A group of Gerry Schwarz’s colleagues and collaborators gathered at the Fields Institute in Toronto for a mathematical festschrift in honor of his 60th birthday. This volume is an outgrowth of that event, covering the wide range of mathematics to which Gerry Schwarz has either made fundamental contributions or stimulated others to pursue. The articles are a sampling of modern day algebraic geometry with associated group actions from its leading experts, with a particular focus on characteristic 0 and modular invariant theory.

This article studies the sequence of iterative degrees of a birational map of the plane. This sequence is known either to be bounded or to have a linear, quadratic or exponential growth. 
The classification elements of infinite order with a bounded sequence of degrees is achieved, the case of elements of finite order being already known. The coefficients of the linear and quadratic growth are then described, and related to geometrical properties of the map. The dynamical number of base-points is also studied. 
Applications of our results are the description of embeddings of the Baumslag-Solitar groups and GL(2,Q) into the Cremona group.

We study the algebraic structure of the n-dimensional Cremona

group and show that it is not an algebraic group of infinite dimension (indgroup)

if n>1. We describe the obstruction to this, which is of a topological

nature.

By contrast, we show the existence of a Euclidean topology on the Cremona

group which extends that of its classical subgroups and makes it a topological

group.

We give a way to construct group of pseudo-automorphisms of

rational varieties of any dimension that fix pointwise the image of a cubic

hypersurface of p^n. These group are free products of involutions, and most of

their elements have dynamical degree > 1. Moreover, the Picard group of the

varieties obtained is not big, if the dimension is at least 3.

We also answer a question of E. Bedford on the existence of birational maps

of the plane that cannot be lifted to automorphisms of dynamical degree > 1,

even if we compose them with an automorphism of the plane.

In this paper, we give a sufficient condition for a set G of polynomials to be a Groebner basis with respect to a given term-order for the ideal I that it generates. Our criterion depends on the linkage pattern of the ideal I and of the ideal generated by the initial terms of the elements of G. We then apply this criterion to ideals generated by minors and pfaffians. More precisely, we consider large families of ideals generated by minors or pfaffians in a matrix or a ladder, where the size of the minors or pfaffians is allowed to vary in different regions of the matrix or the ladder. We use the sufficient condition that we established to prove that the minors or pfaffians form a reduced Groebner basis for the ideal that they generate, with respect to any diagonal or anti-diagonal term-order. We also show that the corresponding initial ideal is Cohen-Macaulay. Our proof relies on known results in liaison theory, combined with a simple Hilbert function computation. In particular, our arguments are completely algebraic.

We give a simple set of generators and relations for the Cre-

mona group of the plane. Namely, we show that the Cremona group is the

amalgamated product of the de Jonquieres group with the group of auto-

morphisms of the plane, divided by one relation .

We characterise smooth curves in P3 whose blow-up produces

a threefold with anticanonical divisor big and nef. These are

curves C of degree d and genus g lying on a smooth quartic, such that

(i) 4d 􀀀 30  g  14 or (g; d) = (19; 12), (ii) there is no 5-secant line,

9-secant conic, nor 13-secant twisted cubic to C. This generalises the

classical similar situation for the blow-up of points in P2.

We describe then Sarkisov links constructed from these blow-ups, and

are able to prove the existence of Sarkisov links which were previously

only known as numerical possibilities.

We give the classication of elements { respectively cyclic subgroups {

of nite order of the Cremona group, up to conjugation. Natural parametri-

sations of conjugacy classes, related to xed curves of positive genus, are

provided.

Ideals generated by pfaffians are of interest in commutative algebra and algebraic geometry, as well as in combinatorics. In this article we compute multiplicity and Castelnuovo-Mumford regularity of pfaffian ideals of ladders. We give explicit formulas for some families of ideals, and indicate a procedure that allows to recursively compute the invariants of any pfaffian ideal of ladder. Our approach makes an essential use of liaison theory.

It is well-known and easy to observe that affine (respectively projective)

simple arrangement of n pseudo-lines may have at most n(n − 2)/3

(respectively n(n − 1)/3) triangles. However, these bounds are reached for

only some values of n (mod 6). We provide the best polynomial bound for the

affine and the projective case, and for each value of n (mod 6).

