The Geometry of Curves in Projective Space via Degeneration and Deformation

多项式方程系统在数学以及相关领域如物理学、密码学和其他科学中无处不在。当这样的系统描述一维对象时，相应的几何对象被称为代数曲线。从广义上讲，这个项目研究代数曲线的几何，无论是单独的还是在家族中变化的。这个项目研究的一个具体问题是插值的概念：什么时候某种类型的曲线可以通过一般的点集合？换句话说，我们可以通过平面上的任意两点画一条直线，也可以通过平面上的任意三点画一个圆，除非这三点在一条直线上。更一般地说，PI将考虑高维空间中其他类型曲线的情况，再次假设这些点不处于任何特殊构型中。该项目还包括对本科生的培训。该项目考虑了研究一般曲线方程结构的各种问题。例如，它们是在什么程度上产生的？在这样的曲线上一般因子的贝蒂表是什么？PI还将研究控制其变形理论的曲线上的某些自然向量束——受限切线束和法向束——并研究它们如何分解成稳定的向量束，这是所有向量束的原子构建块。最后，PI将研究曲线模空间的几何，重点是它们的积分相交理论。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。

Systems of polynomial equations are ubiquitous in mathematics, and related fields such as physics, cryptography, and other sciences. When such a system describes a one-dimensional object, the corresponding geometric object is known as an algebraic curve. Broadly speaking, this project studies the geometry of algebraic curves, both individually and as they vary in families. One specific question this project investigates is the concept of interpolation: When can a certain type of curve be passed through a general collection of points? In other words, one can draw a line through any two points in the plane and draw a circle through any three points in the plane, unless those three points lie on a line. More generally, the PI will consider what happens for other types of curves in higher-dimensional spaces, assuming again that the points do not lie in any special configurations. This project also includes training for undergraduate students. The project considers various questions for the investigation of the structure of the equations of general curves. For example, in what degrees are they generated? What is the Betti table of a general divisor on such a curve? The PI will also investigate certain natural vector bundles on curves that control their deformation theory --- the restricted tangent bundle and the normal bundle --- and study how they break down into stable vector bundles, which are the atomic building blocks of all vector bundles. Finally, the PI will study the geometry of moduli spaces of curves, focusing on their integral intersection theory.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.