




Algebraic Geometry / Géométrie algébrique (Peter Russell, Organizer)
 T. ASANUMA, Toyama University Facuty of Education, Gofuku toyamashi 3190,
9308555 Japan
On generic A^{1}fibrations

This is a joint work with Nobuharu Onoda (Fukui University).
Let (R, pR) be a discrete valuation ring with the quotient field
K = R[p^{1}] and the residue field R/pR = k of characteristic
chk ³ 0. A commutative Rdomain A is called a generic
A^{1}fibration if
So if A is a generic A^{1}fibration, then A[p^{1}] is
a polynomial ring in one variable over R[p^{1}].
Theorem 1
Let A be a finitely generated generic A^{1}fibration over
R. Then we have the following:
REMARK 0.2.
(1) The condition `normal' is necessary in
Theorem 0.1(1) and (2).
(2) There exist examples of generic A^{1}fibrations A
satisfying the condition in Theorem 0.1(1) (resp. Theorem 0.1(3)),
but A/pA are not polynomial ring in one variable over Artin
rings for any chk ³ 0 (resp. for any chk > 0).
(3) We do not know whether Theorem 0.1 (2) is true
in case of chk = p > 0.
 A. BROER, Universite de Montreal, Montreal, Quebec
Normality of nilpotent varieties

In this talk we shall show how we classified normal nilpotent varieties
in a Lie algebra of type E_{6}, using an algorithm for the vanishing
degree of the cohomology of line bundles on the cotangent bundle of
G/B.
 S. D. CUTKOSKY, University of Missouri, Columbia, Missouri, USA
Monomialization of morphisms

Suppose that f: X® Y is a morphism of varieties, over
an algebraically closed field. We consider the problem of monomializing
f by blowing up sequences of nonsingular subvarieties
X_{1}® and Y_{1}® Y to obtain a morphism
X_{1}® Y_{1} which is monomial. We discuss our recent results
on this problem.
 DANIEL DAIGLE, Universite d'Ottawa, Ottawa, Ontario
Affine rulings of weighted projective planes
and actions of (C,+) on C^{3}

We will discuss our investigation (joint with Peter Russell) of affine
rulings of weighted projective planes and some of its consequences for
G_{a}actions on C^{3}. In particular, there is now a proof
of the fact that all homogeneous locally nilpotent derivations of
C[X,Y,Z] can be constructed from some ``basic'' ones, via
the local slice construction of Freudenburg.
 G. FREUDENBURG, University of Southern Indiana, Evansville, Indiana 47712, USA
A nonlinearizable S_{3}action on C^{4}

This talk will discuss recent results obtained jointly by the speaker
and L. MoserJauslin. Examples of nonlinearizable algebraic actions of
certain finite groups on C^{n} (n ³ 4) first appeared in
the 1990s. We give the first such example for the group S_{3}. The
action is on C^{4}, and we also generalize this to show the
existence of nonlinearizable S_{3}actions on C^{n} for
each n ³ 4. Our example is obtained as a restriction of Schwarz's
wellknown action of O(2,C) on C^{4}, and we
thus obtain a new proof that the Schwarz action is nonlinearizable.
The importance of the new example is twofold. First, all known
examples of nonlinearizable reductive group actions can be realized as
the total space of an equivariant vector bundle whose base is a
representation space. In case the group is abelian, it is known that
the bundle must be trivial, and therefore the action on the total space
is trivial. S_{3} is thus the smallest group for which the method of
equivariant vector bundles can be used to construct nonlinearizable
actions. Secondly, the proof is elementary and more transparent than
in other cases. One reason for this is that the action has a line of
fixed points. This is the first example of a nonlinearizable action of
any reductive group on C^{4} having a line of fixed points.
 ANTHONY GERAMITA, Department of Mathematics, Queen's University, Kingston, Ontario
Higher secant varieties of Segre varieties

