I’m Skyler, a Junior at Boston University. I do mathematics, mostly algebraic geometry and commutative algebra. I’m currently particularly interested in computing specific examples with schemes. In the near future I hope to solidify further my foundatons in intersection theory and birational geometry before moving studying moduli theory and derived algebraic geometry with a focus on singularities and intersection theory. I am a fellow at MIT’s Summer Geometry Initiative this summer, where I hope to explore the possible applications of algebraic geometry to computer vision. This is a website for various projects of mine, including some thoughts about math (and other topics) that I have, and have realized people don’t always want to hear about. I’ve also put links to various expository papers I’ve written here. I run the Society of Mathematics at BU.
Toric Surfaces from Fans
Here are examples of 2-dimensional toric varieties with all the affine spectra worked out and the gluing data specified explicitly to the degree necessary to work out e.g. Čech cohomology by hand. I’ve used sage at times, and will include the code I’ve utilized. I’ll also include some visualizations of the cones when possible. I use to denote the convex hull of the , and to denote the set . If is a cone, is the dual of .
Blowups and Fiber Products
This past week, I’ve been working on computing blowups. Classically, the blowup of is the algebraic subset of given by ; now, we blowup along a sheaf of ideals by forming the sheaf of graded rings . In the affine case this is called the blowup algebra. We then take relative of this sheaf of graded rings, which yields the blowup. In particular, this week, I was trying to compute the blowup of the affine cone at the origin (to resolve that singularity). I found this shockingly difficult, mostly due to my inability to work with the graded ring and the proj construction.
Problems
Here are some problems designed to get you started thinking in terms of first order logic.
Properties of the Tensor Product
Here are various results about the tensor product, one of my favorite mathematical constructions.
Chapter 1: Roots of Commutative Algebra
Exercises
A note: I use to denote the submodule generated by .