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\begin{document}
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\title[Sample Article Title]
{Sample Article Title to Illustrate\\
the Use of Birkart.cls}
\author[J.~Author]{James Author}
\address{%%
Department of Mathematics,\br
University of Sampleville,\br
35, Gau\ss\ Avenue\br
U-314159 Sampleville, Utopia}
\email{author@math.unisamp.ut}
\begin{abstract}
Since a projective
variety
$V=\mathcal{Z}(I)\subseteq {\bf P}^n$ is an
intersection of hypersurfaces, one of the most basic problems we can pose in
relation to $V$ is to describe the hypersurfaces containing it. In
particular, one would like to know the maximal number of linearly independent
hypersurfaces of each degree containing $V,$ that is to know the dimension of
$I_d,$ the vector space of homogeneous polynomials of degree
$d$ vanishing on $V$ for various $d.$
\end{abstract}
\maketitle
\section{Introduction}
Since one knows the dimension
$\binom{n+d}{d}$ of the space of all forms of degree
$d$, knowing the dimension of $I_d$ is
equivalent to knowing the {\bf Hilbert function} of the homogeneous
coordinate ring $A=k[X_0,\dots,X_n]/I$ of $V$, which is the vector
space dimension of the degree
$d$ part of $A.$
In his famous paper ``\"Uber die Theorie der algebraischen Formen'' (see \cite
{Hil}) published a century ago, Hilbert proved that a graded module $M$ over a
polynomial ring has a finite graded free resolution, and concluded from this
fact that its Hilbert function is of polynomial type. The {\bf Hilbert
polynomial} of a graded module is thus the polynomial which agrees with the Hilbert
function $H_M(s)$ for all large $s.$ Hilbert's insight was that all the
information encoded in the infinitely many values of the Hilbert function can be
read off from just finitely many of its values.
\subsection{First subsection}
Hilbert's
original motivation for studying these numbers came from invariant theory. Given
the action of a group on the linear forms of a polynomial ring, he wanted to
understand how the dimension of the space of invariant forms of degree $d$ can
vary with $d.$
\subsubsection{First subsubsection}
The Hilbert function of the homogeneous coordinate
ring of a projective variety $V,$ which classically was called the postulation of
$V,$ is a rich source of discrete invariants of
$V$ and its embedding. The dimension, the degree and the arithmetic genus of $V$
can be immediately computed from the generating function of the Hilbert function
of $V$ or from its Hilbert polynomial.
As for the Hilbert polynomial, there are two
important geometric contexts in which it appears. The first is the
Riemann-Roch theorem which plays an enormously important role in Algebraic
Geometry. This celebrated formula arises
from the computation of a suitable Hilbert polynomial. Secondly, the
information contained in the coefficients of the Hilbert polynomial is usually
presented in Algebraic Geometry by giving the Chern classes of the corresponding
sheaf, a set of different integers, which can be deduced from the coefficients,
and from which the coefficients can also be deduced (see \cite {Ei}, pg. 44).
\subsubsection{Second subsubsection}
The
influence of Hilbert's paper on Commutative Algebra has been tremendous. Both
free resolutions and Hilbert functions have fascinated mathematicians for a
long time. In spite of that, many central problems still remain open. The last
few decades have witnessed more intense interest in these objects. I believe this
is due to two facts. One is the arrival of computers and their breathtaking
development. The new power in computation has prompted a
renewed interest in the problem of effective construction in Algebra. The ability
to compute efficiently with polynomial equations has made it possible to
investigate complicated examples that would be impossible to do by hand and has
given the right feeling to tackle more difficult questions.
The second is the work of Stanley in connection with Algebraic
Combinatorics. In 1978 R. Stanley published the fundamental paper
``Hilbert
functions of graded algebras'' (see \cite {St1}) where he related the
study of the
Hilbert function of standard graded algebras to several basic problems in
Combinatorics. An introduction to this aspect of Commutative Algebra is given in
Stanley's monograph \cite {St3} which is well known as the ``green
book''.
From the point of view of the theory of Hilbert functions, one of
the greatest merits of Stanley's work was to restate and explain, in the right
setting, a fundamental theorem of Macaulay which characterizes the Hilbert
functions of homogeneous $k$-algebras. This theorem is a great source of
inspiration for many researchers in the field.
