CAREER:Combinatorial Intersection Theory on Moduli Spaces of Curves

职业：曲线模空间的组合交集理论

• 批准号: 2137060
• 负责人: Emily Clader
• 金额: $ 50.57万
• 依托单位: San Francisco State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2137060&HistoricalAwards=false
• 关键词: CAREER; Combinatorial; Intersection; Theory; Moduli

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.

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。曲线是数学中最基本的几何对象之一；最简单的例子包括像抛物线和圆这样熟悉的形状，但在更深层次上，这些物体在无数数学领域以及弦理论物理中发挥着至关重要的作用。尽管数学家们研究曲线已经有几个世纪了，但随着模空间的出现，在20世纪后期出现了突破。模数空间，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，通过考虑它们在这样的家族，而不是单独研究它们。在这个项目中，PI将承担几个子项目，这些子项目将从理论和计算的角度推进对曲线模空间的理解，并且她将启动模空间的一个新变体的研究，该变体将阐明曲线几何、多体组合学和置换代数之间的联系。除了他们的智力价值，这些项目将为学生参与提供许多途径：一个本科研究项目，通过该项目，PI将在她的母校旧金山州立大学（SFSU）招募和指导本科生攻读硕士学位；编写代数几何教材，为经验不足的硕硕生在PI领域的研究做准备；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。由于旧金山州立大学的本科学生群体高度多样化，这些加强从学士到博士的管道的步骤为扩大数学社区的参与和促进包容性提供了一个独特的机会。从技术上讲，该项目侧重于两个独立但相互关联的研究方向。首先研究曲线的delign - 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，但有一个被称为重言环的子环，它承载了模空间的大部分几何内容，同时允许一个显式的加性生成集。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，长期目标是利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和她的合作者构建的一个新的模空间族，它参数化了具有循环作用的属零曲线。这些空间的动机源于这样一个事实，即在属零情况下，曲线模空间的周氏环承认与环面变体的简单设置有有趣的相似之处，然而，从两族几何的角度来看，它与环面情况的分歧比最初预期的要大。也许这个故事中最著名的部分是Fulton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，但他们的交理论推广了环面变异的理论，因为它是由一个多面复合体编码的。对这些空间的进一步研究将揭示多面体组合方法在环变域之外的适用性，并且最雄心勃勃的是，可能给出一个可以证明f猜想的类似的设置。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Topological Recursion Relations from Pixton’s Formula

• DOI: 10.1307/mmj/20195795
• 发表时间: 2017-04
• 期刊: Michigan Mathematical Journal
• 影响因子: 0.9
• 作者: E. Clader;F. Janda;Xin Wang;D. Zakharov

Permutohedral complexes and rational curves with cyclic action

具有循环作用的全面体复形和有理曲线

• DOI: 10.1007/s00229-022-01419-6
• 发表时间: 2022
• 期刊: manuscripta mathematica
• 影响因子: 0.6
• 作者: Clader, Emily;Damiolini, Chiara;Huang, Daoji;Li, Shiyue;Ramadas, Rohini
• 通讯作者: Ramadas, Rohini

Wonderful compactifications and rational curves with cyclic action

美妙的紧凑化和具有循环作用的理性曲线

• DOI: 10.1017/fms.2023.26
• 发表时间: 2023
• 期刊: Sigma
• 作者: Clader, Emily;Damiolini, Chiara;Li, Shiyue;Ramadas, Rohini
• 通讯作者: Ramadas, Rohini