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\title{\Large\bf Commutative Algebra: Hilbert functions, 

homological methods,\\ and combinatorial aspects}



\author{Aldo Conca, Anna Guerrieri, Claudia Polini and Bernd Ulrich}



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\date{}



\maketitle



\section*{Saturdayday (June 15, 2002)}



\bigskip



\noindent 

{\bf Title:} F-signatures in local rings.



\noindent 

{\bf Author:} Ian Aberbach* and Graham Leuschke.



\noindent 

{\bf Abstract:} 

Let $R$ be a reduced Noetherian ring of

positive prime

characteristic $p$, and let $R^{1/q}$ be the ring of

$q = p^e$th

roots.  Assume that $R^{1/p}$ is a finitely

generated $R$-module.

Then the F-signature of $R$, $s(R)$, introduced by

Huneke and

Leuschke, is an asymptotic measure

of the number of summands in $R^{1/q}$ as an

$R$-module.  In

joint work with Leuschke we show that $s(R) > 0$ if

and only if

$R$ is strongly $F$-regular.



\bigskip

\bigskip



\bigskip



\noindent 

{\bf Title:} Cohomology of dualizing complexes.



\noindent 

{\bf Author:} Luchezar L.\ Avramov*, Ragnar-Olaf Buchweitz and Liana M. \ Sega 



\noindent 

{\bf Abstract:} 

Let $R$ be a commutative noetherian local ring having a dualizing

complex $D$; shifting $D$, if neecessary, assume $\inf\{j\in{\mathbb

Z}\mid \mbox{H}_j(D)\ne0\}=0$.  If $R$ is Gorenstein, then $D$ is

quasi-isomorphic to $R$, hence $\mbox{Ext}^n_R(D,R)$ vanishes for

all $n>0$.



Partly motivated by a long standing conjecture of Tachikawa on the

cohomology of (not necessarily commutative) finite dimensional algebras

over a field, we conjecture that if $\mbox{Ext}^n_R(D,R)=0$ for all $n>0$,

then $R$ is Gorenstein.  Proofs of this conjecture in several significant

cases will be discussed.



\bigskip

\bigskip



\noindent 

{\bf Title:} Unimodular covers of lattice polytopes



\noindent 

{\bf Author:} Winfried Bruns* and Joseph Gubeladze



\noindent 

{\bf Abstract:} 

Let $P$ be a $d$-dimensional lattice polytope. We show that

there

exists a natural number $\mathfrak{c}_d$, only depending on $d$,

such

that

the multiples $cP$ have a unimodular cover for every $c\in\Bbb N$,

$c\ge \mathfrak{c}_d$. Actually, an explicit upper bound for

$\mathfrak{c}_d$ is

provided, together with an analogous result for unimodular

covers

of rational cones. As an auxiliary result we bound on the

complexity of

equivariant resolutions of toric singularities.



\bigskip

\bigskip



\noindent 

{\bf Title:} Castelnuovo-Mumford regularity: Some examples in dimensions one and two.



\noindent 

{\bf Author:} Marc Chardin* and Clare V. D'Cruz 



\noindent 

{\bf Abstract:} 

The behaviour of Castelnuovo-Mumford regularity under

``geometric''

transformations is not well understood. We try to shed some

light

on few questions concerning this behaviour by looking at

examples. Simple

questions as: does the regularity  

improve by passing to radical or removing embedded primes~?

seemed

open. We will show examples that answers (negatively) to 

these questions, and consider other related issues. For example

we

provide examples of defining ideals of 

curves in ${\bf   P}^{3}$ (resp. in   ${\bf  P}^{4}$)  such that

the

regularity of their radical is essentially the square (resp. the

cube)

of the one of the ideal. Removing points (embedded or not)

the regularity improves, so we have to go to surfaces in order

to find

examples where removing an embedded prime increases the

regularity. More surprising is the fact that we find an

irreducible

surface such that we may improve the depth of its coordinate

ring by

embedding a line into it!



