


Commutative Algebra / Algèbre commutative Org: Sara Faridi (Ottawa), Sindi Sabourin (York), Will Traves (US Naval Academy) and/et Adam van Tuyl (Lakehead)
 JASON BELL, University of Michigan, Department of Mathematics,
525 E. University Ave, Ann Arbor, MI 48109
Critical density in integral schemes

We say that an infinite set S of closed points of an integral scheme
X is critically dense if every infinite subset of S is
Zariski dense in X. This property was introduced by Rogalski, who
used it to construct examples of noncommutative algebras with
counterintuitive properties.
Given a closed point p Î X and an automorphism s of X, we
look at the set {s^{n}(p)  n Î Z}. Motivated by a
question of Rogalski, we explain why for sets of this form, critical
density is the same as density for affine nspace over a field of
characteristic 0, Fano varieties over a field of characteristic 0,
and projective nspace over a field of characteristic 0.
Moreover, we explain why these results do not hold in characteristic
p > 0. We conclude with some open questions.
 ALESSANDRA BERNARDI, Università degli studi di Milano, Via Saldini 50
Osculating varieties of Veronesean and their higher secant
varieties

I want to consider the kosculating varieties to the Veronese
duple embeddings of P^{n} and study the dimension of their
higher secant varieties O^{s}_{k,n,d} via Inverse Systems. By
associating certain 0dimensional schemes Y Ì P^{n} to
O^{s}_{k,n,d} and by studying their Hilbert function it's possible, in
several cases, to determine whether those secant varieties are
defective or not.
 RAGNAROLAF BUCHWEITZ, University of Toronto, Dept. of Math., 100 St. George St,
Toronto, ON M5S 3G3
Free Divisors in Quiver Representations

Free divisors are reduced hypersurfaces whose singular locus is
CohenMacaulay of codimension 2 in the ambient space. This concept
was isolated first by K. Saito in the early seventies, who observed
that the classical discriminants of polynomials are of this form and
explained this in terms of deformation theory. In this talk, we
indicate how new examples arise from the representation theory of
quivers.
Parts of this talk are joint work with W. Ebeling (Hanover, Germany)
and D. Mond (Warwick, UK).
 MASSIMO CABOARA, Dip. di Matematica, Università di Pisa, Via Buonarroti 2,
56127 Pisa, Italy
Basic Module Operations For The Dumb

I will deal with basic module theoretic operations (syzygy
computations, module intersection and colon, linear system solving) of
free, finitely generated modules over k[x_{1},...,x_{n}]. I will show
a technique for reducing these operations to the computation of a
suitable component elimination module. The algorithms resulting from
this technique are easy to describe, prove and implement without any
efficiency loss in comparison with the traditional algorithms.
 JAYDEEP CHIPALKATTI, University of Manitoba, Winnipeg, Manitoba
Transvectants, Feynman diagrams and normality of a Brill
variety

Let d=2e denote an even positive integer. Identify the space of
nary dics with P^{N}, where N = \binomd+n1d1.
Now consider the projective subvariety
X = {F Î P^{N} : F = (l_{1} l_{2})^{e} for some linear forms l_{i} }. 

For historical reasons, this may be called a Brill variety. Our main
result says that X is rnormal for r ³ 2 (in characteristic
zero). As a consequence, one can explicitly write down defining
equations for X in the case of binary forms.
The crux of the result turns out to be representationtheoretic, and
is reminiscent of the FoulkesHowe conjecture. It amounts to the
assertion that a certain canonical morphism
Sym^{r} (Sym^{d} C^{n}) ® Sym^{2} (Sym^{re}C^{n}) 

is surjective for any r ³ 2. We show this by reducing to the
binary case, rephrasing it as a question involving transvectants,
followed by a calculation with Feynman diagrams and hypergeometric
series. This is joint work with Abdelmalek Abdesselam from
Universit\' e Paris XIII.
 BRIAN COOLEN, Queen's University, Kingston, Ontario, Canada
Lower Bounds and Hilbert Functions of Level Algebras

