $\z $ should be a local property. That is,
$F(\mathbf{x})$ is semi-stable over $\z $ iff it is semi-stable 
over $\z _{p}$ (the ring of $p$-adic integers) for every $p$.

(1.6.2)   $f$  should have only finitely many integral
models (up to integral equivalence)  which are semi-stable over $\z $.

(1.6.3) Let $\Phi : (\text{polynomials})\to \r ^{+}$ be any function which is
invariant under coordinate changes by $SL$ and satisfies
$\Phi (cf)=c^{r}\Phi (f)$ for some $r=r(\Phi )>0$. Then an integral model $F$ of
$f$ is semi-stable over $\z $ iff $\Phi (F)$ is minimal among all
integral models of $f$.


A procedure to construct semi-stable models for a polynomial 
over $\z $ is described in (\ref{proc4.3}).
In some cases this procedure does  not
produce integral models satisfying all of the above properties (\ref{ex4.4}). On the
other hand, such polynomials turn out to be  rare, as shown by the
following   special case of the main theorem (\ref{thm4.1}):

\setcounter{definition}{6}
\begin{theorem}\label{thm1.5}  
Let $f(\mathbf{x})\in \q [x_{0},\dots ,x_{n}]$ be a homogeneous
polynomial and assume that 
 $(f=0)\subset \cXYZi \p ^{n}$ is a smooth hypersurface.
Then $f$ has semi-stable models over $\z $ which satisfy the properties
(1.6.1--3).
\end{theorem}


A key point of the proof is to connect our problem with the
Hilbert--Mumford geometric invariant theory of hypersurfaces,
 revealing the correct class of polynomials for which semi-stable models over
$\z $  can be defined.
%\end{proof}

\setcounter{subsection}{7}
\subsection{Generalizations}\label{gen1.8}
Instead of $\z $, one can work over other rings.
There are no major changes for principal ideal domains.
It seems to me that the natural setting of the
construction is over arbitrary Dedekind domains, or over one-dimensional
regular schemes. In these cases we obtain polynomials defined on vector
bundles.

One can try to work over an arbitrary normal scheme, but, as far as I can
tell, the theory is restricted to codimension one primes. 

The most general setting is probably the following. Let $C$ be a 
one-dimensional
regular scheme and $G/C$ a group scheme over $C$ which acts on a
scheme $X/C$. Given a closed point $P$ of the generic fiber of $X/C$, we   want
to find a ``simple" section $s:C\to X$ which passes through the $G$-orbit of
$P$. (If
$G=GL$ and $X$ is a symmetric power of its standard representation, we obtain
the case of homogeneous polynomials considered above.)

\subsection{}\label{ques1.9}  \textbf{What is missing?}
In this paper we consider
integral models which are ``simple" at every prime $p$. It is very natural to
ask for a theory which also takes into account the infinite prime.
For definite quadratic forms such an approach  is provided by the reduction
theory of Minkowski--Siegel. Unfortunately, I do not know how to generalize
it to higher degree polynomials.

It is also not clear that the simplest integral model of a polynomial should
be a polynomial. This is more apparent if we look at the corresponding
hypersurface $X_{\q }$ over $\q $ instead. This can be extended to a scheme
$X_{\z }$ in many different ways. In many cases it is advantageous to consider 
a model $X_{\z }$ which is not a hypersurface. One such  example  is in
 \cite[V.5]{Kollar96}.

One can also ask the more general question: given a scheme over $\q $, how can
we extend it to a scheme over $\z $ in the simplest way? This is closely
related to the general geometric invariant theory problem considered in
(\ref{gen1.8}).

Finally, it would  be desirable to have an algorithm which constructs 
semi-stable models. The proof of the existence of semi-stable models 
(\ref{thm4.1}) is
very close to being algorithmic, but so far I could not prove a bound on the
weights occurring in (\ref{proc4.3}), except in some special cases. 

\section*{Acknowledgements}
I would like to thank A. Bertram, H. Clemens, J.-L. Colliot-Th\'{e}l\`{e}ne, S.
Gersten, J. Harris, and R. Lazarsfeld for useful conversations and   e-mails. 
 Partial financial support was provided by the
NSF under grant number  DMS-9622394.
These notes were typeset by %\AmSTeX
$\mathcal{A}\raisebox{-.6ex}{$\mathcal{M}$}\mathcal{S}$-\TeX, 
the \TeX \ macro system of the American Mathematical Society.

