Let be a projective variety over the algebraically closed field , endowed with a basepoint . In the previous post, we saw how to define the **Picard scheme** of : a map from a -scheme into is the same thing as a line bundle on together with a trivialization on . Equivalently, is the sheafification (in the Zariski topology, even) of the functor

so we could have defined the functor without a basepoint.

We’d like to understand the local structure of (or, equivalently, of ), and, as with moduli schemes in general, deformation theory is a basic tool. For example, we’d like to understand the tangent space to at the origin . The tangent space (this works for any scheme) can be identified with