# A Fourier-Mukai equivalence between non trivial component of cubic threefold and degree 14 prime Fano threefold

Consider a cubic threefold $$Y$$ and its associated degree $$14$$ prime Fano threefold $$X$$, we have the equivalences of non-trivial components of $$D^b(Y)$$ and $$D^b(X)$$, i.e, $$\mathcal{A}_X\cong\mathcal{B}_Y$$ and this equivalence is given by a Fourier-Mukai functor $$\Phi:=\Phi_{I_Z(H_Y)}:D^b(X)\rightarrow D^b(Y)$$, where $$Z\subset X\times Y$$ is an irreducible four dimensional subvariety. Consider the projection map $$\pi:Z\rightarrow Y$$, my question is how to describe the fiber of this map $$\pi^{-1}(y)$$ for every point $$y\in Y$$, which is $$Z\cap X_y$$.

Since $$X_y\cong X$$ is a prime Fano threefold, its Picard number is 1, so it can not contain any divisor which is not multiple of the degree 14 $$K3$$ surface, by playing some game with matrices, it seems to me that $$\pi^{-1}(y)$$ is degree 2, so it looks like it is of dimension 1 and degree $$2$$, which looks like a conic on $$X$$, but I have difficulty to show this fiber is connected to exclude the possibility that two disjoint lines.

Indeed, the FM kernel is induced by the HPD kernel, which is, essentially, the locus $$\mathbf{Z}$$ of pairs $$(U,y)$$, where $$U$$ is a 2-dimensional subspace in the fixed 6-dimensional space $$V_6$$ and $$y$$ is a degenerate skew form on $$V_6$$ such that $$U \cap \operatorname{Ker}(y) \ne 0.$$ The fiber $$\mathbf{Z}_y$$ of this locus over a general degenerate $$y$$ is $$\mathbf{Z}_y \cong \operatorname{Cone}(\mathbb{P}^1 \times \mathbb{P}^3).$$ Finally, $$Z_y$$ is a liner section of $$\mathbf{Z}_y$$ of codimension 4, and it is easy to see that it is a rational quartic curve.