\section{A short review of geometric invariant theory}\label{sec:2}

\begin{definition}\label{defn2.1} Let $S$ be a ring.
A {\em weight system} $(\mathbf{x},\mathbf{w})$ on $S[y_{0},\dots ,y_{n}]$
is a 
 choice of coordinates
\begin{equation*}(x_{0},\dots ,x_{n})^{t}=M(y_{0},\dots ,y_{n})^{t},\qtq {where} M\in SL(n+1,S),
\end{equation*}
and weights $x_{i}\mapsto w_{i}:=w(x_{i})\in \r $. 
(I frequently write this in vector notation as $\mathbf{x}=M\mathbf{y}$.) 

It may be more appropriate to allow $M\in GL(n+1,S)$. At the end this leads
to an equivalent theory, but intermediate steps become more complicated
since strong approximation (cf. \cite[Ch.V]{Humpreys80}) does not hold for
$GL$.

A weight system is called {\em trivial}
if all the $w_{i}$ are the same.

Let $t$ be a new variable. For $f\in S[y_{0},\dots ,y_{n}]$   set 
\begin{equation*}f(t^{\mathbf{w}}\mathbf{x}):=f(t^{w_{0}}x_{0},\dots ,t^{w_{n}}x_{n}).
\end{equation*}
The {\em multiplicity} of $f$ at $(\mathbf{x},\mathbf{w})$ is defined as the
exponent of the  lowest $t$-power occurring in  $f(t^{\mathbf{w}}\mathbf{x})$.
It is denoted by $\mult _{t}f(t^{\mathbf{w}}\mathbf{x})$.
\end{definition}


\begin{definition}\label{defn2.2} Let $K$ be a field 
and 
$f\in K[y_{0},\dots ,y_{n}]$ a homogeneous polynomial. Let 
$X:=(f=0)\subset \p ^{n}_{K}$ be the corresponding  hypersurface.
A weight system  $(\mathbf{x},\mathbf{w})$  over $K$ is called
\begin{equation*}\begin{cases}\text{{\em properly stable}} \\
\text{{\em semi-stable}}\\
\text{{\em unstable}}\\
\end{cases}
\text{ on $f$ if } \mult _{t}f(t^{\mathbf{w}}\mathbf{x}) 
\begin{cases}< \\
 \leq \\
>\\
\end{cases}
\frac{\deg f}{n+1}\sum _{i} w(x_{i}).
\end{equation*}
(The value on the right-hand side is the average multiplicity of a  
monomial in $f$. So unstable means roughly that more than half of the monomials
are missing from $f$.)

 These notions are invariant under   affine linear changes of the
weights
$w_{i}\mapsto aw_{i}+b$. 
\end{definition}


\begin{definition}\label{defn2.3}  
Let $K$ be a field with   algebraic closure $\bar K$
and 
$f\in K[y_{0},\dots ,y_{n}]$ a homogeneous polynomial. Let 
$X:=(f=0)\subset \p ^{n}_{K}$ be the corresponding  hypersurface.

(2.3.1) $f$ or $X$ is    called
{\em properly stable} (resp. {\em semi-stable}) iff  every nontrivial weight
system over $\bar K$ is properly stable (resp. semi-stable) on $f$. 

(2.3.2) $f$ or $X$    is called
{\em unstable}  iff  there is an unstable  weight system over $\bar K$ on $f$.

For both of these cases it is sufficient to consider weight systems with
integral weights. 
\end{definition}


\begin{definition}\label{defn2.4} Fix an infinite  field $K$ and let
$V_{d}$ denote the vector space of all degree $d$ homogeneous polynomials
$f\in K[x_{0},\dots ,x_{n}]$. 
$SL(n+1,K)$ acts by coordinate changes on $V_{d}$, thus it also acts
on $K[V_{d}]$, 
 the ring of all polynomials on $V_{d}$.

Any invarant $I\in K[V_{d}]$ of this action is called a {\em polynomial 
$SL$-invariant} of degree $d$ polynomials.