(Joint work with M. V. Catalisano and A. Gimigliano)
The Segre varieties I will consider in this talk are the usual
embeddings of proj^{n1} ×¼proj^{nr} into proj^{N}
(N = P_{i = 1}^{r}(n_{i}+1)  1) given by O_{projn1}(1)×¼×O_{projnr }(1). If
X is any nondegenerate variety in proj^{N}, then a
secantproj^{s1} to X is a proj^{s1} Ì proj^{N} which is spanned by s distinct points of X. The
variety X^{s1}, which is the closure of the union of all
secantproj^{s1}'s to X, is called the
(s1)secant variety of X.
The problem I will discuss in this talk is the following: What is the
dimension of X^{s1}? and, more particularly, when is this
dimension the expected dimension, i.e.

min
 { sdimX + (s1), N }? 

The answers to these questions are wellknown for r = 2 but very little
is known when r > 2. I will explain an approach to this problem using
apolarity (an approach first intimated by Macaulay and recently
brought into sharp focus by the work of A. Iarrobino and V. Kanev). We
convert the problem to one about Hilbert functions of ``fat'' points in
products of projective spaces (essentially a modern reinterpretation of a
classical theorem of Terracini) and solve this problem in a number of cases.
One particularly interesting collection of cases in which we can give a
complete solution to the problem involves the use of combinatorics. I will
explain a connection between the question concerning the dimension of
secant varieties of Segre varieties and, on the one hand, rook coverings of
multidimensional chessboards (which is, in turn, related to the study of
perfect graphs) and on the other hand a problem about monomial ideals in
multigraded polynomial rings. This last collection of examples builds on
recent results of R. Ehrenborg.
 S. KALIMAN, Department of Mathematics, University of Miami, Coral Gables,
Florida 33124, USA
Families of affine planes: the existence of a cylinder

Given a family of complex affine planes, we show that it is trivial
over a Zariski open subset of the base. The proof relies upon a
relative version of the contraction theorem. This result can be viewed
as a step towards proving a conjecture of Dolgachev and Weisfeiler
which asserts that any such family is a fiber bundle.
 JOSEPH KHOURY, University of Ottawa, Ottawa, Ontario K1N 6N5
On a conjecture of Nowicki

Given a field k of characteristic zero, it is wellknown that the
kernel of any linear derivation of k[X_{1},¼,X_{n}] (that is, a
derivation which maps each X_{i} to a linear form in X_{1},¼,X_{n}) is a finitely generated kalgebra. All known proofs of this
result are not constructive in the sense that we don't know a
generating set for the kernel. Nowicki conjectured in 1996 that the
kernel of the derivation d = å_{i = 1}^{n}X_{i}¶/¶Y_{i} of k[X_{1},¼,X_{n},Y_{1},¼,Y_{n}] is generated
over k[X_{1},¼,X_{n}] by the elements
u_{ij} = X_{i}Y_{j}X_{j}Y_{i} for 1 £ i < j £ n. Using the theory
of Groebner bases, we prove this conjecture in the more general case of
the derivation D = å_{i = 1}^{n}X_{i}^{ti}¶/¶Y_{i}
where each t_{i} is a nonnegative integer. Note that in the case of
the derivation D, the finite generation of the kernel is no longer
evident.
 FV. KUHLMANN, Department of Mathematics and Statistics, University of Saskatchewan,
Saskatoon, Saskatchewan S7N 5E6
On local uniformization in arbitrary characteristic

