Let {X} be a projective variety over the algebraically closed field {k}, endowed with a basepoint {\ast}. In the previous post, we saw how to define the Picard scheme {\mathrm{Pic}_X} of {X}: a map from a {k}-scheme {Y} into {\mathrm{Pic}_X} is the same thing as a line bundle on {Y \times_k X} together with a trivialization on {Y \times \ast}. Equivalently, {\mathrm{Pic}_X} is the sheafification (in the Zariski topology, even) of the functor

\displaystyle Y \mapsto \mathrm{Pic}(X \times_k Y)/\mathrm{Pic}(Y),

so we could have defined the functor without a basepoint.

We’d like to understand the local structure of {\mathrm{Pic}_X} (or, equivalently, of {\mathrm{Pic}^0_X}), and, as with moduli schemes in general, deformation theory is a basic tool. For example, we’d like to understand the tangent space to {\mathrm{Pic}_X} at the origin {0 \in \mathrm{Pic}_X}. The tangent space (this works for any scheme) can be identified with

\displaystyle \hom_{0}( \mathrm{Spec} k[\epsilon]/\epsilon^2, \mathrm{Pic}_X). (more…)