The Stacks project

Exercise 111.66.2 (Theorems). Precisely and succintly state one nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

1. the topological space of a scheme $X$,

2. a representability criterion for functors on the category of schemes,

3. the Picard functor for a smooth projective curve $X$ over an algebraically closed field $k$,

4. torsion in the Picard group of a smooth projective curve $X$ over an algebraically closed field $k$,

5. a theorem on the Picard group of a product $X \times Y \times Z$ of smooth projective varieties $X$, $Y$, $Z$ over an algebraically closed field $k$.