The Cremona group is connected in any dimension and, endowed

with its topology, it is simple in dimension 2.

We study the family of ideals generated by minors of mixed size contained in a ladder of a symmetric matrix from the point of view of liaison theory. We prove that they can be obtained from ideals of linear forms by ascending G-biliaison. In particular, they are glicci.

The ideals generated by pfaffians of mixed size contained in a subladder of a skew-symmetric matrix of indeterminates define arithmetically Cohen-Macaulay, projectively normal, reduced and irreducible projective varieties. We show that these varieties belong to the G-biliaison class of a complete intersection. In particular, they are glicci.

The paper starts with the following simple observation. Let V be a representation of a reductive group G, and let f₁,f₂,...,f_n be homogeneous invariant functions. Then the polarizations of f₁,f₂,...,f_n define the nullcone of k ≤ m copies of V if and only if every linear subspace L of the nullcone of V of dimension ≤ m is annhilated by a one-parameter subgroup (shortly a 1-PSG). (This means that there is a group homomorphism h: C* -> G such that lim_{t->0} h(t) x = 0 for all x in L.)

This is then applied to many examples. A surprising result is about the group SL(2,C) where almost all representations V have the property that all linear subspaces of the nullcone are annihilated. Again, this has interesting applications to the invariants on several copies. Another result concerns the n-qubits which appear in quantum computing. This is the representation of a product of n copies of SL(2,C) on the n-fold tensor product C² otimes C²otimes ... otimes C². Here we show just the opposite, namely that the polarizations never define the nullcone of several copies if n ≥ 3.

(An earlier version of this paper, distributed in 2002, was split into two parts; the first part with the title ``On the nullcone of representations of reductive groups'' is published in Pacific J. Math. 224 (2006), 119-140.)

We give a complete classification of maximal algebraic subgroups of the Cremona group Bir(P2) and provide algebraic varieties that parametrize the conjugacy classes.

The authors T.Harima, J.C.Migliore, U.Nagel and J.Watanabe characterize
the Hilbert function of algebras with the Lefschetz property.
We extend this characterization to algebras with the Lefschetz
property m times. We also give upper bounds for the Betti numbers
of Artinian algebras with a given Hilbert function and with the Lefschetz
property m times and describe the cases in which these bounds
are reached

We study the group of birational transformations of the plane that

fix (each point of) a curve of geometric genus 1.

A precise description of the finite elements is given; it is shown in particular

that the order is at most 6, and that if the group contains a non-trivial torsion,

the fixed curve is the image of a smooth cubic by a birational transformation

of the plane.

We show that for a smooth cubic, the group is generated by its elements

of degree 3, and prove that it contains a free product of Z/2Z, indexed by the

points of the curve.

The authors J. Herzog, T. Hibi, N. V. Trung and X. Zheng characterize the vertex cover
algebras which are standard graded. In this paper we give a simple
combinatorial criterion for the standard graded property of vertex
cover algebras in the case of quasi-trees. We also give an example
of how this criterion works and compute the maximal degree of a
minimal generator in that case.

Let G be a finite group and f: V -> W an equivariant morphism of finite dimensional G-modules, classically called a "covariant". We say that f is faithful if G acts faithfully on the image f(V). The covariant dimension of G is the minimum of the dimension of f(V) taken over all faithful covariants f. The essential dimension of G is defined in the same way, but allows for rational equivariant morphisms.

The essential dimension and covariant dimension of G are related to cohomological invariants, generic polynomials and other topics, see the work of Buehler-Reichstein [BuR97]. In this paper we investigate covariant dimension and are able to determine it for abelian groups and to obtain estimates for the symmetric and alternating groups. We also classify the groups of covariant dimension less or equal to 2. It turns out that they are the finite subgroups of GL(2,C). A byproduct of our investigations is the existence of a purely transcendental field of definition of degree n-3 for a generic field extension of degree n > 5.