All the above arguments suffice to
justify the choice of ``Hilbert functions'' as the right topic for a postgraduate
course in Commutative Algebra. I would however add some further motivation. One
of the delightful things about this subject is that one can begin studying it in
an elementary way and, all of a sudden, one can front extremely challenging and
interesting problems. An
example will help to clarify this. The possible Hilbert functions of a
homogeneous Cohen-Macaulay algebra are easily characterized by using
Macaulay's theorem and a standard graded prime avoidance theorem. The
corresponding problem for
a Cohen-Macaulay domain is completely open and we do not even have a
guess as to the possible structure of the corresponding numerical functions.
\subsection{Second subsection}
Further, following what is written at the beginning of Eisenbud's
excellent book \cite{Ei}, ``It has seemed to me for a long time that Commutative
Algebra is best practiced with knowledge of the geometric ideas that played a
great role in its formation: in short, with a view toward Algebraic Geometry''. And
what better topic than Hilbert functions to give a concrete example of this
way of thinking?
Since I had
to choose among the many different scenes which compose the picture, I was very
much influenced in my choice by what I know better and what I have worked on
recently. This means that the chapters I present here are by no means the most
important in the theory of Hilbert functions. However I was also guided by the
possibility of inserting open problems and conjectures more than results.
Besides the basic definitions,
the first two sections are devoted to the problem of characterizing the possible
numerical functions which are the Hilbert functions of some graded algebras with
special properties e.g. reduced, Cohen-Macaulay or Gorenstein. Since the
main problems arise when the algebras are integral domains, we will show that, by
the classical Bertini theorem, the
$h$-vector of a Cohen-Macaulay graded domain of dimension bigger than or equal
to two, is the same as that of the homogeneous coordinate ring of an
arithmetically Cohen-Macaulay projective curve in a suitable projective space.
But, by a more recent result of J. Harris on the generic hyperplane section of a
projective curve, this
$h$-vector is also the
$h$-vector of the homogeneous coordinate ring of a set of points in Uniform
Position. This gives a very interesting shift from a purely algebraic approach to
a more geometric context which has been very useful.
In the
third section we introduce a list of conjectures recently made by Eisenbud, Green
and Harris and which are closely related to the topics introduced in the first
two sections. These
conjectures mainly deal with the problem of finding precise
bounds on the multiplicity of special Artinian graded algebras, but they can also
be read in a more geometric contest. A stronger form
of some of these conjectures is a guess which extends, in a very natural way, the
main theorem of Macaulay we have introduced above. Despite the fact that
we have almost no answer to these questions, they fit very well into the picture,
because they are so easy to formulate and so difficult to solve.
A short fourth section is devoted to a longstanding question on
the possible Hilbert function of generic graded algebras. A solution in the case
of a polynomial ring in two variables is given which uses a nice argument related
to the Gr\"obner basis theory of ideals in the polynomial ring. A very natural
problem in this theory closes the section.
In the fifth section
we discuss some problems related to the Hilbert function of a scheme of fat
points in projective space. We will try to explain how the knowledge of the
postulation of these zero-dimensional schemes can be used to study the Waring
problem for forms in a polynomial ring and the symplectic packing problem for
the four-dimensional sphere.
As for the first problem,
following the approach used by T. Iarrobino in his recent work, we will show how a
deep theorem of Alexander and Hirschowitz on the Hilbert function of the scheme
of generic double fat points gives a complete solution to the Waring problem
for forms, which is the old problem of determining the least integer $G(j)$
such that the generic form of degree $j$ in $k[X_0,\dots,X_n]$ is the sum of $G(j)$
powers of linear forms.
As for the second problem, we just
present the relationship between an old conjecture of Nagata on the postulation of
a set of fat points in ${\bf P}^2$ and the problem on the existence of a full
symplectic packing of the four dimensional sphere, as presented in the work of
McDuff and Polterovich.
In the last section we come to the
Hilbert function of a local Cohen-Macaulay ring and present several results and
conjectures on this difficult topic. Since the associated graded ring of a
local Cohen-Macaulay ring can be very bad (there do exist local complete
intersection domains whose associated graded ring has depth zero), very little is
known on the possible Hilbert functions of this kind of ring, even if this has
strong connections with the well developed theory of singularities.
Starting from the classical results of S. Abhyankar, D. Northcott and J.Sally, we
discuss how Cohen-Macaulay local rings which are extremal with
respect to natural numerical constraints on some of their Hilbert coefficients,
have good associated graded rings and special Hilbert functions. A recent result of
Rossi and Valla, which gives a solution to a longstanding conjecture made by
Sally, is also discussed at the end of the section.
