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Algebraic Geometry / Géométrie algébrique
(Peter Russell, Organizer)

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T. ASANUMA, Toyama University Facuty of Education, Gofuku toyama-shi 3190, 930-8555  Japan
On generic A¹-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 R-domain A is called a generic A¹-fibration if

A ÄR K = K[1].

So if A is a generic A¹-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 A¹-fibration over R. Then we have the following:

(1) If A is normal and the radical Ö{pA} of the ideal pA of A is prime, then there exists a finite field extension k₁/k such that

A/ Ö   pA   = k1 [1]


(2) If A is normal and chk = 0, then there exists a finite set {k_i|i = 1,...,s} of finite field extensions k_i/k (i = 1,...,s) such that

A/ Ö   pA   = (k1×¼×ks) [1]


(3) If chk = 0 and A is generated by two elements over R, then there exists an Artin ring L such that

A/pA = L[1]

REMARK 0.2. (1)  The condition `normal' is necessary in Theorem 0.1(1) and (2).

(2)  There exist examples of generic A¹-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 E₆, 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 X₁® and Y₁® Y to obtain a morphism X₁® Y₁ 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 C³

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 C³. 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 non-linearizable S₃-action on C⁴

This talk will discuss recent results obtained jointly by the speaker and L. Moser-Jauslin. Examples of non-linearizable algebraic actions of certain finite groups on Cⁿ (n ³ 4) first appeared in the 1990s. We give the first such example for the group S₃. The action is on C⁴, and we also generalize this to show the existence of non-linearizable S₃-actions on Cⁿ for each n ³ 4. Our example is obtained as a restriction of Schwarz's well-known action of O(2,C) on C⁴, and we thus obtain a new proof that the Schwarz action is non-linearizable.

The importance of the new example is twofold. First, all known examples of non-linearizable 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. S₃ is thus the smallest group for which the method of equivariant vector bundles can be used to construct non-linearizable 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 non-linearizable action of any reductive group on C⁴ 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^n_r into proj^N (N = P_{i = 1}^r(n_i+1) - 1) given by O_projⁿ¹(1)×¼×O_proj^nr (1). If X is any non-degenerate variety in proj^N, then a secant-proj^s-1 to X is a proj^s-1 Ì proj^N which is spanned by s distinct points of X. The variety X^s-1, which is the closure of the union of all secant-proj^s-1's to X, is called the (s-1)-secant variety of X.

The problem I will discuss in this talk is the following: What is the dimension of X^s-1? and, more particularly, when is this dimension the expected dimension, i.e.

min { sdimX + (s-1), N }?

The answers to these questions are well-known 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 re-interpretation 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 well-known that the kernel of any linear derivation of k[X₁,¼,X_n] (that is, a derivation which maps each X_i to a linear form in X₁,¼,X_n) is a finitely generated k-algebra. 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}ⁿX_i¶/¶Y_i of k[X₁,¼,X_n,Y₁,¼,Y_n] is generated over k[X₁,¼,X_n] by the elements u_ij = X_iY_j-X_jY_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}ⁿX_i^t_i¶/¶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.

F-V. 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 F|K 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 F|K (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 F|K 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 F|K 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 F|F can be chosen to be Galois. Alternatively, F|F 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, 671-2201  Japan
Algebraic G-vector bundles over G-representations

Let G be a reductive algebraic group defined over the ground field C of complex numbers. Let P and Q be G-representations. We denote by VEC_G(P,Q) the set of equivariant isomorphism classes of algebraic G-vector bundles over P whose fiber over the origin is isomorphic to Q. For a non-abelian group G, VEC_G(P,Q) can be non-trivial. 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 one-dimensional. The non-trivial G-vector bundles found by Schwarz led to the first examples of non-linearizable actions on affine space. However, when dimP//G ³ 2, some examples show that VEC_G(P,Q) is not finite-dimensional any more. In this talk, we construct a map Y_P,Q from a certain subspace of VEC_G(P,Q) to a C-module 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 non-triviality of moduli of algebraic G-vector bundles over G-stable affine hypersurfaces of a certain type. In particular, we show that moduli space of algebraic G-vector bundles over G-stable affine quadrics with fixpoints and one-dimensional quotient contains C^p.

M. MIYANISHI, Department of Mathematics, Osaka University, Toyonaka, Osaka  560-0043, 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 G-actions. We say that a morphism f: ([`(V)],[`(D)]) ®([`(W)],[`(G)]) is a G-morphism (or equivariant morphism) if f commutes with the G-actions. We can define the notion of relatively minimal (or minimal) model with respect to the G-birational morphisms.

The objective of the present research is to consider the equivariant classification of such G-relatively 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 H¹(K, G), often called G-torsors or principal homogeneous G-spaces, are naturally identified with certain algebraic objects defined over K. These objects are n-dimensional 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 = G₂, etc.

In this talk, based on joint work with Boris Youssin, I will take the following approach to the study of H¹(K,G). First I will use resolution of singularities techniques to construct a ``good'' birational model X for a given torsor T in H¹(K, G), then read off certain birational invariants of X (as a variety with a G-action) 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 C² and rational surfaces of the form ruled surface\ample section

This is a report on joint work with Pierrette Cassou-Nogues. Given a birational morphism f: X = C² ® C² = 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) counter-example 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 @ C² in surfaces Z = F_n \S, F_n a rational ruled surface and S an ample section. We show that Z is determined by S² = 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 Pⁿ ×P^m

I will begin by presenting a result that will describe the long term behaviour of a Hilbert function of points in Pⁿ xP^m. I will then specialize to the case n = m = 1. I will show that if X Ì P¹ x P¹ 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 Cohen-Macaulay points in P¹ xP¹.

J. WLODARCZYK, Purdue University, West Layfayette, Indiana, USA
Algebraic Morse theory and factorization of birational maps

We develop a Morse-like 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.