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Let $X$ be an abelian variety over $k$, $X^t = \text{Pic} _{X/k}^0$ its dual, and $\mathscr{P}$ be the Poincare bundle on $X \times X^t$.

View $\mathscr{P}$ as a family of line bundles on $X^t$ parameterized by $X$. This gives a morphism $\kappa : X \to X^{tt}$

How exactly does $X$ parameterize the family of line bundles? Do we take $x \in X$ and consider $\mathscr{P} \mid _{\{x\} \times X^t}$?

Another possible interpretation I have in mind, although less intuitive:

$\text{Pic}_{X/k}$ is actually a functor. By shorthand notation, if it is representable, then the same notation is used to denote the group variety representing it. Making this explicit, we have that giving a morphism $Y \to \text{Pic}_X$ the representing object is equivalent to giving an element of the $\text{Pic}_X (Y)$, where now it is being considered as a functor. The latter is essentially $\text{Pic}(X \times Y)$, the group of line bundles on $X \times Y$. So given a line bundle $L$ on $X \times Y$, there is a corresponding $Y \to \text{Pic}_{X/k}$. Do we then say that viewing $L$ as a family of line bundles on $X$ parameterized by $Y$ gives $Y \to \text{Pic}_{X/k}$? This seems the likely approach based on how the theory is set up. However, it is not conducive to computations that one would like to make, especially eventually showing that $\kappa$ is an isomorphism. See the following paragraph.

Finally, is there a more explicit description of $\kappa$? Namely, either a description of what it does on points or perhaps a geometric intuition for it. An example I would like to compute is that $\kappa (0) = 0$, i.e. that it is a homomorphism. Using my previous interpretation of "parameterization" this seems equivalent to saying that $\mathscr{P} \mid _{0 \times X^t}$ is trivial.

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It turns out that the two approaches I had in mind are equivalent. Picking a fiber over a point is equivalent to choosing such an isomorphism for the representation. So every line bundle on a product gives a map on one of the factors to $\text{Pic}$ by the description in the question. Tracing through, this is exactly pulling back the bundle over a fiber of a point $x_0$, namely taking the restriction to $\{x_0\} \times Y$. The latter interpretation is obviously better for explicit computations.

With this in mind, we can compute $\kappa ( 0 )$ by taking $\mathscr{P} \mid _{0 \times X^t} $ which is indeed just the structure sheaf. This is what it means for $\kappa (0) = 0$.

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