\medskip
I am
personally grateful to the organizers of the school for giving me the possibility
to teach on my favourite topic. A number of people helped me a great deal in the
development of this manuscript. In particular, the section on Waring's problem is
very much influenced by a series of talks Tony Geramita gave in Genova last year
(see \cite{tony}). The last section on the Hilbert function in the local case grew
out of a long time cooperation with M. E. Rossi.
Finally I apologize to those whose work I may have failed to cite
properly. My feelings are best described by the following sentence which I found
in
\cite {Ha1}:
``Certainly, the absence of a reference for any particular discussion
should be taken simply as an indication of my ignorance in this regard, rather
than as a claim of originality.''
\section {Second Section}
Our standard assumption will be that $k$ is a field of characteristic zero, $R$
is the polynomial ring
$k[X_1,\dots,X_n]$ and $M$ a finitely generated graded
$R$-module such that $M_i=0$ if
$i<0.$ If $M$ is such a graded $R$-module, the homogeneous components
$M_n$ of $M$ are $k$-vector spaces of finite dimension.
\begin{definition} Let $M$ be a finitely generated graded
$R$-module. The numerical function
$$
H_M:{\bf N}\to {\bf N}
$$
defined as
$$
H_M(t)=dim_k(M_t)
$$
for all $t\in {\bf N}$ is the {\it Hilbert function}
of
$M.$
The power series
$$
P_M(z)=\sum_{t\in {\bf N}}H_M(t)z^t
$$
is called the {\it Hilbert Series} of
$M.$
\end{definition}
For example, for every $t\ge 0$ we have
$$
H_R(t)={\binom{n+t-1}{t}}
\qquad\mbox{and}\qquad
P_R(z)=\frac{1}{(1-z)^n}.
$$
The most relevant
property of the Hilbert series is the fact that it is additive on
short exact sequences of finitely generated $R$-modules. A classical result of
Hilbert says that the series
$P_M(z)$ is rational and even more,
\begin{theorem}\label{Hilbert-Serre}{\bf (Hilbert-Serre)} Let $M$
be a finitely generated graded $R$-module. Then there exists a polynomial
$f(z)\in {\bf Z}[z]$ such that
$$
P_M(z)=\frac{f(z)}{(1-z)^n}.
$$
\end{theorem}
It is easy to see that if $M\not= 0$ then the multiplicity of 1 as a root of
$f(z)$ is less than or equal to $n$ so that we can find a unique polynomial
$$
h(z)=h_0+h_1z+\cdots+h_sz^s\in {\bf Z}[z]
$$
such that
$h(1)\not = 0$ and for some integer $d,$ $0\le d\le n$
$$
P_M(z)=\frac{h(z)}{(1-z)^d}.
$$
The polynomial $h(z)$ is called {\it the
h-polynomial } of $M$ and the vector
$(h_0,h_1,\dots,h_s)$ {\it the h-vector} of $M.$
The integer $d$ is the
{\it Krull dimension} of $M.$
Now for every $i\ge 0,$ let
$$
e_i:=\frac{h^{(i)}(1)}{i!}
$$
and
$$
{\binom{X+i}{i}}:=\frac {(X+i)\cdots (X+1)}{i!}.
$$
Then it is easy to see that
the polynomial
$$
p_M(X):=\sum_{i=0}^{d-1}(-1)^ie_i{\binom{X+d-i-1}{d-i-1}}
$$
has rational
coefficients and degree $d-1;$
further for every $t>>0$
$$
p_M(t)=H_M(t).
$$
The polynomial $p_M(X)$ is called the
{\it Hilbert polynomial} of $M$ and its leading coefficient is
$$
\frac {h(1)}{(d-1)!}.
$$
This implies that
$e_0(M):=h(1)$ is a positive integer which is usually denoted simply by $e(M)$
and called {\it the multiplicity} of $M.$
If $d=0$ we define
$e(M)=dim_k(M).$
Another relevant property of Hilbert series is the so called
{\it sensitivity to regular sequences.} We recall that a sequence $F_1\dots,F_r$ of
elements of the polynomial ring $R$ is a {\it regular sequence} on a
finitely generated graded $R$-module $M$ if $F_1,\dots,F_r$ have positive degrees
and
$F_i$ is not a zero-divisor on $M$ modulo $(F_1,\dots,F_{i-1})M$ for
$i=1,\dots,r.$
If
$J$ is the ideal generated by the homogeneous polynomials $F_1,\dots,F_r,$ of
degrees
$d_1,\dots,d_r,$ one can prove that
$$
P_M(z)\le \frac{P_{M/JM}(z)}{\prod_{i=1}^r(1-z^{d_i})}
$$
with equality holding if
and only if the elements $F_1,\dots,F_r$ form a regular sequence on $M.$ This means
that if $L\in R_1$ is regular on $M,$ we have
$$
P_{M/LM}(z)=P_M(z)(1-z).
