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.

If V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is called separating if the following holds: If two elements v,v' from V can be separated by an invariant function, then there is an f from S such that f(v) is different from f(v').
It is known that there always exist finite separating sets. Moreover, if the group G is finite, then the invariant functions of degree ≤ |G| form a separating set. We show that for a non-finite linear algebraic group G such an upper bound for the degrees of a separating set does not exist.
If G is finite, we define b(G) to be the minimal number d such that for every G-module V there is a separating set of degree less or equal to d. We show that for a subgroup H of G we have b(H) ≤ b(G) ≤ [G:H] b(H), and that b(G) ≤ b(G/H) b(H) in case H is normal. Moreover, we calculate b(G) for some specific finite groups.

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 show that a linear subspace of a reductive Lie algebra g that consists of nilpotent elements has dimension at most equal to the number of positive roots, and that any nilpotent subspace attaining this upper bound is equal to the nilradical of a Borel subalgebra of g. This generalizes a classical theorem of Gerstenhaber which states this fact for the algebra of n x n matrices.

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.

Let G be a complex reductive group, V a G-module and f a nonconstant homogenous invariant polynomial on V. We investigate relations between the following properties:
- The differential df: V -> V* is dominant;
- The invariant f is homaloidal, i.e., df induces a birational map P(V) -> P(V*);
- V is a stable representation, i.e., the generic G-orbit in V is closed.
If f generates the invariants, we show that the properties are equivalent, generalizing results of Sato-Kimura on prehomogeneous vector spaces.

We consider subrings S of rings R such that R = S oplus V with V either two sided invariant under multiplication by S or invariant under the commutator with S. We show that some important properties of R are inherited by S under such conditions. One is the FCR-property which says that every finite dimensional representation is completely reducible. Another application gives a characterization (in characteristic zero) of reductive subgroups of reductive groups.

Invariant theory was a major subject of research in the 19th century. One of the highlights was Gordan's famous theorem from 1868 showing that the invariants and covariants of binary forms have a finite basis. His method was constructive and led to explicit degree bounds for a system of generators (Jordan 1876/79).

In 1890, Hilbert presented a very general finiteness result using completely different methods such as his famous ``Basissatz''. He was heavily attacked because his proof didn't give any tools to construct a system of generators. In his second paper from 1893 he again introduced new techniques in order to make his approach more constructive. This paper contains the ``Nullstellensatz'', ``Noether's Normalization Lemma'', and the ``Hilbert-Mumford Criterion''!

We shortly overview this development, discuss in detail the degree bounds given by Popov, Wehlau and Hiss and describe some exciting new development relating these bounds with the (geometric) degree of projective varieties and with the Eisenbud-Goto conjecture. The challenge is still the fact that the degree bounds for binary forms given by Jordan are much better than those obtained from the work of Popov and Hiss.

PS. Very recently, Harm Derksen was able to give polynomial bounds for the generators of the invariant ring for any representation of a reductive group.

Complex affine n-space C^n, the basic object of algebraic geometry, offers a number of exciting and striking problems. The most famous one, the Jacobian Conjecture is the still unsolved. Others are the Cancellation Problem (Does Y times C^k simeq C^{n+k} imply that Y simeq C^n?), the Linearization Problem (Is every automorphism of C^n of finite order conjugate to a linear automorphism?), or the Embedding Problem (Are there other embeddings of C^{n-1} into C^n than the standard ones?). It turns out that these questions and several others are intimately related and have very interesting connections with problems arising from algebraic group actions and orbit spaces. We give a survey on these problems and discuss some recent progress and examples.