Algebraic Geometry Examples

This is a collection of examples in Algebraic Geometry I find noteworthy or in some other way interesting. I will try to make them more or less user-friendly, and to include references to my notes on AG whenever possible, but I offer you no promises.

• Assorted Singular Plane Curves

  The Union of the Axiis is Singular

  Recall that, as a scheme, we can write the union of the axiis as $X := \spec\left[k[x, y]/(xy)\right]$. Consider the local ring $\mc O_{(x, y), X}$, a local ring with maximal ideal $(\bar x, \bar y)$. Consider $(\bar x, \bar y)/(\bar x, \bar y)^2$. Note that $(\bar x, \b y)^2$ is generated by $\b x^2, \b x\b y,$ and $\b y^2$, but $\b x\b y=0$, so $(\b x, \b y)^2 = (\b x^2, \b y^2)$. Then $\dim_k (\b x, \b y)/(\b x^2, \b y^2) = 2$. Suppose $\dim \mc O_{(x, y), X}\neq 1$. We have a nontrivial prime ideal, so the dimension must be positive; thus there must be some tower of prime ideals $0\subsetneq P \subsetneq Q \subsetneq (x, y)$. Elements of $(x, y)$ are of the form $ax + by$ for $a\in k[x]$ and $b\in k[y]$ (if $a$ had any $y$ factor it would vanish, and conversely for $b$). Consider an ideal which contains $ax+by$ for both $a$ and $b$ nonzero; quotienting by such an ideal would yield $\b{x}\b{ax} = \b{x} \b{-by} = 0$ as $xy=0$. Thus such an ideal cannot be prime. No prime ideal can thus contain $ax+by$ for $a, b$ nonzero; moreover, if an ideal contains $ax$ and $by$ it must contain $ax+by$ by closure, so each ideal must contain only multiples of $x$ or multiples of $y$. Moreover, it is clear that $a, b\in k$ for the ideal to be prime, and so any prime ideal in $(x, y)$ is of the form $(x)$ or $(y)$. These ideals don’t contain each other, and so we have found the dimension of $\mc O_{(x, y), X}$ is 1.
• 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.
• Blowup Computations

  I did my first blowup computations this semester, and found it tremendously satisfying. Those were entirely blowups of affine plane curves at the origin, using what I’ll call the “classical” blowing up procedure (take the subset of $\mb A^2 \times \mb P^1$ with coordinates $(x, y, [z:w])$ cut out by $xw=zy$, and look at where the curve pulls back to under the projection map to $\mb A^2$). I might write up those examples eventually, but this page is mostly dedicated to blowing up a sheaf of ideals.
• Intersections on Surfaces

  The quadratic and an axis.

  (Taken from my notes on intersection theory)
• Toric Geometry Examples

  In the words of Prof. Melody Chan, toric geometry is interesting because it yields examples entirely controlled by combinatorics; we can associate toric varieties to polyhedral fans by gluing together the prime spectra of polynomial rings given by adjoing semigroups associated to those fans to a ground field (usually $\mb C$). Rarely is such robust gluing data given in such a conscise and accessable manner.

  • 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 $\hull(\{v_i\})$ to denote the convex hull of the $v_i$, and $\cone(\{v_i\})$ to denote the set $\{a_1v_1+a_2v_2+\cdots\ \text{s.t.}\ a_i\in \mb R_{\geq 0}\}$. If $\sigma$ is a cone, $\sigma^\nu = \{v\in\mathbb R^n \text{ s.t. } v\cdot w\geq 0\ \ \forall w\in \sigma\}$ is the dual of $\sigma$.
• "Fibrations"

  These are various examples or counterexamples to the slogan “total space = base $\times$ fiber”.