CAREER:Combinatorial Intersection Theory on Moduli Spaces of Curves

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。曲线是数学中一些最基本的几何对象；最简单的例子包括抛物线和圆等熟悉的形状，但在更深层次上，这些对象在数学的无数领域以及弦理论的理论物理中扮演着至关重要的角色。虽然数学家对曲线的研究已经有几个世纪了，但在二十世纪末，随着模空间的出现，出现了一个突破。粗略地说，模空间是给定类型的所有曲线的集合，这是一个开创性的认识，人们通常可以通过在这样的族中考虑曲线而不是单独研究它们来更有效地理解曲线。在这个项目中，PI将承担几个子项目，从理论和计算的角度促进对曲线模空间的理解，她将启动对模空间的新变体的研究，该变量阐明了曲线几何、多面体组合学和置换代数之间的联系。除了智力上的优势，这些项目还将为学生参与提供多种途径：本科生研究项目，通过该项目，PI将在她所在的旧金山州立大学(SFSU)招收和指导本科生达到硕士水平；编写一本代数几何教科书，旨在为缺乏经验的硕士学生在PI领域的研究做准备；以及研究项目和社区建设，PI将通过该项目指导硕士水平的研究人员过渡到博士学位。由于旧金山州立大学服务的本科生群体非常多样化，这些加强从学士到博士的渠道的步骤为扩大参与和促进数学界的包容性提供了一个独特的机会。从技术上讲，这个项目专注于两个独立但相互关联的研究领域。第一部分研究了曲线的Deligne-Mumford模空间的交理论。虽然这个模空间的Chow环总体上是笨拙的，但有一个被称为重言式环的子环，它承载了模空间的大部分几何内容，同时允许一组显式的加法生成元。继续一项长期的研究计划，PI将调查这些生成器之间的关系，长期目标是使用它们来确定超椭圆轨迹的Chow类的公式。第二个研究方向是由PI和她的合作者构造的一族新的模空间，它用循环作用来参数化亏零曲线。产生这些空间的动机是，在亏格为零的情况下，曲线模空间的Chow环承认与更简单的环簇设置有有趣的相似之处，但从二元几何的角度来看，它与环簇的偏离比最初预期的要大得多。也许这个故事最著名的部分是富尔顿的F猜想，这是一个关于模空间的Mori锥的陈述，至今仍未解决。PI和她的合作者引入的新的模空间不是环面的，但他们的交集理论推广了环面簇的理论，因为它是由一个多面体复合体编码的。对这些空间的进一步研究将阐明多面体组合方法在环面变数领域之外的适用性，最雄心勃勃的是，可能会给出一个可以证明F猜想类似的环境。这个奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Curves are some of the most fundamental geometric objects in mathematics; the simplest examples include such familiar shapes as the parabola and the circle, but on a deeper level, these objects play a crucial role in myriad fields of mathematics as well as the theoretical physics of string theory. Although mathematicians have studied curves for centuries, a breakthrough occurred in the late twentieth century with the advent of moduli spaces. A moduli space, roughly speaking, is the collection of all curves of a given type, and it was a groundbreaking realization that one can often more effectively understand curves by considering them in such families rather than studying them individually. In this project, the PI will undertake several sub-projects that will advance understanding of the moduli space of curves from both a theoretical and a computational standpoint, and she will initiate the study of a new variant of the moduli space that illuminates a connection between the geometry of curves, the combinatorics of polytopes, and the algebra of permutations. Alongside their intellectual merit, these projects will provide numerous avenues for student engagement: an undergraduate research program through which the PI will recruit and mentor undergraduates at her home institution of San Francisco State University (SFSU) to the Master’s level; the authoring of an algebraic geometry textbook geared toward preparing less-experienced Master’s students for research in the PI’s field; and research projects as well as community-building via which the PI will mentor Master’s-level researchers through the transition to a PhD. Because SFSU serves a highly diverse undergraduate student body, these steps toward strengthening the pipeline from Bachelor’s to PhD present a unique opportunity for broadening participation and promoting inclusivity in the mathematics community.More technically speaking, this project is focused on two separate but interrelated lines of research. The first involves studying the intersection theory of the Deligne-Mumford moduli space of curves. Although the Chow ring of this moduli space is unwieldy in general, there is a subring known as the tautological ring that carries much of the moduli space’s geometric content while admitting an explicit set of additive generators. Continuing a longstanding research program, the PI will investigate the relations among these generators, with the long-term goal of using them to determine a formula for the Chow class of the hyperelliptic locus. The second line of research pursues a new family of moduli spaces constructed by the PI and her collaborators, which parameterize genus-zero curves with cyclic action. The motivation for these spaces arises from the fact that, in the genus-zero case, the Chow ring of the moduli space of curves admits intriguing parallels to the simpler setting of toric varieties and yet, from a birational geometry perspective, it diverges from the toric case more than was originally expected. Perhaps the most famous part of this story is Fulton’s F-conjecture, a statement about the Mori cone of the moduli space that remains unsolved. The new moduli spaces introduced by the PI and her collaborators are not toric, yet their intersection theory generalizes that of toric varieties in that it is encoded by a polytopal complex. Further investigation of these spaces will shed light on the applicability of polyhedral combinatorial methods outside the domain of toric varieties, and most ambitiously, may give a setting in which the analogue of the F-conjecture can be proven.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.