Assume we are given a $ K3$ -surface $ X$ with $ Pic(X)=\mathbb{Z}\mathcal{O}_X(1)$ such that $ \mathcal{O}_X(1)$ ample and of degree 2.

Then we have the linear system $ |\mathcal{O}_X(1)|=\mathbb{P}^2$ and a general member is a smooth curve of genus 2.

Let $ f: C\rightarrow |\mathcal{O}_X(1)|$ be the universal curve, that is $ C=\{(x,D)\,|\, x\in D\}\subset X\times |\mathcal{O}_X(1)|$ .

Denote by $ \overline{J}^d$ the relative compactified Jacobian of degree $ d$ of the family $ f: C\rightarrow |\mathcal{O}_X(1)|$ .

If $ d$ is even, then $ \overline{J}^d$ is a fine moduli space and hence there is a Poincare sheaf $ U$ over $ C\times_{|\mathcal{O}_X(1)|}\overline{J}^d$ flat over $ \overline{J}^d$ .

$ \textbf{Question:}$ Is $ U$ flat over $ C$ ?

We also have $ C\times_{|\mathcal{O}_X(1)|}\overline{J}^d\subset X\times \overline{J}^d$ . Is $ U$ seen as a sheaf on $ X\times \overline{J}^d$ flat over $ X$ ?