Let R be a ring. The Special Automorphism Group SAut(R)R[x(1),..., x(n)] is the set of all automorphisms with determinant of the Jacobian equal to 1. It is shown that the canonical map of SAut(R[t]) R[t][x(1),...,x(n)] to SAutR(m) R-m[x(1),...,x(n)] where R-m := R[t]/(t(m)) and Q subset of R is surjective. This result is used to study a particular case of the following question: if A is a subring of a ring B and f is an element of A vertical bar n vertical bar is a coordinate over B does it imply that f is a coordinate over A? It is shown that if A = R[t(m), t(m+1),...] subset of R[t] = B then the answer to this question is ``yes``. Also, a question on the Venereau polynomial is settled, which indicates another ``coordinate-like property`` of this polynomial. (c) 2006 Elsevier B.V. All rights reserved.

We describe the conjugacy classes of affine automorphisms in the group

Aut(n,K) (respectively Bir(Kn)) of automorphisms (respectively of birational

maps) of Kn. From this we deduce also the classification of conjugacy classes of

automorphisms of Pn in the Cremona group Bir(Kn).

We give necessary and sufficient conditions on a commutative ring with unity R so that the following holds: any polynomial f in R[x, y] such that R[x, y]/(f) similar or equal to R-[1] (a line for short) is a variable (= alpha(x) for some automorphism alpha:R[x, y] -> R[x, y]). The main ingredient here is the construction of new bad lines (lines which are not variables). (c) 2006 Elsevier Inc. All rights reserved.

We study the geometry of the nullcone N(V^k) for several copies of a representation V of a reductive group G and its behavior for different k. We show that for large k there is a certain "stability" with respect to the irreducible components. In the case of the so-called theta-representations, this can be made more precise by using the combinatorics of the weight system as a subset of the root system. All this finally allows to calculate explicitly and in detail a number of important examples, e.g. the cases of 3- and 4-qubits which play a fundamental role in quantum computing.

The Abhyankar-Sathaye Embedded Hyperplane Problem asks whether any hypersurface of U isomorphic to Cn-1 is rectifiable, i.e., equivalent to a linear hyperplane up to an automorphism of C-n. Generalizing the approach adopted by Kaliman, Venereau, and Zaidenberg, which consists in using almost nothing but the acyclicity of Cn-1, we solve this problem for hypersurfaces given by polynomials of C[x, y, z(1),..., z(k)] as in the title.

We prove that an automorphism of order 5 of the Del Pezzo surface S5 of degree 5 is conjugate through a

birational map S 99K P2 to a linear automorphism of P2. This completes the classification of conjugacy classes of

elements of prime order in the Cremona group.

The Abhyankar-Sathaye Problem asks wether any biregular embedding
of C^k into Cⁿ can be rectified. Here we study
this problem for the embeddings of C³ into C⁴ whose image X is
given in C⁴ by an equation p = f(x; y)u+g(x; y; z) = 0. Under certain additional assumptions we show that, indeed,
the polynomial p is a variable of the polynomial ring C^[4] = C[x; y; z; u] (i.e., a
coordinate of a polynomial automorphism of C⁴). This is an analog of a theorem
due to Sathaye (1976) which concerns the case of embeddings of C² into C³.
Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial
p as above, a criterion for when X is isomorphic to C³.

An algebra A over a field k is FCR if every finite dimensional representation of A is completely reducible and the intersection of the kernels of these representations is zero. We give a useful characterization of FCR-algebras and apply this to C*-algebras and to localizations. Moreover, we show that ``small'' products and sums of FCR-algebras are again FCR.

The pole placement problem belongs to the classical problems of linear systems theory. It is often assumed that the ground field is the real numbers R or the complex numbers C. The major result over the complex numbers derived in 1981 by Brockett and Byrnes states that arbitrary static pole placement is possible for a generic set of m-inputs, p-outputs and McMillan degree n system as soon as mp>=n. Moreover the number of solutions in the situation mp=n is an intersection number first computed by Hermann Schubert in the 19th century. In this paper we show that the same result with slightly different proofs holds over any algebraically closed field.