Another important issue is to understand the limits of validity

of

Kodaira type vanishing theorems. The canonical module of a

reduced

curve have  Castelnuovo-Mumford regularity 2. This type of

results

extends in particular to higher dimensionnal varieties with

isolated

singularities (in characteristic zero), thanks to Kodaira

vanishing, and

as a corollary one may derive bounds for Castelnuovo-Mumford

regularity. In ``Pathologies III'', Mumford proves that for an

ample line

bundle ${\mathcal L}$ on a normal surface ${\mathcal S}$,

$H^{1}({\mathcal S} ,{\mathcal   O}_{{\mathcal S}}\otimes

{\mathcal

  L}^{-1})=0$. He remarks that this is false if ${\mathcal S}$

is not

Cohen-Macaulay and asks if ever the $S_{2}$ condition is

sufficient. A

first counter-example was given by Arapura and Jaffe. We here

show

that it is sufficient to construct counter-examples satisfying

$S_{1}$, and provide some monomial surfaces whose canonical

module

have rather big Castelnuovo-Mumford regularity, so that this

vanishing

fails. We also give a simple proof that if ${\mathcal S}$

satisfies

$R_{1}$, then  $H^{1}({\mathcal S},{\omega}_{{\mathcal

S}}\otimes

{\mathcal L})=0$  and  $H^{1}({\mathcal S} ,{\mathcal

O}_{{\mathcal

S}}\otimes {\mathcal L}^{i})=0$ have constant dimension

for $i<0$ if ${\mathcal L}$ is very ample. This dimension

measures the

defect of Macaulyness of ${\mathcal S}$.



These monomial surfaces also provides examples of complete

intersection surfaces with two reduced irreducible components

such

that one of them have a regularity bigger than the complete

intersection, this doesn't happend in dimension one. 





\bigskip

\bigskip



\noindent 

{\bf Title:} Koszul homology and integral closure of ideals.



\noindent 

{\bf Author:} Alberto Corso



\noindent 

{\bf Abstract:} 

If $I$ is an equidimensional ideal of a Cohen--Macaulay local

ring, then the annihilators of the nonzero Koszul homology

modules

$H_i(I)$ are contained in $\sqrt{I}$. A number of preliminary

results by

C. Huneke, D. Katz, W.V. Vasconcelos and ourselves suggest that

these

annihilators are always contained in the integral closure of

$I$. In the

talk, we will discuss some of these results (focusing on the

annihilator

of $H_1(I)$).



\bigskip

\bigskip





\noindent 

{\bf Title:} Ulrich Modules -- their existence and uses.



\noindent 

{\bf Author:} David Eisenbud



\noindent {\bf Abstract:} 

An Ulrich Module over a standard graded commutative algebra R =

S/I,

with S a polynomial ring, is a finitely generated graded maximal

Cohen-Macaulay R-module M whose free resolution as an S-module

is

linear. There are many other characterizations, and the

condition

is rather natural when one regards the module as a sheaf on the

associated projective variety.



Schreyer and I discovered that Ulrich modules play a key role in

producing the nicest determinantal formulas for Chow forms and

resultants, and I will explain this connection, which

generalizes

the notion of matrix factorizations over hypersurfaces.



The existence of Ulrich modules in general is open, but there

are many

cases in which it is known, and I will report on recent progress

in

this by Hanes, myself, Schreyer, and Weyman.



\bigskip

\bigskip





\noindent 

{\bf Title:} Arf characters of algebroid curves.



\noindent 

{\bf Author:} Valentina Barucci, Marco D'Anna and Ralf Fr\"oberg*



\noindent 

{\bf Abstract:}

An algebroid

branch is 

a one-dimensional ring of the form $R=k[[x_1,\ldots,x_n]]/P$,

where $P$ is 

a prime ideal. The blowup of $R=R_0$ is $R_1=\cup(m^n:m^n)$,

where $m$ is the

maximal ideal of $R$. The blowup $R_1$ is a local overring of

$R$ and, if

$R_{i+1}$ denotes the blowup of $R_i$, then $R_j=k[[t]]$ if

$j>>0$. The

multiplicity sequence of $R$ is $e_0=e(R_0),e_1=e(R_1),\ldots$.

Two

algebroid branches are said to be equivalent if they have the

same

multiplicity sequence. (For plane curves, this is topological

equivalence.)

There is a smallest so called Arf ring $R'$ containing $R$, the

Arf closure

of $R$, and $R'$ is equivalent to $R$. Since the integral

closure of $R$

is $k[[t]]$, every element of $R$ has a value, and the set of

values 

constitute a semigroup $v(R)$. Now $v(R')$ is a so called Arf

semigroup.