In general such invariants are  hard to write down since they have very
high degree. The best known invariant is the {\em discriminant} $D(f)$. By
definition this is the unique (up to a multiplicative constant) lowest
degree invariant such that $D(f)=0$ iff the corresponding hypersurface
$(f=0)$ is singular (cf.  \cite{Macaulay16}).
\end{definition}


We need the following basic result (cf. \cite[pp. 49
and 79]{Mumford-Fogarty82}):

\begin{theorem}[Hilbert--Mumford criterion]\label{thm2.5}  Let $K$ be an
infinite  field and
$f\in K[x_{0},\dots ,x_{n}]$ a degree $d$ homogeneous polynomial. 

(2.5.1) There is a polynomial
$SL$-invariant $I$  such that $I(f)\neq 0$ iff $f$ is semi-stable.

(2.5.2) If $(f=0)$ is a smooth hypersurface and $\deg f\geq 3$, then $f$ is
properly stable.\end{theorem}


\section{The main definitions}\label{sec:3}

The theories using affine and projective equivalences are very similar, but I
did not find any convenient way to treat them together. The following
definitions all use projective equivalence, which is more natural from the
point of view of algebraic geometry.

\subsection{Notation}\label{not3.1} In this section
$R$ denotes a principal ideal domain with quotient
field $K$. Let 
$p\in R$ be a prime element and $k=R/(p)$ the residue field.  
For a polynomial  $f_{R}\in R[y_{0},\dots ,y_{n}]$,  $f_{k}$ denotes its reduction
modulo $(p)$. 
\setcounter{definition}{1}
\begin{definition}\label{defn3.2} Notation as in (\ref{not3.1}).  
A weight system $(\mathbf{x},\mathbf{w})$ is called {\em integral} if $w_{i}\in \z $
for every $i$.

Let $f\in K[y_{0},\dots ,y_{n}]$ be a  polynomial. One can always find 
 $s\in \z $ and $p'\in R$   prime to $p$ such that  
$p^{-s}\cdot p'\cdot f\in R[y_{0},\dots ,y_{n}]$.
The largest such $s$ is called the  {\em multiplicty} of $f$ at $p$; it is
denoted by  
$\mult _{p}f$.
\end{definition}


\begin{definition}\label{defn3.3} Notation as in (\ref{not3.1}).     Let 
$f_{R}\in R[y_{0},\dots ,y_{n}]$ be a homogeneous polynomial and $X_{R}\subset \p ^{n}_{R}$
 the hypersurface defined by the equation $(f_{R}=0)$.  

(3.3.1) An integral  weight system $(\mathbf{x},\mathbf{w})$ over
$R$ is called  (projectively)
\begin{equation*}\begin{cases}\text{\em properly stable}\\
\text{\em semi-stable} \\
\text{\em unstable} \\
\end{cases}
\text{ on $f_{R}$ at $p$ if }  \mult _{p}f_{R}(p^{\mathbf{w}}\mathbf{x}) 
\begin{cases}<\\
\leq \\
>\\
\end{cases}
\frac{\deg f_{R}}{n+1}\sum _{i} w_{i}.
\end{equation*}

(3.3.2) $f_{R}$ (or $X_{R}$)  is called {\em properly stable} (resp. {\em semi-stable}) at $p$ over $R$ if every weight system is properly stable (resp. 
semi-stable) on
$f_{R}$ at $p$. 

(3.3.3)   $f_{R}$ (or $X_{R}$) is called {\em unstable} at $p$ over $R$ 
if there is an unstable weight system on $f_{R}$ at $p$. 

(3.3.4) $f_{R}$ (or $X_{R}$)  is called {\em properly stable} (resp. {\em semi-stable})   over $R$ if it is properly stable (resp.  semi-stable) at $p$ 
over $R$ for every prime $p\in R$. 

(3.3.5)   $f_{R}$ (or $X_{R}$) is called {\em unstable}   over $R$ 
if it is   unstable  at $p$ over $R$ for some $p$. 

It is important to note that, unlike in (\ref{defn2.2}--\ref{defn2.3}), 
here  all coordinate changes
must take place in
$R$.
\end{definition}


\begin{comments}\label{com3.4}  
This is clearly a formal generalization of (\ref{defn2.3}). In
order to understand the precise relationship, let us  consider the special case
when $R=k[t]$, $p=t$ and  $f_{R}=\sum _{J} a_{J}(t)x^{J}$.

Since the value of the   weight of $t$ is fixed to be 1, the other weights
$w_{i}$ cannot be changed to $\lambda w_{i}$ without changing the concept.

