Quotients in algebraic geometry

不变量理论的一个主要主题是将正则作用$G×X\to X$的G-不变正则函数与适当的商态射$\pi：X\to Y$联系起来。如果我们采用传统的方法，我们接受对象$Y=Spec(k[X]^G)$，以及自然映射$\pi：x\to Y$，作为研究的必然。如果$G$是约化的，而$X$是仿射的，则$Y$是“闭轨的簇”，并且它具有预期的普适性质，即使$Y$不是轨道空间.如果$X$是极化射影簇，也可以得到类似的结果.用这种方法构造了许多重要的模空间. 这项提议的目的是考虑另一种方法，其中重点是最大维度的轨道，而不是封闭轨道。在这个场景中，我们考虑X的足够小的开G-子集U，使得每个$G\乘以U\to U$具有情况所允许的尽可能多的期望性质。如果我们然后通过等式$\mathscr{O}(U/G)=\mathscr{O}(U)^G$定义$U/G$，那么我们可以要求如下。(1)$\mathscr{O}(U/G)$有限生成。(2)$\pi：U\to U/G$没有特殊因子。我们这样做是为了只丢弃X的一小部分。 这里的主要思想植根于希尔伯特、扎里斯基、长田和罗森利赫特的作品。我们的主要目标是在不依赖半不变量的情况下，评估拟不变有理函数和$G$不变因子对识别正则作用为$G×X\to X$的有用商对象的影响。 假设$G乘以X到X$是一个作用，其中$G$是仿射的，而$X$是拟仿射的。如果$G$是约化的，且$X$是仿射的，则$k[X]^G$是有限生成的，并且它通过声明$k[X/G]=k[X]^G$将闭轨参数化为范畴商$X\到X/G$。如果没有足够的闭合轨道，或者如果群不是约化的，那么这种方法不会在轨道和不变量之间产生吸引人的关系。当$G$是约化时，几何不变理论得到了应用，并且在许多情况下，它产生了一个非常有用的商。 答：几何不变量理论(G.I.T.)有什么“错误”？ 1.G.I.T.依赖于半不变量。但是，如果没有足够的半不变量怎么办？如果没有半不变量，我们如何才能从吸引人的局部信息中粘合在一起呢？ 2.G.I.T.专注于分离封闭轨道，因为当我们坚持使用$G$-不变仿射面片和约化群时，就会发生这种情况。但在罗森利希特定理的详细阐述中，没有任何进展，罗森利希特定理致力于使用不变量理论作为分离最大维度轨道的方法。 3.GIT有刻意的模棱两可。根据我们选择的线路捆绑，它通常会产生不同的答案。我们从GIT获得的商数并不完全是本土的。 我们应该怎么做呢？ 1.关注最大维度的轨道，而不是封闭的轨道。 2.考虑“不变环应试图将轨道闭合而不是闭合轨道参数化”。 3.不用担心$k[X]^G$是否有限生成。这个问题总是可以回避的。 4.进一步发展了$G$-因子和拟不变有理函数理论。在某种程度上，调查员已经做到了这一点。 C.谁真的在乎呢？ 这一研究计划完成后，将带来与不变理论和商空间相关的新篇章。任何在代数几何中遇到问题的人，都会非常高兴有人在认真地研究这个问题，因为他们需要从一般意义上对对象进行分类，直到来自线性群作用的等价关系。

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.