There is a smallest semigroup $T$, such that $v(R')$ is the

smallest Arf

semigroup containing $T$. The generators of $N$ was called the

characters

of the curve by Arf. In the semigroup setting these set of

generators

has been studied by Rosales, Garcia-Sanches, Garcia-Garcia, and

Branco.

We generalize the results above to algebroid curves, i.e.

one-dimensional

reduced rings $k[[x_1,\ldots,x_n]]/P_1\cap\cdots\cap P_d$. 



\bigskip

\bigskip



\noindent 

{\bf Title:} Toric varieties with huge Grothendieck

group.



\noindent 

{\bf Author:} Joseph Gubeladze



\noindent 

{\bf Abstract:}

In each dimension $n\geq3$ there are many projective simplicial

toric varieties whose Grothendieck groups of vector bundles are at

least as big as the ground field. In particular, the conjecture

that the Grothendieck groups of locally trivial sheaves and

coherent sheaves on such varieties are rationally isomorphic fails

badly.





\bigskip

\bigskip





\noindent 

{\bf Title:} On the regularity of simplicial semigroup rings with isolated singularity.



\noindent 

{\bf Author:} J\"{u}rgen Herzog* and Takayuki Hibi



\noindent 

{\bf Abstract:}

Let $S = K[x_1, \ldots, x_n]$ be the polynomial ring in $n \geq

2$

variables over a field $K$ and $m$ its graded maximal ideal.

Let $f_1,\ldots, f_m \in S$ be homogeneous polynomials of degree

$d-1\geq 2$ generating an

$m$-primary ideal, and let  $g_1,\ldots,g_r\in S$ be arbitrary

homogeneous polynomials

of degree $d$. It will be proved that

the Castelnuovo--Mumford regularity of the standard graded

$K$-algebra

$A=K[\{f_ix_j\}_{i=1,\ldots,m\atop  j=1,\ldots,n}, g_1,\ldots,

g_r]$

is at most $(d-2)(n-1)$.  By virtue of this result,

it follows that the regularity of a simplicial semigroup ring

$K[C]$ with 

isolated singularity is bounded by $e(K[C]) - codim(K[C])$.

Hence in this 

particular case the Eisenbud-Goto conjectured bound for the

regularity

of a standard graded domain holds. 





\bigskip

\bigskip





\noindent

{\bf Title:} Rings with countable representation type.



\noindent

{\bf Author:} Craig Huneke* and Graham Leuschke



\noindent

{\bf Abstract:}

This talk is a preliminary report on Cohen-Macaulay local rings

having countably many isomorphism classes of maximal

Cohen-Macaulay

modules. Such rings are said to have countable CM representation

type.

Schreyer conjectured that such rings have singular locus of

dimension

at most one. We prove this conjecture for complete rings or

rings

with uncountable residue field. We further prove that this

property

localizes and discuss some questions related to the structure of

such rings.



\bigskip

\bigskip



\noindent 

{\bf Title:} Action of the Borel group on monomial ideals.



\noindent 

{\bf Author:} Alessandro Logar



\noindent 

{\bf Abstract:}

Let $\mathbb T \subseteq {\mathrm{GL}(n,K)}$ be the Borel group

of

upper triangular

matrices. In this paper we want to study the action of $\mathbb

T$

on the set of monomial ideals of ${K[x_1,\dots,x_n]}$

($K$ a field of characteristic

zero) from a computational point of view. More specifically, we

show that the stabilizer of a monomial ideal $M \subseteq

{K[x_1,\dots,x_n]}$ in $\mathbb T$

is a purely combinatorial object and we give

an algorithm for computing it. Then we characterize the

subgroups

of $\mathbb T$ that are stabilizers of monomial ideals,

we give an algorithm

which finds if a given ideal $J$ is in the orbit of a monomial

ideal

$M$ under the action of $\mathbb T$ and in the affirmative case,

finds

the matrices $W\in \mathbb T$ such that $W\cdot J=M$. We

show that the entries of $W$ can be directly obtained

from the coefficients of the generators of $J$, so in particular

no

solutions of polynomial equations are required.





