CAREER:Combinatorial Intersection Theory on Moduli Spaces of Curves

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。 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，但在更深的层次上，这些对象在无数数学领域以及弦理论的理论物理中起着至关重要的作用。 尽管数学家们研究曲线已有几个世纪的历史，但随着模空间的出现，在20世纪后期出现了一个突破。 模空间，粗略地说，是给定类型的所有曲线的集合，这是一个突破性的认识，人们通常可以更有效地理解曲线，通过考虑他们在这样的家庭，而不是研究他们单独。 在这个项目中，PI将进行几个子项目，从理论和计算的角度来推进对曲线模空间的理解，她将开始研究模空间的一个新变体，阐明曲线几何，多面体组合学和置换代数之间的联系。 除了他们的智力价值，这些项目将提供学生参与的许多途径：通过该PI将招募和指导她的家乡机构旧金山弗朗西斯科州立大学（SFSU）的本科生到硕士水平的本科生研究计划;代数几何教科书的编写，面向准备在PI的领域研究经验不足的硕士生;和研究项目以及社区建设，通过这些项目，PI将指导硕士水平的研究人员过渡到博士学位。 由于SFSU为高度多样化的本科生群体提供服务，这些加强学士到博士课程的步骤为扩大数学界的参与和促进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Deligne-Mumford模空间的交理论。 虽然这个模空间的Chow环一般来说是笨拙的，但有一个子环被称为重言式环，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。 继续一个长期的研究计划，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。 第二条研究路线追求由PI和她的合作者构造的一个新的模空间族，它是具有循环作用的参数生成零曲线。 这些空间的动机来自于这样一个事实，即在亏格为零的情况下，曲线的模空间的Chow环承认有趣的相似之处，以更简单的设置环面品种，但从双有理几何的角度来看，它偏离环面的情况比原来预期的。 也许这个故事中最著名的部分是富尔顿的F猜想，一个关于模空间的森锥的陈述，至今仍未解决。 PI和她的合作者引入的新模空间不是环面的，但他们的交集理论推广了环面变种，因为它是由多面体复形编码的。 这些空间的进一步调查将揭示多面体组合方法的复曲面品种以外的域的适用性，最雄心勃勃的，可能会给一个设置中的F-猜想的模拟可以证明。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.