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…)