It is nonetheless very useful to see what happens in the limits
$\lambda \to 0,\infty $.

(3.4.1) If $\lambda \to 0$, then any monomial $a_{J}(t)x^{J}$ such that $t|a_{J}(t)$
becomes very large. Thus in the limit only the monomials with $t\not |a_{J}(t)$
matter, that is, we are looking at the stability properties of $f_{k}$.

(3.4.2) If $\lambda \to \infty $, then $\mult _{t}a_{J}(t)$ becomes very small
compared  with the
$t$ powers coming from $x^{J}$.
 Thus in the limit only the vanishing or nonvanishing of the  monomials
matters, that is, we are looking at the stability properties of
$f_{R}$ over $K$.

Thus  (\ref{defn3.3}) gives a notion that interpolates between the stability
properties of the central and general fibers. 
\end{comments}


\begin{proposition}\label{prop3.5}  Notation as in \eqref{not3.1}.  

(3.5.1)
 Assume that
$f_{k}\in k[y_{0},\dots ,y_{n}]$ is semi-stable (resp. properly stable) over $\bar k$. Then 
$f_{R}$ is semi-stable (resp. properly stable) at $p$.

(3.5.2)  Let $\hat R$ denote the completion of $R$ at $(p)$. Then 
$f_{R}\in R[\mathbf{y}]$ is properly stable (semi-stable,  unstable)  at $p\in R$
iff
$f_{R}\in \hat R[\mathbf{y}]$ is properly stable (semi-stable,  unstable)  at
$p\in \hat R$. 
\end{proposition}


\begin{proof} The first part is easy. The second part is also clear once we note
the following. By strong approximation (cf. \cite[Ch.V]{Humpreys80}), the
natural map 
$SL(n,R)\to SL(n,\hat R)$ has dense image, thus every weight system over
$\hat R$ can be approximated by weight systems over $R$.\end{proof}


 \section{The main theorem over $\z $}\label{sec:4}

The results of this section hold for affine and for projective equivalence as
well. The first result settles the question of existence of semi-stable
models. 

\begin{theorem}\label{thm4.1} 
 Let $f\in \q [x_{0},\dots ,x_{n}]$ be a homogeneous
polynomial. 

(4.1.1) $f$ has a 
semi-stable model over $\z $ iff $f$ is semi-stable
over $\cXYZi $.

(4.1.2) If $f$ is properly stable
over $\cXYZi $, then $f$ has only finitely many semi-stable models over $\z $
(up to the action of $SL(n+1,\z )$). 
\end{theorem}


\begin{remarks}\label{rem4.2}  If  the stabilizer of $f$ in  
$GL(n+1,\q )$ is finite,  then the converse of (4.1.2) also holds.

The implication $\Leftarrow $ of (4.1.1) holds if $\z $ is replaced by any
principal ideal domain $R$; the converse  implication holds if $\chr R=0$.

(4.1.2) holds if  all the residue
fields of $R$ are finite.
\end{remarks}


The main part of the proof is the existence of semi-stable models over $\z $.
\setcounter{subsection}{2}
\subsection{A procedure to find semi-stable models}\label{proc4.3} 
We start with a homogeneous
polynomial  $f_{\q }\in \q [y_{0},\dots ,y_{n}]$.
\begin{description}
\item[Step 1] 
 Find any $\z $-model $F^{1}$ of $f_{\q }$. For instance, $F^{1}:=N\cdot f_{\q }$
will do for  $N$ sufficiently divisible.

\item[Step 2]  Assume that we already have $F^{j}$. If $F^{j}$ is semi-stable at
every prime $p$, then we are done.

\item[Step 3]  Otherwise  there is a prime $p$ and an integral  weight system 
$(\mathbf{x},\mathbf{w})$ which is unstable on $F^{j}$. Set 
\begin{equation*}F^{j+1}:=p^{-s} F^{j}(p^{w_{0}}x_{0},\dots ,p^{w_{n}}x_{n}),
\qtq {where} s:=\mult _{p} F^{j}(p^{\mathbf{w}}\mathbf{x}),
\end{equation*}
and go back to Step 2.
\end{description}

If the procedure ever stops, we obtain a semi-stable model of $f_{\q }$.
Unfortunately, in some cases the procedure never halts:
\setcounter{definition}{3}
\begin{example}\label{ex4.4}  For any prime $p$ consider the family of polynomials
\begin{equation*}F_{i}:=p^{i}x_{0}^{4}+x_{2}(x_{0}^{3}+x_{1}^{3})\qtq {for}  i\geq 0.
\end{equation*} 
Over $\cXYZi $ this gives an irreducible plane quartic curve with a triple point at
$(0:0:1)$. Essentially the only unstable weight system at $p$ is  given by the 
weights
$w_{0}=w_{1}=1, w_{2}=0$.

