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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.

If I were a Springer-Verlag Graduate Text in Mathematics, I would be Robin Hartshorne's Algebraic Geometry.

My creator studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. After receiving his Ph.D. from Princeton in 1963, he became a Junior Fellow at Harvard, then taught there for several years. In 1972 he moved to California where he is now Professor at the University of California at Berkeley. My siblings include "Residues and Duality" (1966), "Foundations of Projective Geometry (1968), "Ample Subvarieties of Algebraic Varieties" (1970), and numerous research titles. My creator's current research interest is the geometry of projective varieties and vector bundles. He has been a visiting professor at the College de France and at Kyoto University, where he gave lectures in French and in Japanese, respectively.

My creator is married to Edie Churchill, educator and psychotherapist, and has two sons and one daughter. He has travelled widely, speaks several foreign languages, and is an experienced mountain climber. He is also an accomplished musician, playing flute, piano, and traditional Japanese music on the shakuhachi.

Which Springer GTM would you be? The Springer GTM Test

Intersection Theory on Surfaces

Intersection theory is a cornerstone of modern algebraic geometry, and the case of surfaces is the simplest and most classical case thereof. This paper develops the intersection pairing and intersection multiplicity, following Hartshorne’s “Algebraic Geometry”, and presents some examples.

Blowup Computations

The doubled cone

The blowup of the affine plane at the origin, and self-intersection of the exceptional divisor.

(Taken from my notes on intersection theory)

Intersections on Surfaces

The quadratic and an axis.

(Taken from my notes on intersection theory)

Assorted Rings and Ideals

Polynomial Rings as Graded rings, and some ideals.

  1. The ring k[x,y]k[x, y] is a graded ring, with grading given by total degree. The homogeneous elements are homogeneous polynomials (polynomials where the total degree of each term is the same).
  2. The ideal (x,y)(x, y) is a homogeneous, prime, and maximal ideal. It is also the irrelevent ideal.
  3. The ideal (x,y2)(x, y^2) is homogeneous.
  4. The ideal (x+y2)(x + y^2) is not homogeneous. It cannot be generated by homogeneous elements. The same is true of (x−a,y−b)(x-a, y-b) for any a,ba, b.

Projectivization, Homogenization, and Dehomogenezation

Simple Dehomogenization in a Polynomial Ring

I’ve endeavored to be as general with this example as possible so that it may generalize to the proof of Proposition (Proj is a Scheme). I also wrote this before I wrote those notes, and so the notes might be a little more well-organized. It might be good pedagogy to struggle through this example first because I included some more motivational remarks here, then read those notes, and then come back to this example with a better idea of the structure of the thing.