Quotients in algebraic geometry

不变量理论的一个主要主题是将正则作用$G\乘以X\到X$的G不变正则函数，与一些合适的商态射$\pi: X\到Y$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$\pi: X\到Y$，作为不可避免的研究对象。如果$G$是约化的，$X$是仿射的，那么$Y$就是“闭轨道的种类”，它具有预期的全称性质，即使$Y$不是轨道空间。如果$X$是一个极化的射影变量，也可以得到类似的结果。使用这种方法构造了许多重要的模空间。本建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。在这种情况下，我们考虑足够小的开放G子集U (X)，使得每个$G\乘以U\到U$具有尽可能多的理想性质。如果我们用等式$\mathscr{O}(U/G)=\mathscr{O}(U)^G$来定义$U/G$，那么我们可以要求如下。(1) $\mathscr{O}(U/G)$是有限生成的。(2) $\pi: U\to U/G$没有例外因数。这里的主要思想来源于希尔伯特、扎里斯基、永田和罗森莱特的工作。我们的主要目标是评估拟不变有理函数和$G$不变因子对识别有用商对象问题的影响，对于正则动作$G\乘以X\到X$，不依赖于半不变量。假设$G \乘以X \到X$是一个作用，其中$G$是仿射的，$X$是准仿射的。如果$G$是约化的，$X$是仿射的，则$k[X]^G$是有限生成的，它通过声明$k[X/G]=k[X]^G$将闭轨道参数化为一个范畴商$X\to X/G$。如果没有足够的闭合轨道，或者如果群不是约化的，那么这种方法就不能产生轨道和不变量之间的吸引人的关系。几何不变理论在$G$是约化的情况下找到了不变理论的应用，并且在很多情况下，它产生了一个非常有用的商。几何不变量理论（G.I.T.）“错”在哪里？it依赖于半不变量。但是如果没有足够的半不变量呢？我们如何在没有半不变量的情况下，从吸引人的局部信息中粘合出一个分离商？G.I.T.专注于分离闭合轨道，因为当我们坚持使用$G$不变仿射补丁和约化群时，就会发生这种情况。但是这里没有发展，没有详细阐述罗森莱特定理，它致力于用不变理论作为分离最大维轨道的方法。G.I.T.故意让人觉得模棱两可。根据我们选择的线束，它通常会产生不同的答案。我们从it得到的商数并不完全是本地的。我们应该怎么做呢？关注最大维度的轨道，而不是闭合轨道。考虑“不变环应该尝试参数化轨道闭包而不是闭合轨道”。3. 不要担心$k[X]^G$是否是有限生成的。这个问题总是可以回避的。进一步发展了G -因子和拟不变有理函数的理论。在某种程度上，调查者已经这样做了。谁会在乎呢？本研究项目的完成，将为不变量理论和商空间的研究开辟新的篇章。任何在代数几何中遇到问题的人，如果他们需要根据线性群作用的等价关系对物体进行一般分类，他们会很高兴有人在认真地研究这个问题。

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.