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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 Hull({vi})\hull(\{v_i\}) to denote the convex hull of the viv_i, and Cone({vi})\cone(\{v_i\}) to denote the set {a1v1+a2v2+ s.t. aiR0}\{a_1v_1+a_2v_2+\cdots\ \text{s.t.}\ a_i\in \mb R_{\geq 0}\}. If σ\sigma is a cone, σν={vRn s.t. vw0  wσ}\sigma^\nu = \{v\in\mathbb R^n \text{ s.t. } v\cdot w\geq 0\ \ \forall w\in \sigma\} is the dual of σ\sigma.

Blowups and Fiber Products

This past week, I’ve been working on computing blowups. Classically, the blowup of An\mb A^n is the algebraic subset of An×Pn1\mb A^n\times \mb P^{n-1} given by {(a1,...,an,[p1,...,pn])aipi=ajpj}\{(a_1, ..., a_n, [p_1, ..., p_n]) | a_ip_i=a_jp_j\}; now, we blowup along a sheaf of ideals I\ms I by forming the sheaf of graded rings B=OXII2\ms B = \mc O_X\oplus \ms I\oplus \ms I^2\oplus \cdots. In the affine case this is called the blowup algebra. We then take relative Proj\proj 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 x2y2z2x^2-y^2-z^2 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 (f1,...)(f_1, ...) to denote the submodule generated by f1,...f_1,....