Given a polynomial ring R = k[x_{1},...,x_{n}], graded by
N, and a homogeneous artinian ideal I of R, then the
quotient A=R/I is a graded ring that is a finite dimensional vector
space over k. In particular, A can be decomposed as A = kÅA_{1}Å¼ÅA_{c}, where A_{c} ¹ 0. We let the
hvector of A be the vector (1,h_{1},...,h_{c}), where h_{i} = dim_{k} A_{i}.
It is a natural question to ask which sequences of numbers
(1,b_{1},...,b_{l}) can be the hvector of a graded artinian ring.
This question was answered by Macaulay, who gave a test one can apply
to the sequence to figure out whether it is the hvector of a graded
artinian ring or not.
However, if one imposes the condition that A must be a level
algebra, then question becomes more difficult and it has not been
solved yet. In general, provided one knows n, c and h_{c}, then
one can find an upper bound on the size of the h_{i}. However, there
is no corresponding lower bound.
In this talk, we shall examine this issue and compute a lower bound
for the dimension of A_{c1}. We shall then show that while this
lower bound is not sharp, we can get very close to it. In particular,
we will show that Zanello's theorem on the size of h_{c1} does not
extend to the case n > 7.
 SUSAN COOPER, Queen's University, Kingston, Ontario K7L 3N6
Hilbert Functions Of Subsets Of Complete Intersections

A characterization of which sequences can be the Hilbert function of a
finite set of distinct points in projective nspace P^{n}
(called valid Hilbert functions for P^{n}) follows from
the work of Macaulay, Hartshorne, and others. Although Hilbert
functions of complete intersections are wellknown, Hilbert functions
of subsets of complete intersections have not yet been classified,
even for the simplest cases. Let 1 £ d_{1} £ d_{2} £ ¼ £ d_{n} be positive integers and H be a valid Hilbert function for
P^{n}. We wish to determine if there exists some reduced
zerodimensional complete intersection C.I.(d_{1}, ..., d_{n}) which
contains a subset whose Hilbert function is H.
The special case of this problem where the ideal of the complete
intersection is generated by products of linear forms follows from the
combinatorial work of Clements and Lindström. In general, this
problem was completely answered for the case n=2 in my M.Sc. thesis
and I currently have partial results for n=3. We will discuss these
results as well as consider the complications that have arisen in
working on this research problem. We will conclude with applications
to subsets with the CayleyBacharach Property and extremal subsets
similar to Sub_{d}(H). This work will be included in my Ph.D. dissertation.
 DANIEL DAIGLE, University of Ottawa
Endomorphisms of R[X,Y] which are generically automorphic

Let R be an integral domain, K its field of fractions and X,Y
indeterminates. An Rendomorphism R[X,Y] ® R[X,Y] is said to be
generically automorphic if its extension K[X,Y] ® K[X,Y]
is an automorphism. Let \operatornameEnd^{*}_{R} (R[X,Y]) be the set of
Rendomorphisms of R[X,Y] which are generically automorphic. For
each r Î R\{0}, define m^{(r)} Î \operatornameEnd^{*}_{R} (R[X,Y])
by m^{(r)}(X) = X and m^{(r)}(Y) = rY.
We say that R has the weak GAproperty if every element
of \operatornameEnd^{*}_{R} (R[X,Y]) is a composition of Rautomorphisms of
R[X,Y] and of endomorphisms of type m^{(r)}. We partially solve
the problem of identifying the class of domains having the weak
GAproperty. In particular, we show that an affine domain over
C has the weak GAproperty if and only if it is a
principal ideal domain.
Joint work with Stéphane Vénéreau.
 ANTHONY GERAMITA, Queen's University, Kingston, Ontario
SegreVeronese Embeddings of P^{1} × P^{1}: Secant Varieties and Fat Point Schemes in
P^{2}