\bigskip

\bigskip



\noindent 

{\bf Title:} Movable components of intersection cycles.



\noindent 

{\bf Author:} Mirella Manaresi



\noindent 

{\bf Abstract:}

St\"uckrad-Vogel intersection cycle for projective varieties over

an

algebraic closed field k contains some fixed parts, defined over

the field

k, and some movable parts, defined over  trascendental

extensions of k.

Van Gastel proved that the fixed parts are the distinguished

varieties of

Fulton-McPherson.  Flenner and Manaresi showed that movable

parts appear as

ramification loci of geneic projections and, using this, gave

some

estimations of their variation (in terms of their transcendence

degree over

k)  and their contribution to the intersection cycle.

The talk will be a survey on some recent results of

Flenner-Manaresi,

Achilles-Manaresi and Achilles-St\"uckrad on movable components

and their

intersection numbers.



\par

References

\par



[1] R.Achilles-M.Manaresi: Multiplicities of bigraded rings

and intersection theory.  Math. Ann. 309,  n. 4, 573-591 (1997)



\par



[2]     R.Achilles - M.Manaresi:  Self-intersection of surfaces

and Whitney

stratifications.: Self-intersections of surfaces and Whitney

stratifications.  In: F. Norguet (ed.), S. Ofman (ed.), Complex

Geometry

1998, Proceedings of the Conference, Paris June 29-July 3,

International

Press, Boston, MA, to appear.



\par



[3]     R.Achilles-J.St\"uckrad: Generic residual intersections

and

intersection numbers of movable components. Preprint 2002



\par



[4]     H.Flenner-M.Manaresi:  Intersections of projective

varieties and

generic projections.  Manuscripta Math. 92, 273-286 (1997)



\par



[5]     H.Flenner-M.Manaresi: Variation of the ramification loci

of generic

projections.  Math. Nachr. 194 (1998), 79-92



\par



[6]     H.Flenner - M.Manaresi: A numerical characterization of

reduction

ideals   Math. Zeitschrift  238 (2001), 205-214.



\par



[7]     H.Flenner - M.Manaresi:  A length formula for the

multiplicity of

distinguished components of intersections    J. Pure and Applied

Algebra

165 (2001) 155-168.



\bigskip

\bigskip



\noindent 

{\bf Title:} Some new results on the Hilbert coefficients.



\noindent 

{\bf Author:} Maria Evelina Rossi



\noindent 

{\bf Abstract:}

Let $I$ be an ${\mathfrak m}$-primary ideal of a local

Cohen--Macaulay ring $(R, {\mathfrak m})$ of dimension $d.$  The

{\it 

Hilbert--Samuel function} of $I$

is the numerical function that measures the growth of the length

of $R/I^n$ for all $n \geq 1$. For $n \gg 0$ this function

$\lambda(R/I^n)$ is a polynomial $P_I(n) $ in $n$ of degree $d$

  $$ P_I(n) = e_0(I) {{n+d-1}\choose d} - e_1(I) {{n+d-2}\choose

d-1} +

  \ldots + (-1)^d e_d(I),

  $$

where $e_0(I), e_1(I), \ldots, e_d(I) $ are the  {\it Hilbert 

coefficients}  of $I$. These

integers give asymptotic information on the Hilbert function of

$I$ 

and hence geometric information

on the singular locus of $R.$ In particular $e_0(I) $ is the 

multiplicity, $ e_d(I) $ and $ (-1)^d

(P_I(1) - \lambda(R/I)) $ are the so called genus and arithmetic

genus of $I.$



Classically one has sought knowledge about the Hilbert

coefficients 

in presence of good depth

properties of the   associated graded ring   $G= \oplus_{n \ge

0} 

I^n/I^{n+1} $ of $I.$

Conversely,  numerical information on $e_i(I)'s, $  have

been used to obtain information on the depth of $G,$ for example

this

has been a constant theme in the work of J.D. Sally.



