Quotients in algebraic geometry

代数几何中的商

• 批准号: RGPIN-2014-06117
• 负责人: Renner, Lex
• 金额: $ 1.31万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635466
• 关键词: Quotients; algebraic; geometry

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.

不变理论的主题之一是将规则动作 $G\times X\to X$ 的 G 不变规则函数与某些合适的商态射 $\pi : X\to Y$ 联系起来。如果我们采用传统方法，我们就会接受对象 $Y = Spec(k[X]^G)$ 以及自然映射 $\pi : X\to Y$，作为不可避免的研究对象。如果 $G$ 是还原性的，而 $X$ 是仿射性的，那么 $Y$ 就是“闭轨道簇”，即使 $Y$ 不是轨道空间，它也具有预期的普适性。如果 $X$ 是极化射影簇，也可以得到类似的结果。许多重要的模空间已经用这种方法构造出来了。这个建议的目的是考虑另一种方法，其中重点是最大维数轨道而不是闭轨道 轨道。在这种情况下，我们考虑 X 的足够小的开放 G 子集 U，使得每个 $G\times U\to U$ 具有情况所能容忍的尽可能多的所需属性。如果我们通过方程 $\mathscr{O}(U/G)=\mathscr{O}(U)^G$ 定义 $U/G$，那么我们可以要求以下结果。 (1) $\mathscr{O}(U/G)$ 是有限生成的。 (2) $\pi : U\to U/G$ 没有例外除数。我们这样做是为了仅丢弃 X 的一小部分。这里的主要思想源于 Hilbert、Zariski、Nagata 和 Rosenlicht 的工作。我们的主要目标是评估准不变有理函数的影响和 关于识别有用商对象问题的 $G$ 不变除数，对于常规动作 $G\times X\to X$，不依赖于半不变量。假设 $G \times X \to X$ 是一个动作，其中 $G$ 是仿射的，$X$ 是拟仿射的。如果 $G$ 是还原性的，而 $X$ 是仿射性的，那么 $k[X]^G$ 是有限生成的，并且 通过声明 $k[X/G]=k[X]^G$ 将闭轨道参数化为分类商 $X\to X/G$。如果没有足够的闭合轨道，或者群不是还原性的，那么这种方法就不会在轨道和不变量之间产生有吸引力的关系。当 $G$ 被还原时，几何不变量理论找到了不变量理论的应用，并且在很多情况下，它 产生一个非常有用的商。 A. 几何不变量理论 (G.I.T.) 有什么“错误”？ 1．胃肠道取决于半不变量。但是如果没有足够的半不变量怎么办？我们怎样才能在没有半不变量的情况下，从吸引人的局部信息中将分离的商粘合在一起？2。胃肠道专注于分离闭合轨道，因为这就是 当我们坚持使用 $G$ 不变仿射补丁和还原群时就会发生。但这里没有详细阐述罗森利希特定理，该定理致力于使用不变理论作为分离最大维度轨道的方法。3。胃肠道故意含糊其辞。根据我们选择的线束，它通常会产生不同的答案。我们得到的商 胃肠道并不完全是本土的。B.对此我们该怎么办？ 1．关注最大维度的轨道，而不是封闭的轨道。2．考虑“不变环应该尝试参数化轨道闭合而不是闭合轨道”。 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。4。理论的进一步发展 $G$-除数和准不变有理函数。在某种程度上，研究者已经做到了这一点。C.谁真正在乎呢？这个研究计划完成后，将带来有关不变理论和商空间的新章节。任何在代数几何中遇到问题的人，他们需要对对象进行一般性分类，直到来自线性群作用的等价关系，都会很高兴有人 认真地致力于解决这个问题。