## 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 […]

## What Topological Spaces are π₀?

This was a fun question I thought about once.  My answer is at the end, in case you’d like to try solving the problem yourself.  The question is likely more interesting than my solution. A well known theorem says that every group occurs as for some topological space .  It’s not a hard construction; is […]

## Isometries of a product of Riemannian manifolds

Theorem. Let and be two compact Riemannian manifolds with irreducible holonomy groups. Let . Then This result seems to be a folklore, probably well known to the specialists, although it is hard to find it in the literature. The only discussion which I managed to find on Mathoverflow contains an answer by Igor Rivin (due […]

## 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 […]

## Finite flat commutative group schemes embed locally into abelian schemes

Let be a finite flat commutative group scheme over a fixed locally noetherian base scheme . In this brief note, I want to explain the proof of the following theorem due to Raynaud. Theorem. There exists, Zariski-locally on , an abelian scheme such that embeds as a closed -subgroup of . This theorem is rather […]

## 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 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 […]

## Some families of finite flat group schemes

When people say finite flat group scheme, what exactly do they mean? Sometimes, they just mean finite flat group scheme, presumably over some prefixed base. But often people meant finite flat *commutative* group scheme. It’s confusing. In this blog, I shall always mean the former, and add the adjective “commutative” when I mean it. Anyway, […]