$$
So
that the Hilbert function of the module $M/LM$ is given by the so called
{\it first difference function}
$\Delta H_M$ of $H_M$ which is defined by the formula
$$
\Delta H_M(t)=\left \{\begin{array}{ll}1 & \mbox{if $t=0$}\\H_M(t)-H_M(t-1) &
\mbox{if $t\ge 1.$ }\end{array}\right.
$$
Now we present a fundamental theorem, due to Macaulay (see \cite{M2}), describing
exactly those numerical functions which occur as the Hilbert function
$H_A(t)$ of a standard homogeneous
$k$-algebra
$A.$ Macaulay's theorem says that for each $t$ there is an upper bound for
$H_A(t+1)$ in terms of
$H_A(t)$, and this bound is sharp in the sense that any numerical function
satisfying it can be realized as the Hilbert function of a suitable
homogeneous
$k$-algebra.
Let $d$ be a positive integer. One can easily see
that any integer
$a$ can be written uniquely in the form
$$
a={\binom{k(d)}{d}}+{\binom{k(d-1)}{d-1}}+\cdots +{\binom{k(j)}{j}}
$$
where
$$
k(d)>k(d-1)>\cdots >k(j)\ge j\ge 1.
$$
For example, if
$a=49,$ $d=4,$ we get
$$
49={\binom{7}{4}}+{\binom{5}{3}}+{\binom{3}{2}}+{\binom{1}{1}}.
$$
Given
the integers $a$ and $d$, we let
$$
a^{}:={\binom{k(d)+1}{d+1}}+{\binom{k(d-1)+1}{d}}+\cdots
+{\binom{k(j)+1}{j+1}}.
$$
Hence, for example,
$$
49^{<4>}={\binom{8}{5}}+{\binom{6}{4}}+{\binom{4}{3}}+{\binom{2}{2}}.
$$
\begin{theorem}\label{Macaulay theorem}{\bf (Macaulay)} Let $H:{\bf
N}\to
{\bf N}$ be a
numerical function. There exists a standard homogeneous $k$-algebra $A$ with
Hilbert function
$H_A=H$ if and only if
$H(0)=1$ and $H(t+1)\le H(t)^{}$ for every $t\ge 1.$
\end{theorem}
A numerical function verifying the conditions of the above theorem is called an
{\it admissible} numerical function.
The following example demonstrates the effectiveness of Macaulay's theorem.
Let us check that
$1+3z+4z^2+5z^3+7z^4$ is not the Hilbert series of a homogeneous $k$-algebra. We
have
$$
5={\binom{4}{3}}+{\binom{2}{2}}
$$
$$
5^{<3>}={\binom{5}{4}}+{\binom{3}
{3}}=6.
$$
In the same paper Macaulay produced an algorithm to construct,
given an admissible numerical function $H,$ an homogeneous $k$-algebra having it as
Hilbert function.
Let $n=H(1);$ we fix in the set of monomials of
$R=k[X_1,\cdots,X_n]$ a total order compatible with the semigroup structure of this
set. Let us fix for example the {\it degree lexicographic order.} This is
the order given by
$$
X_1^{a_1}X_2^{a_2}\cdots X_n^{a_n}>X_1^{b_1}X_2^{b_2}\cdots X_n^{b_n}
$$
if and
only if either $\sum a_i>\sum b_i$ or $\sum a_i=\sum b_i$ and for some integer
$jb_{j+1}.$
Macaulay proved that if for every $t\ge 0$ we delete the smallest
$H(t)$ monomials of degree
$t,$ the remaining monomials generate an ideal $I$ such that
$H_{R/I}(t)=H(t)$ for every $t\ge 0.$
The difficult part of the
proof is to show that, due to the upper bound
$H(t+1)\le H(t)^{},$ if a monomial $M$ is in $I_t$, which means that it has not
been erased at level $t,$ then $MX_n$ is not erased at level $t+1$\ so that
$MX_1,\dots,MX_n\in I$ as we need.
