The Picard group of an algebraic variety

Invertible sheaves are important objects in algebraic geometry because their sections tell us how to map varieties into projective space. It thus becomes a natural question to ask what are all the invertible sheaves up to isomorphism. It turns out that the set of such isomorphism classes is a group called the Picard group. We introduce this group in this video. We show how to study it by relating it to the theory of Cartier divisors using the first Chern class map. We then compute the Picard group of projective space which also allows us to enunciate all the line bundles on it. Finally, we introduce the degree of a divisor or invertible sheaf on a smooth projective curve. This is a very important invariant which helps us to study both divisors, invertible sheaves and the Picard group.

[danielvidal]

Superb video! Very clear explanations :)