If R = k[x_{0},x_{1}] = Å_{d ³ 0} R_{d} is the usual graded
decomposition of the polynomial ring, then the map f_{(d1,d2)}: P^{1} ×P^{1} ® P^{d1 d2+d1+d2}
induced by the canonical multilinear map
R_{d1} ×R_{d2} ® R_{d1} ÄR_{d2} 

is called the (d_{1},d_{2}) SegreVeronese embedding of P^{1} ×P^{1}.
The question considered in this talk is: What are the dimensions of
all the higher secant varieties to this embedding of P^{1} ×P^{1}?
We show how this question is related to that of finding the Hilbert
function, in degree d_{1} + d_{2}, of the subscheme of P^{2} defined
by an ideal of the form
I = Ã_{1}^{d1} ÇÃ_{2}^{d2} ÇÃ_{3}^{2} Ç¼ÇÃ_{s}^{2} 

where: the Ã_{i} are the ideals of points in P^{2}; X
={P_{1},...,P_{s}} is a generic set of s points in P^{2}; and
s2 indicates which higher secant variety we are considering.
We completely solve this problem by giving a description of the
entire Hilbert function of schemes defined by ideals like I
above. As a byproduct of our investigations we verify the
HarbourneHirschowitz conjecture for this family of ideals.
This is joint work with M. V. Catalisano (Genoa) and A. Gimigliano
(Bologna).
 GRAHAM LEUSCHKE, Syracuse University, Syracuse, NY 13210
Endomorphism Rings of Finite Global Dimension

For a commutative local ring R, consider (possibly noncommutative)
Ralgebras L of the form L = End_{R}(M), where M is
a reflexive Rmodule with nonzero free direct summand. Such
algebras L of finite global dimension can be viewed as
possible subsitutes for, or analogues of, a resolution of
singularities of SpecR. For example, Van den Bergh has shown that
a threedimensional Gorenstein normal Calgebra with
terminal singularities has a crepant resolution of singularities if
and only if it has such an algebra L with finite global
dimension and which is maximal CohenMacaulay over R (a
"noncommutative crepant resolution of singularities"). We produce
algebras L = End_{R}(M) having finite global dimension in two
contexts: when R is a complete onedimensional local ring, or when
R is a CohenMacaulay ring of finite CohenMacaulay type. If, in
the latter case, R is Gorenstein, then the construction gives a
noncommutative crepant resolution of singularities in the sense of Van
den Bergh.
 PING LI, Queen's University, Kingston, Ontario, Canada
The Coincidence Between the CohenMacaulay Property and
Seminormality

In the early 1970s, Hochster proved that normal semigroup rings
generated by monomials are CohenMacaulay. When we weaken normal to
seminormal, naturally, a question is raised.
Question: When is a seminormal affine semigroup ring
CohenMacaulay? Conversely, when is a CohenMacaulay affine semigroup
ring seminormal?
However, this question is not easily answered because for affine
semigroup rings, many examples show that CohenMacaulay and seminormal
may not coincide. For example, A=k[t^{2},t^{3}] is CohenMacaulay but
not seminormal, conversely, A=k[x,y,z^{2},xz,yz] is seminormal but not
CohenMacaulay.
In this talk I will explore the coincidence between the CohenMacaulay
property and seminormality of arbitrary affine semigroup rings, and
demonstrate that under certain circumstances, an affine semigroup ring
is seminormal if it is CohenMacaulay. Furthermore, a conjecture is
given for a seminormal affine semigroup ring k[S] to be
CohenMacaulay, and is proved when rank(S) \leqslant 3.
 MOIRA McDERMOTT, Gustavus Adolphus College, 800 W. College Ave, St. Peter, MN
16082, USA
HilbertKunz functions of normal local rings

This talk will report on work, joint with Craig Huneke and Paul
Monsky, regarding the existence of a second coefficient in the
HilbertKunz function of an mprimary ideal in a normal local ring
of positive characteristic. In particular, let (R, m, k) be an
excellent, local, normal ring of characteristic p with a perfect
residue field and dimR=d. We let n be a varying nonnegative
integer and let q=p^{n}. If I is an mprimary ideal of R, and
M is a finitely generated Rmodule, then there exists a real
number b such that the length of M/I^{[q]}M can be written as
aq^{d} +bq^{d1} +O(q^{d2}).
 CLAUDIA MILLER, Syracuse University, Department of Math, 215 Carnegie,
Syracuse, NY 13244
Extremal Algebras