A very useful technique is to consider the generating function

of 

$\lambda(R/\widetilde {I^n}) $

or $\lambda(R/\overline {I^n}) $

instead of $\lambda(R/{I^n}) $ where $ \widetilde {I^n} $

denotes the 

Ratliff-Rush ideal and

$ \overline {I^n} $ the integral closure of $  {I^n} $ (under

the 

assumption $R $ is analytically

unramified). The advantage is that $ \widetilde{G}=\oplus_{n\ge

0}(\widetilde{I^{n}}/\widetilde{I^{n+1}})$ and $

\overline{G}=\oplus_{n\ge

0}(\overline{I^{n}}/\overline{I^{n+1}})$ have positive depth,

but unfortunately

in general they are not  standard graded algebras.



In this talk we present some result obtained with A.Corso,

C.Polini 

and G.Valla in different papers

involving the use of the above techniques. In the first paper we

study the interplay between the

integral closedness -- or the normality -- of an ideal $I$ and 

conditions on the Hilbert

coefficients of $I$. We relate these properties to the

Cohen--Macaulayness of the associated graded ring of $I.$ We 

generalize some results of Itoh and

Narita and, for normal ideals, we give a positive answer to a 

conjecture stated by Valla

in the case

  $e_2(I)  $ is minimal.



If $I$ is a ${\mathfrak m}-$primary ideal of a  one-dimensional 

Cohen-Macaulay local ring, Valla and

myself

   proved   that the Hilbert function of $\widetilde{G} $ is

strictly increasing up to reach the multiplicity

  $e_0(I), $  the same behaviour which the Hilbert function of

$G$ has 

in the case $G$ is

Cohen-Macaulay.  As a  numerical consequence of the described 

properties of $\widetilde{G}, $

  we  improve   the upper bound for $e_1(\mathfrak m) $  proved

by 

Elias by showing that for a

$\mathfrak m $-primary ideal of a one-dimensional Cohen-Macaulay

local ring $R, $ one has

$$e_1(I) \le     {e_0(I) \choose 2}

-{v(\widetilde{I})-1\choose 2}-\lambda(R/I)+1 $$ where $ v(\ \ )

$ 

denotes the minimal number of

the ideal. This result can be used to  give strict constraints

on the 

Hilbert function of  a

$\mathfrak m

$-primary ideal in a one-dimensional Cohen-Macaulay local ring.

This 

inequality can be extended to

the higher dimensional  case when $I$ is an integrally closed 

$\mathfrak m-$primary ideal.





\bigskip

\bigskip





\noindent 

{\bf Title:} A class of birational transformations.



\noindent 

{\bf Author:} Aron Simis



\noindent 

{\bf Abstract:}

We will briefly review some aspects of birational maps

(mainly

Cremona transformations). A special class to be envisaged is

that of

monomial rational representations of projective space. This

brings a

definite combinatorial flavor into the subject along with

various

questions.



\bigskip

\bigskip



\noindent 

{\bf Title:} Multiplicities and  the Number of Generators

of Cohen--Macaulay Ideals.



\noindent 

{\bf Author}: Wolmer V. Vasconcelos



\noindent 

{\bf Abstract:} 

Let $(R, \mathfrak{m})$ be a Cohen--Macaulay local ring of dimension

$d$ and let $I$ be a Cohen--Macaulay ideal of codimension $g$. There

 are several

invariants of $R$

 and of $I$ (multiplicities, embedding dimension, index of nilpotency, etc)

that may affect the number of generators $\nu(I)$ of $I$. After reviewing

results of Sally, Valla and others, we derive  estimates for

$\nu(I)$ depending on another  mix of some of these invariants.







\bigskip

\bigskip



\noindent 

{\bf Title:} Symmetric and Rees

algebras of Koszul cycles.



\noindent 

{\bf Author}: J\"{u}rgen Herzog, Zhongming Tang and Santiago Zarzuela*



\noindent 

{\bf Abstract:} 

Let $k$ be a field, $P=k[x_1, \dots , x_n]$ the polynomial ring in

$n$ variables, and $E_i$ the $i$-th syzygy module of $k$. We study

the symmetric algebra $S(E_i)$ and the Rees algebra

$R(E_i)=S(E_i)/T_P(S(E_i))$ of $E_i$. By using SAGBI basis

techniques the case $E_2$ is fully described. In particular, we

prove that $x_1, \dots , x_n$ is an unconditioned $d$-sequence on

$S(E_2)$ and $R(E_2)$. We also show that the same result holds for

the case $E_{n-2}$. Preliminary report.



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