Bringing Frobenius to Bear on Birational Algebraic Geometry

该研究计划提出了几个问题的接口代数几何和特征p技术在交换代数。给定一个变量，为了简单起见，假设由整数系数的多项式定义，我们可以 “reduce mod p”对于每一个素数p，并在具有不同特征的有限域上得到一族簇。该计划的总体目标是了解通过解决这些主要特征模型中的“纯代数”问题的复杂品种的奇点或积分定义的现象之间的关系。例如，继续与她的博士后Karl Schwede开始的一个项目，PI建议证明复杂的Log Fano品种将mod p减少到全局F-正则品种。 她还提出了一个可能的攻击的猜想，对数正则奇点减少模p到F-纯奇点，这涉及到直接计算的“F-阈值”，一个主要的特征模拟的对数正则阈值，超曲面。她的博士生丹尼尔埃尔南德斯在这方面取得了很大的进展。代数几何是研究由多项式方程定义的几何对象。就像直线可以用方程y = 3 x + 1来描述一样，圆也可以用方程x^2 + y^2 = 4，可以描述许多更复杂的几何对象，例如，在高维空间中，用多项式。 这个项目试图通过观察定义它们的方程的代数特征来理解这些复杂物体的几何形状。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（如它的斜率，或“它上升的速度”）， 或几何性质（如半径）的圆x^2 + y^2 = 4在其方程的代数（常数项的平方根，或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青年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.