Bringing Frobenius to Bear on Birational Algebraic Geometry

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

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.