Bringing Frobenius to Bear on Birational Algebraic Geometry

本研究计划提出了交换代数中代数几何与特征p技术相结合的几个问题。给定一个变量，简单地说，是由具有整数系数的多项式定义的，我们可以为每个素数整数p“减少模p”，并在具有不同特征的有限域上得到一个变量族。该计划的总体目标是了解在这些主要特征模型中具有“纯代数”问题的复杂品种的奇点或积分的解决所定义的现象之间的关系。例如，继续她的博士后Karl Schwede开始的一个项目，PI提议证明复杂的Log Fano品种将mod p降低到全局f规则品种。她还提出了对对数正则奇点将mod p简化为F-纯奇点这一猜想的一种可能的攻击，这涉及到对超曲面的“F-阈值”的直接计算，这是对数正则阈值的一个素数特征模拟。她的博士生丹尼尔·埃尔南德斯在这方面取得了很好的进展。代数几何是研究由多项式方程定义的几何对象的学科。正如直线可以用y = 3 x + 1这样的方程来描述，圆可以用x^2 + y^2 = 4这样的方程来描述，也可以用多项式来描述许多更复杂的几何物体，例如，在高维空间中。这个项目试图通过观察定义这些复杂物体的方程的代数特征来理解它们的几何形状。例如，我们可以在方程y = m x + b（斜率是x的系数——数字m）中看到直线的几何性质（比如它的斜率，或者“它上升的速度有多快”），或者在方程的代数中看到圆x^2 + y^2 = 4的几何性质（比如半径）（常数项的平方根，或者2，是半径）。这个项目提出的正是：通过仔细观察多项式本身来研究由多项式定义的更复杂物体的几何形状。有几个项目是考虑到年轻数学家的指导而提出的。

The research program proposes several problems at the interface of algebraic geometry and characteristic p techniques in commutative algebra. Given a variety, for simplicity say defined by polynomials with integer coefficients, we can ``reduce mod p" for each prime integer p, and arrive at a family of varieties over finite fields of varying characteristic. The overall goal of this program is to understand the relationships between phenomena defined by resolution of singularities or integration for complex varieties with "purely algebraic" issues in these prime characteristic models. For example, continuing with a project started with her post-doc Karl Schwede, the PI proposes to prove that complex Log Fano varieties reduce mod p to globally F-regular varieties. She also proposes a possible attack on the conjecture that log canonical singularities reduce mod p to F- pure singularities, which involves direct computation of the "F- threshold", a prime characteristic analog of the log canonical threshold, for hypersurfaces. Her PhD student Daniel Hernandez is making excellent progress on this.Algebraic Geometry is the study of geometric objects which are defined by polynomial equations. Just as lines are described by equations like y = 3 x + 1 or circles by equations like x^2 + y^2 = 4, it is possible to describe many more complicated geometric objects, for example, in higher dimensional spaces, with polynomials. This project attempts to understand the geometry of these complicated objects by looking at the algebraic features of the equations that define them. For example, we can see geometric properties of the line (such as its slope, or "how fast it rises") in the equation y = m x + b (the slope is the coefficient of x--the number m), or geometric properties (such as the radius) of the circle x^2 + y^2 = 4 in the algebra of its equation (the square root of the constant term, or 2, is the radius). This project proposes exactly that: study the geometry of much more complicated objects defined by polynomials by looking carefully at the polynomials themselves. There are several projects proposed with the mentorship of young mathematicians in mind.