For example the ideal $I$ in
$R=k[X_1,X_2,X_3]$ such that
$$
P_{R/I}(z)=\frac{1+z-2z^4+z^5}{(1-z)^2}
$$
and constructed by this algorithm, is
the ideal
$$
I=(X_1^2,X_1X_2^3,X_1X_2^2X_3).
$$
The ideal constructed following this
method is called a {\it lex-segment } ideal, in the sense that a $k$-basis of its
homogeneous part of degree $t$ is an initial segment of monomials in the given
order. Since it is clearly uniquely determined by the given admissible function,
it is called
{\it the lex-segment ideal} associated to the admissible numerical function.
This ideal has some very interesting extremal properties. For example it has the
biggest Betti numbers among the perfect ideals with the same multiplicity and
codimension (see
\cite{V}).
Macaulay's theorem is valid for any homogeneous $k$-algebra. It is not surprising
that additional properties yield further constraints on the Hilbert function. We
will discuss this feature for reduced, Cohen-Macaulay, Gorenstein and domain
properties. We start with the reduced case.
\begin{theorem}\label{reduced} Let $H:{\bf N}\to {\bf N}$ be a numerical
function. There exists a reduced homogeneous $k$-algebra $A$ with Hilbert function
$H_A=H$ if and only if either $H=\{1,0,0\dots\}$ or $\Delta H$ is admissible.
\end{theorem}
\begin{Proof}\ If $A=R/I$ is reduced it is clear that either $depth(A)>0,$ or
$I=(X_1,\dots,X_n).$ In the second case all is clear; in the first one we can find
a linear form $L$ which is a regular element on $A.$ Hence the difference of the
Hilbert function of $A$ is the Hilbert function of the graded algebra $A/LA.$
Conversely, if we are given a function whose difference is admissible, we can
construct the lex-segment ideal $J$ in the polynomial ring $S$ such that
$H_{S/J}=\Delta H.$ This is a monomial ideal which can be deformed to a radical
ideal by a general construction due to Hartshorne (see \cite{Hart} and \cite
{GGR}). This can be achieved in the following way. If
$m=X_1^{a_1}X_2^{a_2}\cdots X_n^{a_n}$ is a monomial, we introduce a new variable
$X_0$ and set
$$
l(m):=\prod_{j=1}^n\prod_{p=0}^{a_j-1}(X_j-pX_0).
$$
If $J=(m_1,\dots,m_s)$ is an
ideal generated by monomials, the ideal
$$
l(J):=(l(m_1),\dots,l(m_s))
$$
is a radical ideal in the polynomial ring
$R=k[X_0,\dots,X_n],$ such that $X_0$ is a regular element on $R/l(J)$ and
$$
S/J\simeq (R/l(J))/X_0(R/l(J)).
$$
Thus if $I\subseteq R$ is such a radical
deformation of
$J,$ we have
$$
P_{S/J}(z)=P_{(R/I)/X_0(R/I)}(z)=P_{R/I}(z)(1-z)
$$
so that $H_{S/J}=\Delta
H_{R/I}.$ Since $H_{S/J}=\Delta H=\Delta H_{R/I}$ we get $H=H_{R/I}.$
\end{Proof}
For example the numerical function $H=\{1,2,1,1,1,\dots\}$ is admissible
so it is the Hilbert function of a suitable graded $k$-algebra, but it is not the
Hilbert function of a reduced $k$-algebra since its difference function
$\{1,1,-1,0,0,\dots\}$ is not admissible.
We pass now to the
Cohen-Macaulay case and obtain the following characterization of the Hilbert
Series of Cohen-Macaulay homogeneous algebras.
\begin{theorem}\label{CM} Let $h_0,\dots,h_s$ be a finite sequence of positive
integers. There exists an integer
$d$ and a Cohen-Macaulay homogeneous
$k$-algebra $A$ of dimension $d$ such that
$$
P_A(z)=\frac{h_0+h_1z+\cdots +h_sz^s}{(1-z)^d}
$$
if and only if
$(h_0,\dots,h_s)$
is admissible.
\end{theorem}
\begin{Proof} If
$R/I$ is Cohen-Macaulay, by prime avoidance in the graded case, we can find a
maximal regular sequence of linear forms
$L_1,\dots,L_d$ such that $R/I$ and its Artinian reduction
$(R/I)/(L_1,\dots,L_d)(R/I)$ share the same
$h$-polynomial.
\end{Proof}
In particular the $h$-vector of a Cohen-Macaulay graded algebra has
positive coordinates.
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