Weird example: finite maps don’t preserve projectivity

In this post we discuss a weird example of a finite map of varieties which doesn’t preserve projectivity.

A computation a day: a pullback pushforward

In this post we compute the Galois representation where is the natural inclusion and is the inclusion of the origin.

Reductive groups: a rapid introduction

The goal of this post is to introduce, in a very informal way, the notion of a reductive group, and discuss some examples.

Maps from simply connected projective varieties to curves

In this post we prove a general result that shows, in particular, that any map from a simply connected to a curve of genus at least is constant.

p-divisible groups, formal groups, and the Serre-Tate theorem

In this post we discuss the basic theory of p-divisible groups, their relationship to formal groups, and the Serre-Tate theorem.

Algebraic de Rham cohomology and the Degeneration of the Hodge spectral sequence

In this post we will discuss various properties of the algebraic de Rham cohomology of a variety . We will focus, in particular, on various aspects of when the Hodge-to-de Rham spectral sequence on the first page, the most interesting case of which happens in positive characteristic.

Hodge Symmetry in characteristic Zero

In this post we prove that varieties over characteristic fields are all ‘Hodge symmetric’, meaning that .

Weird Example: Pullback of Very Ample by finite is not Very Ample

In this post we give a counterexample to the claim that the pullback of a very ample sheaf under a finite map is very ample, and prove that the result is true with ‘very ample’ replaced by ‘ample’.

Flat Morphisms and Flatness

In this post we will review some of the basic properties of flat/faithuflly flat modules, define flat morphisms of schemes, and discuss some of the nice properties that these morphisms have.