Quotients in algebraic geometry

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One of the main themes of invariant theory is to relate the G-invariant regular functions, of a regular action $G\times X\to X$, to some suitable quotient morphism $\pi : X\to Y$. If we embrace the conventional approach, we accept the object $Y = Spec(k[X]^G)$, along with the natural map $\pi : X\to Y$, as the inevitable thing to study. If $G$ is reductive and $X$ is affine then $Y$ is ``the variety of closed orbits" and it has the anticipated universal property, even if $Y$ is not the orbit space. Similar results can be obtained if $X$ is a polarized projective variety. Many important moduli spaces have been constructed using this approach. The purpose of this proposal is to consider another approach, where the emphasis is on orbits of maximal dimension rather than on closed orbits. In this scenario we consider sufficiently small open G-subsets U of X such that each $G\times U\to U$ has as many desirable properties as the situation will tolerate. If we then define $U/G$ by the equation $\mathscr{O}(U/G)=\mathscr{O}(U)^G$, then we can ask for the following. (1) $\mathscr{O}(U/G)$ is finitely generated. (2) $\pi : U\to U/G$ has no exceptional divisors. We do this so as to discard only a small portion of X. The main ideas here have their roots in the work of Hilbert, Zariski, Nagata, and Rosenlicht. Our major objective is to assess the influence of quasi-invariant rational functions and $G$-invariant divisors on the problem of identifying a useful quotient object, for the regular action $G\times X\to X$, without relying on semi-invariants. Assume that $G \times X \to X$ is an action where $G$ is affine and $X$ is quasi-affine. If $G$ is reductive and $X$ is affine, then $k[X]^G$ is finitely generated and it parametrizes the closed orbits as a categorical quotient $X\to X/G$ by declaring $k[X/G]=k[X]^G$. If there are not enough closed orbits, or if the group is not reductive, then this approach does not produce an appealing relationship between orbits and invariants. Geometric Invariant Theory finds applications for invariant theory when $G$ is reductive, and in a great many cases, it produces an extremely useful quotient. A. What is ``wrong'' with geometric invariant theory (G.I.T.)? 1. G.I.T. depends on semi-invariants. But what if there aren't enough semi-invariants? How can we glue together a separated quotient, from appealing local information, without semi-invariants? 2. G.I.T. focusses on separating closed orbits, because that is what happens when we insist on using $G$-invariant affine patches and reductive groups. But there is no development here, elaborating on Rosenlicht's Theorem, that is dedicated to using invariant theory as a method of separating orbits of maximal dimension. 3. G.I.T. has deliberate ambiguities built into it. It often produces different answers depending on which line bundle we choose. The quotient we get from G.I.T. is not entirely indigenous. B. What should we do about it? 1. Focus on orbits of maximal dimension, rather than closed orbits. 2. Consider that ``invariant rings should attempt to parametrize orbit closures not closed orbits''. 3. Do not worry whether or not $k[X]^G$ is finitely generated. This issue can always be side-stepped. 4. Further develop the theory of $G$-divisors and quasi-invariant rational functions. This has already been done, to some extent, by the investigator. C. Who really cares? This research program, when completed, will bring about a new chapter relating invariant theory and quotients spaces. Anyone who encounters problems in algebraic geometry, where they need to classify objects generically up to an equivalence relation coming from a linear group action, will be very happy that someone is seriously working on this problem.