Applying Step 3 of (\ref{proc4.3}) to $F_{i}$ yields $F_{i+1}$, thus we never reach a
semi-stable model over $\z $. In fact, one is tempted to say that among the
above equations
$F_{0}$ is the simplest, and our unsuccessful attempt to reach semi-stability
results in more and more complicated equations. 

A closer inspection  of this example reveals that the problem stems from the
fact that the $F_{i}$ are unstable over $\cXYZi $. 
\end{example}


 The proof that (\ref{proc4.3}) halts in most cases, relies on the following
lemma which was first used in this context by Laxton and Lewis
\cite{Laxton-Lewis65}:

\begin{lemma}\label{lem4.5}   Let $I(f)$ be a polynomial
$SL$-invariant of degree $d$ polynomials in $n+1$ variables which has 
 degree
$r$ in the coefficients   of $f$.  If 
$M\in GL(n+1,\q )$ is a matrix and $c\in \q $, then
\begin{equation*}I(c\cdot f(M\mathbf{x}))=c^{r}\cdot (\det M)^{rd/(n+1)}I(f(\mathbf{x})).
\end{equation*}
In particular,
\begin{equation*}I(p^{-s}f(p^{w_{0}}x_{0},\dots ,p^{w_{n}}x_{n}))=p^{r(-s+\frac{d}{n+1}\sum w_{i})}I(f).
\end{equation*}
\end{lemma}


\begin{proof} It is sufficient to check this over $\cXYZi $. There $M=c'M'$, 
where
$M'\in SL(n+1,\cXYZi )$. $I$ is invariant under $SL(n+1,\cXYZi )$ and for
scalar matrices the formula is clear.\end{proof}

\setcounter{subsection}{5}
\subsection{Termination of the procedure}\label{term4.6}
Assume now that $f$ is semi-stable. By (\ref{thm2.5}) there is a polynomial
$SL$-invariant $I$ such that $I(f)\neq 0$. We may assume that $I$ has
integral coefficients, thus $I(F)\in \z $ if $F\in \z [x_{0},\dots ,x_{n}]$.

By (\ref{lem4.5}) we see that $I(F^{j+1})$ is a proper divisor of $I(F^{j})$; 
hence the
procedure (\ref{proc4.3}) will stop after finitely many steps.


The concept of proper stability over $\z $ is connected with the uniqueness of
semi-stable models. The following special case is the easiest to formulate:
\setcounter{definition}{6}
\begin{theorem}\label{thm4.7} 
Let $F(\mathbf{y})\in \z [y_{0},\dots ,y_{n}]$ be a homogeneous
polynomial which is semi-stable over $\z $. 
Assume that $\deg F$ and $n+1$ are relatively prime.
The following are equivalent:

(4.7.1) $F(\mathbf{y})$ is properly stable over $\z $.

(4.7.2) If $A\in GL(n+1,\q )$ is a matrix such that $F(A\mathbf{y})$
is semi-stable over $\z $, then $A\in GL(n+1,\z )$.
\end{theorem}


\begin{proof}  Assume first that $F$ is not properly stable. Then there is a
prime $p$ and a weight system $(\mathbf{x}=M\mathbf{y},\mathbf{w})$ such that
\begin{equation*}\mult _{p}F(p^{\mathbf{w}}\mathbf{x}) =\frac{\deg F}{n+1}\sum _{i} w_{i}.
\end{equation*}
(Equality holds since $F$ is semi-stable over $\z $.) $\deg F$ divides
$ \mult _{p}F(p^{\mathbf{w}}\mathbf{x})$ since $\deg F$ and $n+1$ are relatively
prime, thus we can introduce new weights 
\begin{equation*}w'_{i}:= w_{i}-\frac{\mult _{p}F(p^{\mathbf{w}}\mathbf{x})}{\deg F},
\qtq {so that} \sum w'_{i}=0.
\end{equation*}
Set $A=\diag (p^{w'_{0}},\dots ,p^{w'_{n}})\cdot M$. Then $F(A\mathbf{x})$
is also semi-stable.   $A\in GL(n+1,\z )$ iff $w'_{i}\geq 0$ for every $i$.
Since $\sum w'_{i}=0$, this holds iff $w'_{i}=0$ for every $i$, hence the weight
system is trivial.

