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Some Recent Posts:

Computations of Cannonical Divisors and Sheaves

There is an intrinsic classification of (smooth) curves by cannonical divisor. In particular, if the anticannonical class is ample, the curve is $\mathbb P^1$; if neither the anticannonical class nor the cannonical class is ample the curve is an elliptic curve; and if the cannonical class is ample the curve is “of general type”. Note that these conditions correspond to the genus of the curve being $0$, $1$, and $>1$, respectively; furthermore, they are equal to the possible values for the Kodaira dimension. This classification for curves is in direct analogy with the classification of surfaces, which I hope to write my thesis on. Here we compute some examples of the cannonical divisor to get a feel for it.

Hilbert Polynomials and Schemes

Here I collect a set of computations of hilbert polynomials of various varieties.

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 $\sigma$.

Blowups and Fiber Products

This past week, I’ve been working on computing blowups. Classically, the blowup of $\mb A^n$ is the algebraic subset of $\mb A^n\times \mb P^{n-1}$ given by $\{(a_1, ..., a_n, [p_1, ..., p_n]) | a_ip_i=a_jp_j\}$; now, we blowup along a sheaf of ideals $\ms I$ by forming the sheaf of graded rings $\ms B = \mc O_X\oplus \ms I\oplus \ms I^2\oplus \cdots$. In the affine case this is called the blowup algebra. We then take relative $\proj$ 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 $x^2-y^2-z^2$ 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.