## 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

## 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 …

## 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 …

## 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 …

## Chern connections in the context of CR-geometry

This post is a result of several discussions with Rodion Déev.

#### 1. Preliminaries: Chern’s theorem

Fix a complex manifold and a complex vector bundle over . Recall that a structure of holomorphic bundle on is given by an operator

which satisfies -Leibniz identity

and the integrability condition

For the …

## An example of a non-reduced Picard scheme

Let be a smooth projective connected scheme over an algebraically closed field (experts will notice that several of these hypotheses can be weakened in what follows). Attached to is the Picard scheme , a locally finite type -scheme defined functorially as sending a -scheme to the group . (This uses …

## An explicit construction of indecomposable vector bundles over an elliptic curve

In the celebrated paper “Vector bundles over an elliptic curve,” M. Atiyah classifies indecomposable vector bundles, namely he provides a bijection between indecomposable bundles of arbitrary rank and degree (denoted by ) and (where ). The latter is described explicitly: there is a distinguished element such that for any other …

## A proper scheme with infinite-dimensional fppf cohomology

In algebraic geometry, very often one encounters theorems of the following flavor:

Theorem: Let be a proper morphism of spaces. Then for every sheaf on that is finite, so is its pushforward .

Notice how I was being deliberately vague in the theorem above. What are and ? What does …