In 1939, Zariski proved the Local Uniformization Theorem for places of
algebraic function fields over ground fields of characteristic 0.
Later, he used this theorem to prove resolution of singularities for
surfaces in characteristic 0. Apart from Abhyankar's results for
dimension up to 3 and de Jong's desingularization by alteration, not
much has been known for positive characteristic.
We prove that every place of an algebraic function field FK of
arbitrary characteristic admits local uniformization, provided that the
sum of the rational rank of its value group and the transcendence
degree of its residue field over K is equal to the transcendence
degree of FK (we call such places Abhyankar places). Further,
we show that finitely many such places admit simultaneous local
uniformization if they have isomorphic value groups. Since Abhyankar
places lie dense in the Zariski space of all places of FK with
respect to the patch topology, simultaneous local uniformization of any
finite number of them might open a way to pass from local
uniformization to resolution of singularities.
Further, we prove that every place of an algebraic function field FK
of arbitrary characteristic admits local uniformization in a finite
extension F of F. This fact actually follows from de
Jong's result. But we can show in addition that FF can be
chosen to be Galois. Alternatively, FF can be chosen to
satisfy a valuation theoretical condition which is very natural in
positive characteristic. Our proofs are based solely on valuation
theoretical theorems, which are of fundamental importance in positive
characteristic.
We also indicate certain analogues of our results for the arithmetic
case.
The corresponding preprints can be downloaded from the Valuation Theory
Home Page at http://math.usask.ca/fvk/Valth.html.
 K. MASUDA, Department of Mathematics, Himeji Insititute of Technology,
2167 Shosha, Himeji, 6712201 Japan
Algebraic Gvector bundles over Grepresentations

Let G be a reductive algebraic group defined over the ground field
C of complex numbers. Let P and Q be
Grepresentations. We denote by VEC_{G}(P,Q) the set of equivariant
isomorphism classes of algebraic Gvector bundles over P whose
fiber over the origin is isomorphic to Q. For a nonabelian group
G, VEC_{G}(P,Q) can be nontrivial. In fact, Schwarz first showed
that VEC_{G}(P,Q) is isomorphic to an additive group C^{p}
for a nonnegative integer p when the algebraic quotient P//G is
onedimensional. The nontrivial Gvector bundles found by Schwarz
led to the first examples of nonlinearizable actions on affine space.
However, when dimP//G ³ 2, some examples show that
VEC_{G}(P,Q) is not finitedimensional any more. In this talk, we
construct a map Y_{P,Q} from a certain subspace of VEC_{G}(P,Q)
to a Cmodule possibly of infinite dimension when dimP//G ³ 2. The map Y_{P,Q} can be a surjection or even an
isomorphism under some condition on P and Q.
We also show nontriviality of moduli of algebraic Gvector bundles
over Gstable affine hypersurfaces of a certain type. In particular,
we show that moduli space of algebraic Gvector bundles over
Gstable affine quadrics with fixpoints and onedimensional quotient
contains C^{p}.
 M. MIYANISHI, Department of Mathematics, Osaka University, Toyonaka, Osaka 5600043,
Japan
Equivariant classification of open algebraic surfaces

Let G be a finite group. Consider the set of log projective surfaces
([`(V)],[`(D)]) defined over a fixed, algebraically closed,
gound field of characteristic zero which admit effective algebraic
Gactions. We say that a morphism f: ([`(V)],[`(D)]) ®([`(W)],[`(G)]) is a Gmorphism (or equivariant
morphism) if f commutes with the Gactions. We can define the
notion of relatively minimal (or minimal) model with respect to the
Gbirational morphisms.
The objective of the present research is to consider the equivariant
classification of such Grelatively minimal log projective surfaces
in the case where the log Kodaira dimension of ([`(V)],[`(D)])
is ¥, i.e., [`(k)]([`(V)][`(D)]ÈSing[`(V)]) = ¥. Our attempt is achieved under some technical
hypotheses which enable us to make use of the Mori theory, but it still
reveals some phenomena which are particular to the equivariant
settings.
 Z. REICHSTEIN, Oregon State University, Oregon, USA
Birational geometry of algebraic group actions

