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

# Algebraic Geometry

# 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.

(more…)# 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.