I will speak on joint work concerning ring homomorphisms R ® S
that provide S with an Rmodule structure that has an extremal
resolution. If S is not finitely generated, we have to generalize
the definition of Betti numbers in an appropriate way. (More
generally, we consider and develop the homological algebra of an
Smodule over the homomorphism rather than the special case N=S.)
In particular, we get an example of a surprising phenomenon: that the
Rmodule structure of S can actually give information about the
rings R and S.
The urexample of an extremal homomorphism is the Frobenius
homomorphism in characteristic p > 0, but there are many others; in
fact, the original motivation of this work was to show the extremality
of the Frobenius.
 REZA NAGHIPOUR, Department of Mathematics, University of Tabriz
Asymptotic behavior of integral closures in modules

Let R be a commutative Noetherian Nagata ring, let M be a nonzero
finitely generated Rmodule, and let I be an ideal of R such
that \operatornameheight_{M} I > 0. In this paper there is a definition of the
integral closure N_{a} for any submodule N of M extending Rees'
definition for the case of a domain. As a main result it is shown
that for any submodule N of M, the sequences \operatornameAss_{R} M/(I^{n} N)_{a}
and \operatornameAss_{R} (I^{n} M)_{a} / (I^{n}N)_{a}, n = 1,2,..., of associated
prime ideals, are increasing and ultimately constant for large n.
This talk will describe joint work with Peter Schenzel.
 LESLIE ROBERTS, Queen's University, Kingston, Ontario K7L 3N6
NonCohenMacaulay Projective Monomial Curves

We compare several ways of measuring how far a projective monomial
curve is from being CohenMacaulay, and use these results to show that
as the degree d goes to infinity the number of CohenMacaulay
projective monomial curves of degree d grows exponentially, but the
fraction of all projective monomial curves of degree d that are
CohenMacaulay approaches 0.
This is joint work with Les Reid.
 MICHAEL ROTH, Queen's University, Kingston, Ontario
A conjecture in nonmodular invariant theory.

It seems somewhat unbelievable that there would be a fact about the
invariant theory of finite groups in the nonmodular case which is not
well understood, but I think that there is an essential principle
missing. The principle is simply the invariant theory version of
Brauer's theory of relative invariants over a DVR or Dedekind domain.
The natural language to express this principle is that of algebraic
geometry, and the talk will focus on explaining what the conjectured
principle is, as well as some evidence for its truth, and some of its
consequences.
 MONIREH SEDGHI, Department of Mathematics, University of Tabriz
Asymptotic behaviour of monomial ideals on regular sequences

Let R be a commutative Noetherian ring, and let x = x_{1},...,x_{d} be a regular Rsequence contained in the Jacobson
radical of R. An ideal I of R is said to be a monomial ideal
with respect to x if it is generated by a set of monomials
x_{1}^{e1}¼x_{d}^{ed}. The monomial closure of I, denoted by
[(I)\tilde], is defined to be the ideal generated by the set of
all monomials m such that m^{n} Î I^{n} for some n Î N.
It is shown that the sequences \operatornameAss_{R} R/[(I^{n})\tilde] and \operatornameAss_{R}[(I^{n})\tilde]/I^{n}, n = 1,2,..., of associated prime ideals are
increasing and ultimately constant for large n. In addition, some
results about the monomial ideals and their integral closures are
included.
 GREGORY SMITH, Queen's University
Toric varieties as fine moduli spaces

We'll discuss how to construct projective simplicial toric varieties
as the fine moduli space of representations of a quiver.
 DAVID WEHLAU, Royal Military College of Canada
Separating Invariants

The concept of separating invariants, as opposed to generating
invariants, was introduced a few years ago. One of the most important
goals in invariant theory is to find invariants which separate group
orbits. For this purpose, it is not necessary to find all invariants
but only enough invariants to separate orbits. Such a set of
invariants is called a set of separating invariants. Separating
invariants enjoy a number of good properties and are often more well
behaved then a full set of generating invariants. I will discuss some
of these good properties, giving examples. I will also describe a new
result (proved jointly with with H. E. A. Campbell and G. Kemper)
which gives a characteristic free version of Weyl's theorem on
polarization for separating invariants.

