picard-group Archives

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 …

A proper scheme with infinite-dimensional fppf cohomology

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

Theorem: Let $f : X \to S$ be a proper morphism of spaces. Then for every sheaf $\mathcal{F}$ on $X$ that is finite, so is its pushforward $Rf_\ast \mathcal{F}$ .

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