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Wall-crossing on universal compactified Jacobians

通用压缩雅可比行列式的跨墙

基本信息

批准号：
EP/P004881/1
负责人：
Nicola Pagani
金额：
$ 12.88万
依托单位：
University of Liverpool
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FP004881%2F1
关键词：
Wall crossing universal compactified Jacobians

项目摘要

Enumerative geometry is one of the most ancient fields of mathematics, and it aims at counting the number of geometric objects having a certain property. For example, we may ask how many straight lines pass through two given points in the plane. It is Euclid's very first axiom that asserts that there is a unique such line. Another example is to count how many points belong simultaneously to two lines in the plane. Here Euclid's fifth axiom essentially implies that the answer is one if and only if the lines are not parallel. For a slightly more interesting example, one could consider a parabola and a circle in the plane, and see that the number of points belonging to both could be any number between 0 and 4 (depending on the relative position of the line and the circle.The examples above hopefully demonstrate how such questions can be basic and pervasive in geometry, and they give a glimpse onto geometry's early historical developments. Today the field is still existing and very active, and it employs techniques coming from different fields of mathematics. In the last 25 years, revolutionary ideas in the field have arrived from physics, in particular from theories originating from the quest of unifying the four fundamental forces, like string theory.The main modern approach to counting theories today uses moduli spaces. How many quadrics pass through 5 general points in the plane? A possible approach is to consider the 5-dimensional (projective) space that parametrizes plane quadrics, and to realize that the constraint of passing through a point corresponds to cutting a hyperplane in such space. By intersecting the 5 hyperplanes, we find out that the answer to the counting question is 1. The proposed research follows this paradigm to approach some questions in algebraic geometry. The moduli spaces studied in this proposal are moduli of line bundles of some fixed degree over projective algebraic curves, and the constraints are given (for example) by imposing that such line bundles have a given number of linearly independent global sections (Brill-Noether loci). In order to construct such moduli spaces one has to introduce an "extra" parameter, not a-priori imposed by the problem of parameterizing the aforementioned geometric objects, called stability. This parameter is a continuous parameter, but the moduli space actually varies only when the parameter crosses some hyperplanes (called walls) in the space where it lives. Our point of view is that the geometric picture should simplify when one considers all stability parameters, rather than only one. For example, there is usually one "easy" parameter, for which the given constraints and their geometric nature can be easily understood and there is one "interesting" parameter that has received lots of attention from several mathematicians. The novelty of our approach consists in finding results for the moduli space corresponding to the "interesting" parameter by first solving the same problem for the "easy" parameter, and then investigating how the moduli spaces vary with the stability parameter when a wall is crossed. The different moduli spaces should be related to each other by flips (and going into a wall should correspond to a contraction).

计数几何是最古老的数学领域之一，其目的是对具有一定性质的几何物体的数量进行计数。例如，我们可以问平面上有多少条直线经过两个给定的点。这是欧几里得的第一个公理，它断言存在唯一的这样一条线。另一个例子是计算有多少个点同时属于平面上的两条直线。这里欧几里得的第五公理本质上意味着，当且仅当两条线不平行时，答案是1。举一个稍微有趣一点的例子，我们可以考虑平面上的一条抛物线和一个圆，并看到属于两者的点的数量可以是0到4之间的任何数字（取决于线和圆的相对位置）。上面的例子希望展示这些问题在几何中是如何基本和普遍的，它们让我们瞥见了几何的早期历史发展。今天，这个领域仍然存在并且非常活跃，它采用了来自不同数学领域的技术。在过去的25年里，该领域的革命性思想来自物理学，特别是来自于寻求统一四种基本力的理论，如弦理论。当今主要的现代计数理论方法是使用模空间。平面上有多少次二次曲线经过5个一般点？一种可能的方法是考虑参数化平面二次曲面的5维（射影）空间，并认识到通过点的约束对应于在该空间中切割超平面。通过5个超平面相交，我们发现计数问题的答案是1。本研究遵循这一范式来探讨代数几何中的一些问题。本文所研究的模空间是射影代数曲线上若干固定度的线束的模，并且约束条件（例如）通过施加这样的线束具有给定数量的线性无关的全局截面（Brill-Noether轨迹）来给出。为了构造这样的模空间，必须引入一个“额外的”参数，而不是由上述几何对象的参数化问题先验地强加的，称为稳定性。这个参数是一个连续参数，但是模空间实际上只有当参数穿过它所在空间中的一些超平面（称为壁）时才会变化。我们的观点是，当考虑所有的稳定性参数时，几何图形应该简化，而不是只有一个。例如，通常有一个“简单”的参数，对于给定的约束和它们的几何性质可以很容易地理解，还有一个“有趣”的参数，已经受到了许多数学家的关注。我们的方法的新颖之处在于，通过首先解决“容易”参数的相同问题，找到与“有趣”参数对应的模空间的结果，然后研究当穿过墙壁时模空间如何随稳定性参数变化。不同的模空间应该通过翻转相互关联（并且进入墙壁应该对应于收缩）。