Conversely, assume that  $F(\mathbf{y})$ is properly stable over $\z $
and $F(A\mathbf{y})$ is  semi-stable over $\z $. By the theorem on elementary
divisors (cf. \cite[12.2]{v.d.Waerden91}) we can 
write $A=M_{1}DM_{2}$, where $M_{i}\in SL(n+1,\z )$ 
and $D=\diag (r_{0},\dots ,r_{n})$ for some $r_{i}\in \q ^{*}$. 
If $A\not \in GL(n+1,\z )$, then $r_{i}\not \in \z $ for some $i$, 
and we can construct a
nontrivial weight system with   weights $w_{i}:=\mult _{p}r_{i}$ 
for some $p$ which
shows that
$F$ is not properly stable. This contradiction shows (\ref{thm4.7}).\end{proof}


Some standard results of geometric invariant theory and (\ref{prop3.5})
imply the following result, which is an essential step in proving (4.1.2).

\begin{theorem}\label{thm4.8} Let $f(\mathbf{y})\in \q [y_{0},\dots ,y_{n}]$ be a homogeneous
polynomial which is properly stable over $\cXYZi $. Then $f$ is
 properly stable at $p$ for all but finitely many primes $p$.
\end{theorem}


\section{Integral models of hypersurfaces}\label{sec:5}

In this section I formulate the general versions of the theorems of the
previous section. 

Let $C$ be an integral, regular scheme of dimension 1.
Let $K$ denote the field of rational functions on $C$. If $p\in C$ is a
closed point, then $\oXYZii _{p,C}$ denotes its local ring  
and $C_{p}:=\spec \oXYZii _{p,C}$  the corresponding local scheme.

\begin{theorem}\label{thm5.1}   Notation as above. Let $X_{K}\subset \p
^{n}_{K}$ be a hypersurface and assume that $X_{K}$ is semi-stable  (over $\bar
K$). 

Then there is a vector bundle $E$ of rank $n+1$ over $C$ and a
hypersurface $X_{C}\subset \p _{C}(E)$ such that

(5.1.1) The generic fiber of $X_{C}$ is isomorphic to $X_{K}$.

(5.1.2)  For every closed point $p\in C$ the induced hypersurface
$X_{C_{p}}$  is semi-stable over $\oXYZii _{p,C}$.
\end{theorem}


Any such $X_{C}$ is called a {\em semi-stable model of $X_{K}$ over $C$}.

Finiteness and uniqueness results hold in the  properly stable case:

\begin{theorem}\label{thm5.2} Retain the above notation 
and assume that  $X_{K}$ is 
 properly stable (over $\bar K$). 
Let $X_{C}$ be a semi-stable model of $X_{K}$ over $C$.
Then:

(5.2.1) There is a unique largest Zariski open set $C^{0}$, depending only on
$X_{K}$,  such that 
$X_{C}$ is properly stable at $p$  for every $p\in C^{0}$.

(5.2.2) If $X'_{C}$ is another semi-stable model of $X_{K}$ over $C$, then
$X'_{C^{0}}\cong X_{C^{0}}$.
 
(5.2.3)  If all  residue fields of $C$ are finite, 
then $X_{K}$  has only finitely many 
semi-stable models over $C$ (up to isomorphism over $C$).
\end{theorem}


\section{Examples}\label{sec:6}

In this section  we consider semi-stable models in various special
cases.

