## Flatness properties of Poincare sheaf on relative compactified Jacobians

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 $$X$$?