Divisor Line Bundle Correspondence on a Surface Let $X$ be an elliptic curve over $\mathbb{C}$ and let $S = X \times X$. Suppose $D_1$ is the divisor corresponding to $X\times \{0\}$ in $S$ and $D_2$ the divisor corresponding to the diagonal. Let $\mathcal{L}_1$ and $\mathcal{L}_2$ be the line bundle associated to $D_1$ and $D_2$ respectively, and $\mathcal{L}$ the line bundle corresponding to $D_1 - D_2$.  If $p: S\to X$ is the projection onto the first factor, how can I think about $p_{\ast}(\mathcal{L}$)?
Some context: I think I (sort of) understand the divisor-line bundle correspondence on curves in some easy cases, but I don't really see how to picture it more generally. I am aware that $\mathcal{L}(D_1 + D_2) = \mathcal{L}(D_1)\otimes \mathcal{L}(D_2)^{\ast}$ for arbitrary divisors $D_1$ and $D_2$.

Informally, for any morphism $p\colon Y\to X$ of schemes and any quasi-coherent sheaf $\mathcal{F}$ on $Y$, a section of $p_*\mathcal{F}$ can be thought of as a family of sections of $\mathcal{F}$ over fibers of $p$ varying "continuously".
In your setting, the fiber of $p$ at a closed point $x\in X$ is $\{x\}\times X$, and we have
$$
\mathcal{L}(D_1-D_2)|_{\{x\}\times X}\simeq \mathcal{L}(D_1|_{\{x\}\times X}-D_2|_{\{x\}\times X})\simeq \mathcal{L}(\{0\}-\{x\}).
$$
Therefore a section of $p_*\mathcal{L}$ can be thought of as a family of sections of $\mathcal{L}(\{0\}-\{x\})$ varying continuously in $x\in X$.
It is known that the line bundle $\mathcal{L}(\{0\}-\{x\})$ has no non-trivial section unless $x = 0$, so we expect that $p_*\mathcal{L}$ is a skyscraper sheaf with support $\{0\}$, and this is in fact the case.