\subsection{Binary forms}\label{ssec:6.1}
The condition of semi-stability for binary forms over $\z $ is especially
simple. Over a field, an odd degree semi-stable form is properly stable, and
the
 same unusual situation holds for binary forms over  $\z $ (or over any
principal ideal domain):

\begin{prop}\label{prop6.1.1}  Let $F(x,y)\in \z [x,y]$ be a  homogeneous
 polynomial of degree $d$.
$F$ is semi-stable (resp. properly stable) over $\z $ iff $F$ is not divisible
by any prime
$p$ and the following holds:
\begin{equation*}\left .
\begin{matrix}p^{\frac{d+1}{2}}\\
p^{\frac{d}{2}}\\
\end{matrix}
\right \}
\not | F(pax+pby, cx+dy)\quad \forall \text{  prime $p$, }
 \forall \left (
\begin{matrix}a & b\\
c& d
\end{matrix}
\right )
\in SL(2,\z ).
\end{equation*}
\end{prop}


\begin{thm}\label{thm6.1.2}  Let $f(x,y)\in \q [x,y]$ be a homogeneous polynomial
of degree
$d$. Assume that $d$ is odd and that $f$ is not divisible by the
$\frac{d+1}{2}$th power of a linear form. Then $f$ has a properly stable
model $F\in \z [x,y]$ over
$\z $.
$F$ is unique  up to projective equivalence over $\z $.
\end{thm}


\begin{proof}  Let $F$ be a semi-stable model over $\z $. Since $d/2$ is not an
integer, $F$ is actually properly stable over $\z $. Thus it is unique by
(\ref{thm4.7}).\end{proof}


\subsection{Norm forms}\label{ssec:6.2}
There is   a close 
connection between semi-stable models of norm forms and ideal class groups.
For quadratic extensions  such a relationship is classical, see for instance 
\cite[VII.2]{Frohlich-Taylor91}.

\begin{defn}\label{defn6.2.1}  
Let $L\supset \q $ be a field  extension of degree $n$
and $S\subset L$ the ring of algebraic integers.
Let
$u_{1},\dots ,u_{n}$ be a $\z $-basis of $S$ and
\begin{equation*}N_{L/\q }(\mathbf{x}):=N_{L/\q }(\sum x_{i}u_{i})
\end{equation*}
 the corresponding {\em norm form} (cf. \cite[V.1]{Frohlich-Taylor91}),
viewed as a degree $n$ homogeneous  polynomial  in the variables
$x_{1},\dots ,x_{n}$.
$N_{L/\q }(\mathbf{x})$ is unique up to affine equivalence over $\z $. 

If  $u_{1},\dots ,u_{n}$ is any $\q $-basis of $L$ and $c\in \q ^{*}$, then we obtain a
{\em generalized norm form} 
\begin{equation*}c\cdot N_{L/\q }(\sum x_{i}u_{i}).
\end{equation*}
A generalized norm form is projectively equivalent over $\q $ to the norm form
of
$L/\q $.
\end{defn}


\begin{lem}\label{lem6.2.2} 
A generalized norm form $c\cdot N_{L/K}(\sum x_{i}u_{i})$
is semi-stable over $\z $ iff $u_{1},\dots ,u_{n}$  is a $\z $-basis
of a fractional
$S$-ideal $I\subset L$ and $c=\pm N_{L/\q }(I)^{-1}$,
where $N_{L/\q }(I)$ is the ideal norm (cf. \cite[II.4]{Frohlich-Taylor91}).
\end{lem}


From this one obtains the following:

\begin{thm}\label{thm6.2.3} Let $L\supset \q $ be a Galois  extension  
with  norm form $N_{L/\q }(\mathbf{x})$.
There is a one-to-one correspondence
\begin{equation*}\left \{
\begin{matrix}\text{semi-stable models of $N_{L/\q }(\mathbf{x})$}\\
\text{modulo projective equivalence over $\z $}
\end{matrix}
\right \}
\quad \Leftrightarrow \quad \left \{
\begin{matrix}\text{ideal class group of $L$}\\
\text{modulo  $\gal (L/\q )$}
\end{matrix}
\right \}.
\end{equation*}
\end{thm}


Similar results hold for non-Galois extensions as well.

\subsection{Quadratic forms}\label{ssec:6.3}
Let $q(\mathbf{x})$ be a quadratic form over the $p$-adic field $\q _{p}$, and
assume that $p\neq 2$.
The  concept of
``maximale Gitter" of \cite[p. 50]{Eichler52} and 
the notion of ``$f$-norm" as defined in \cite[II]{Springer55}   provide a
quadratic form   $Q(\mathbf{x})$  over $\z _{p}$ which  is equivalent to $q(\mathbf{x})$ over
$\q _{p}$. The theory of semi-stable models provides a generalization of these,
which works in full generality.
In this case the results are nicer if we work with affine equivalence.

\begin{nota}\label{not6.3.1}  
Let $R$ be a local Dedekind domain with quotient
field $K$,  maximal ideal
$(p)$ and residue field $k$. Let $q(\mathbf{x})\in K[x_{0},\dots ,x_{n}]$ be a
quadratic form such that $(q=0)\subset \p ^{n}_{K}$ is a smooth quadric
and let $Q(\mathbf{x})\in R[x_{0},\dots ,x_{n}]$ be affine equivalent to $q$.
\end{nota}


It should be emphasized that I even allow the case $\chr K=2$.

