## 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

## 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 …

## 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 …

## 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 …

## 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 …

## 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 …

## 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 …

## 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 …

## Brieskorn resolutions via algebraic spaces

I’d like to discuss simultaneous resolutions of surfaces from a moduli-theoretic perspective, following Michael Artin’s paper on Brieskorn resolutions.

Artin’s approach to moduli begins with the most desirable aspect of a moduli space, its universal property. That is to say, define a functor the space should represent and then check …

## The torsion component of the Picard scheme

This post is a continuation of Sean Cotner’s most recent post [see An example of a non-reduced Picard scheme]. Since writing that post, Bogdan Zavyalov shared some notes of his proving the following strengthened version of the results described there.

Main Theorem. Let be a noetherian local ring and …