Let G be an algebraic group. Recall that elements of H^{1}(K, G),
often called Gtorsors or principal homogeneous Gspaces, are
naturally identified with certain algebraic objects defined over K.
These objects are ndimensional nonsingular quadratic forms if
G = O_{n}, degree n etale algebras if G = S_{n}, degree n central simple
algebras if G = PGL_{n}, Cayley algebras when G = G_{2}, etc.
In this talk, based on joint work with Boris Youssin, I will take the
following approach to the study of H^{1}(K,G). First I will use
resolution of singularities techniques to construct a ``good''
birational model X for a given torsor T in H^{1}(K, G), then read
off certain birational invariants of X (as a variety with a
Gaction) from this model. Several applications of this technique
will be discussed in the talk.
 PAUL ROBERTS, University of Utah, Salt Lake City, Utah 84112, USA
Intersection theory and commutative algebra

The problem of defining intersection multiplicities in Algebraic
Geometry has given rise to a number of fundamental questions in
Commutative Algebra. Serre gave a definition using homological methods
which satisfies many of the desired properties, and he stated several
other properties as conjectures. This talk will present the background
of this problem, discuss how these questions led to various homological
conjectures in Commutative Algebra, and outline the current
developments on these conjectures.
 PETER RUSSELL, Department of Mathematics, McGill University, Montréal,
Québec H3A 2K6
Birational endomorphisms of C^{2} and rational surfaces of the
form ruled surface\ample section

This is a report on joint work with Pierrette CassouNogues. Given a
birational morphism f: X = C^{2} ® C^{2} = Y, the
``missing curves'' (curves in Y with generic point not in f(X))
form a configuration of rational curves with one place at infinity,
formally similar to the ramification locus in a (potential)
counterexample to the Jacobian problem. We classify the f where the
missing curves consist of k concurrent lines together with an
additional curve D. This leads to the determination of all open U @ C^{2} in surfaces Z = F_{n} \S, F_{n} a rational ruled
surface and S an ample section. We show that Z is determined by
S^{2} = k + 1 up to isomorphism and discuss Aut (Z).
 A. SATHAYE, University of Kentucky, Lexington, Kentucky 40506, USA
Planes over a two dimensional base

Given a finitely generated two dimensional algebra A over a base R,
it is of interest to determine when A is isomorphic to R^{[}2], the
polynomial ring in two variables over R. In case, R is a DVR
containing the rationals, a traditional technique consists of assuming
that A has two (generic) ring generators over the quotient field of
R. Then under the necessary condition that A tensored with the
residue field of R stays a polynomial ring in two variables (over the
residue field), one develops a modification techniqu to repair the
generic generators into genuine ring generators over R. The technique
demands new modification techniques over a two dimensional base R. We
will discuss the difficulties and needed modifications.
 ADAM VAN TUYL, Queen's University, Kingston, Ontario
Hilbert functions of points in P^{n} ×P^{m}

I will begin by presenting a result that will describe the long term
behaviour of a Hilbert function of points in P^{n} xP^{m}. I will then specialize to the case n = m = 1. I will
show that if X Ì P^{1} x P^{1} is a set
of s distinct points, then the eventual behaviour of the Hilbert
function can be determined from the combinatorial information of
X. After introducing some results on partitions, I will
also demonstrate a connection between the Hilbert functions of points
and a classical theorem of Ryser on the properties of
(0,1)matrices. If time permits, I will demonstrate a new
characterization of arithmetic CohenMacaulay points in P^{1} xP^{1}.
 J. WLODARCZYK, Purdue University, West Layfayette, Indiana, USA
Algebraic Morse theory and factorization of birational maps

We develop a Morselike theory which plays a crucial role in our proof
of the Weak Factorization Theorem: Any birational map between two
complete nonsingular varieties over an algebraically closed field of
characteristic zero can be factored into a sequence of blowups and
blowdowns with smooth centers. In the algebraic Morse theory the Morse
function is replaced by a K^{*}action. The critical points of the
Morse function correspond to connected fixed point components. Passing
through the fixed points induces birational tarnsformations (blowups,
blowdowns and flips) which are analogous to spherical modifications.

