I am considering blow up of $\mathcal{C}\subset(\mathbb{P}^1)^3$, $X=\operatorname{Bl}_{\mathcal{C}}(\mathbb{P}^1)^3$, where $\mathcal{C}$ is a curve given by $$\{s^2u=0\}\subset\mathbb{P}^1_{s:t}\times\mathbb{P}^1_{u:t}$$ which is a copy of $\mathcal{C}_0:V(u)\cong\mathbb{P}^1$ intersecting with the thickened branch $V(s^2)\cong\mathbb{P}^1\times\frac{\mathbb{C}[s]}{s^2}$ at a fat point. We consider $(\mathbb{P}^1)^3$ containing $\mathcal{C}$ as a locally complete intersection and $\mathcal{C}$ is the vanishing locus $V(x,s^2u)$. Then, $X$ is singular with a line of surface nodes compounded with a threefold nodal singularity at one point, thus the singularity is non-isolated. My question is: what is the correct definition of intermediate Jacobian of $X$? From some categorical consideration, I would expect the "intermediate Jacobian" of $X$ is trivial under the "correct" definition. I welcome any reasonable definition which does not give the trivial intermediate Jacobian.
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