For quadratic forms  affine semi-stability turns out to be equivalent to the
following very simple notion, which we adopt as our definition:

\begin{defn}\label{defn6.3.3.} 
$Q$ is affine semi-stable iff there is no change of
coordinates $\mathbf{y}=M\mathbf{x}$ (where $M\in SL(n+1,R)$) such that
$Q(p^{-1}y_{0},y_{1},\dots ,y_{n})\in R[x_{0},\dots ,x_{n}]$.
\end{defn}


\begin{thm}\label{thm6.3.3}  Notation as in \eqref{not6.3.1}. 
Let $q(\mathbf{x})\in K[x_{0},\dots ,x_{n}]$ 
be a quadratic form such that $(q=0)\subset \p ^{n}_{K}$ is a
smooth quadric. Then
$q$ has a unique affine semi-stable model $Q\in R[x_{0},\dots ,x_{n}]$ 
(up to affine
equivalence over $R$).  

Moreover, if $Q'\in R[x_{0},\dots ,x_{n}]$ is equivalent to $q(\mathbf{x})$ over $K$,
then there is a matrix $T\in M(n+1,R)$ such that $Q'(\mathbf{x})=Q(T\mathbf{x})$. 
\end{thm}


\subsection{Cubic  forms}\label{ssec:6.4}

Corti \cite{Corti96} proposed a program of constructing what he calls ``standard
models" of cubic  surfaces   over local Dedekind domains \cite[Conjectures 2.11
and 2.14]{Corti96}.  He established several steps of the program, and his proofs are
complete   for  families of  cubic surfaces over a smooth curve in 
characteristic zero. 
Although he does not explicitly state it, his conjectures can be
easily reformulated for cubic hypersurfaces.  Our results imply that his
conjectures are true, even for cubic hypersurfaces:

\begin{prop}\label{prop6.4.1} Conjectures 2.11 and 2.14 of \cite{Corti96}
are true.
\end{prop}


The main point of our discussion is, however, the observation, that the
models proposed by Corti are not always the optimal ones. In our terminology,
he uses only weight systems where all the weights are 0 or 1. 
As \cite[5.8]{Corti96} shows, this does not give as much uniqueness as 
(\ref{thm5.2}).

At least for cubic surfaces, one needs to use two more weight systems:

\begin{prop}\label{prop6.4.2} In order to check semi-stability of a family of
cubic surfaces  over a one-dimensional regular scheme,  it is sufficient to use
weight systems with the following five weight sequences:
\[
(0,0,0,1),\ (0,0,1,1),\ (0,1,1,1),\ (0,1,2,2),\ (0,2,2,3).
\]
\end{prop}


This implies that for cubic forms in four variables over $\z $, (\ref{proc4.3})
becomes an effective algorithm.

\begin{exa}\label{ex6.4.3} Let $R=\cXYZi [[t]]$ and consider the cubic form
$F_{1}=x_{0}^{3}+x_{1}^{3}+x_{2}^{2}x_{3}+t^{6}x_{3}^{3}$. It is easy to see that it is semi-stable
with respect to every weight system where all the weights are 0 or 1. 
$F_{1}$ is unstable with weights $(2,2,3,0)$, and we obtain the properly
stable cubic form $F_{2}=x_{0}^{3}+x_{1}^{3}+x_{2}^{2}x_{3}+x_{3}^{3}$, which is the unique
semi-stable model by (\ref{thm5.2}).  In this example $F_{1}$ is not representable by
$F_{2}$, that is, there is no matrix $T\in M(4,\cXYZi [[t]])$ such that
$F_{1}(\mathbf{x})=F_{2}(T\mathbf{x})$. Thus $F_{1}$ is not representable by any
semi-stable model.
\end{exa}

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\end{document}