A proof of a general slice-Bennequin inequality

In this blog post, I’ll provide a slick proof of a form of the slice-Bennequin inequality (as outlined by Kronheimer in a mathoverflow answer.) The main ingredient is the adjunction inequality for surfaces embedded in closed 4-manifolds. To obtain the slice-Bennequin inequality (which is a statement about surfaces embedded in 4-manifolds with boundary) we use […]

The Picard number of a Kummer K3 surface

Let be a separably closed field of characteristic not , and an abelian surface. Then it is a basic fact (e.g. see Example 1.3 (iii) of Huybrechts’ “K3 Surfaces”) that one can make a K3 surface out of . The construction is as follows. Consider the involution given by The fixed locus of this involution […]

A curiosity: “supersmooth” varieties

I want to share a curious condition on varieties for which I have found no use. Let be a field and let be a locally finite type -scheme. Recall that is said to be smooth if, for every Artin local -algebra with a proper ideal and every -morphism , there exists a -morphism extending it, […]

The rank of \(y^2 = x^3 - 2\) via Mazur-Tate methods

When I was a young kid, I heard the mathematical fact that the only (positive) integer that is one more than a square and one less than a cube is . Said differently, the only integer solutions to are given by . There are elementary methods to prove this, using the fact that the ring […]

Some examples of algebraic groups

In this post I want to give a few examples of the known “pathological” behavior of algebraic groups defined over general bases. In particular, this post contains examples of the following. A smooth group scheme over a DVR with generic fiber and special fiber , An affine smooth group scheme over a regular dimension base […]