algebraic geometry Archives

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” …

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 $\mathbf{G}_m$ and special fiber $\mathbf{G}_a$ ,
An affine smooth group scheme

…

Finite flat commutative group schemes embed locally into abelian schemes

Let $G$ be a finite flat commutative group scheme over a fixed locally noetherian base scheme $S$ . In this brief note, I want to explain the proof of the following theorem due to Raynaud.

Theorem. There exists, Zariski-locally on $S$ , an abelian scheme $A$ such that $G$ embeds as a closed $S$ -subgroup of …

A curiosity: “supersmooth” varieties

I want to share a curious condition on varieties for which I have found no use. Let $k$ be a field and let $X$ be a locally finite type $k$ -scheme. Recall that $X$ is said to be smooth if, for every Artin local $k$ -algebra $A$ with a proper ideal $I \subset A$ and every $k$ -morphism $\Spec A/I \to X$ , there …

The Picard number of a Kummer K3 surface

Let $k$ be a separably closed field of characteristic not $2$ , and $A/k$ 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 $A$ . The construction is as follows. Consider the involution $\iota : A \to A$ given by $x \mapsto -x.$ 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 $S$ 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 $X$ and a complex vector bundle $E$ over $X$ . Recall that a structure of holomorphic bundle on $E$ is given by an operator

$\overline{\partial}_{\mathcal{E}} \colon \Gamma (E) \to \Gamma(E \otimes \Lambda^{0,1}X)$

which satisfies $\overline{\partial}$ -Leibniz identity

$\overline{\partial}_{\mathcal{E}}(fs) = f\overline{\partial}_{\mathcal{E}}(s) + \overline{\partial}(f) \otimes s$

and the integrability condition

$\overline{\partial}_{\mathcal{E}}^2 = 0.$

For the …

An example of a non-reduced Picard scheme

Let $X$ be a smooth projective connected scheme over an algebraically closed field $k$ (experts will notice that several of these hypotheses can be weakened in what follows). Attached to $X$ is the Picard scheme $\mathrm{Pic}_{X/k}$ , a locally finite type $k$ -scheme defined functorially as sending a $k$ -scheme $T$ to the group $\mathrm{Pic}(X_T)/\mathrm{Pic}(T)$ . (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 $\alpha_{(r, d)}$ between indecomposable bundles of arbitrary rank $r$ and degree $d$ (denoted by $\mathcal{E}_{(r, d)}$ ) and $\mathcal{E}_{(h, 0)}$ (where $h = \text{gcd}(d, r)$ ). The latter is described explicitly: there is a distinguished element $F_r \in \mathcal{E}_{(r, 0)}$ such that for any other …