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Consider the smooth intersection of two $4$-dimensional quadrics $Y = Q \cap Q' \subset P^5$. To the Fano threefold $Y$ we can associate a genus $2$ curve as follows. Take the pencil of quadrics $\{ Q_\lambda \}_{\lambda \in P^1}$ generated by $Q$ and $Q'$. Then there are $6$ points in $P^1$ where the quadric $Q_\lambda$ is singular. If we take the double cover of $P^1$ branched at those six points, we get a genus $2$ curve. Each $Y$ in this deformation family of Fano threefolds can be associated to a genus $2$ curve this way.

Now consider the intersection of three $5$ dimensional quadrics $X = Q \cap Q \cap Q'' \subset P^6$. Is there some kind of analogous geometric construction for Fano threefolds $X$, where we can associate some certain type of variety to each $X$?

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  • $\begingroup$ You get a pencil of quadrics parameterized by $\mathbb P^2$. The singular ones give a curve of degree 7 in that projective plane, and you get a double covering of that curve. $\endgroup$
    – Will Sawin
    Commented Oct 12, 2023 at 22:24
  • $\begingroup$ And there is a beautiful paper of Tjurin discussing this correspondence: Tjurin, A. N. The intersection of quadrics. (Russian) Uspehi Mat. Nauk 30 (1975), no. 6(186), 51--99. $\endgroup$
    – Sasha
    Commented Oct 13, 2023 at 4:51
  • $\begingroup$ Thank you both for your comments. It is known that in the case of $Y$, we have $\langle O_Y, O_Y(1) \rangle^\bot \simeq D^b(C_2)$. In the case of $X$, can we say that $\langle O_X \rangle^\bot$ is related to any category related to the double cover? Also, Will, do you not mean ramified in $C_7$? Otherwise, why is it a double cover of $C_7$ (the thing parametrizing singular quadrics) in the $X$ case, but in the $Y$ case the double cover is ramified over the thing (points) parametrizing singular quadrics? $\endgroup$
    – alg_et_geom
    Commented Oct 13, 2023 at 9:09
  • $\begingroup$ The crucial difference is the parity of dimension (or rather of the rank) of the quadrics. The point is that a quadric of even rank has two families of maximal isotropic subspaces, this leads to various double coverings. In the case of $Y$, general quadric in the pencil has even rank, hence there is a double covering over the open subset of $\mathbb{P}^1$ (and it extends to the entire $\mathbb{P}^1$). In the case of $X$ general quadric has odd rank, but those over the discriminant curve have rank 6, hence a covering of the discriminant. $\endgroup$
    – Sasha
    Commented Oct 13, 2023 at 20:28
  • $\begingroup$ In this case, what is the covering of $C_7$ ramified in? Or is it unramified? $\endgroup$
    – alg_et_geom
    Commented Oct 14, 2023 at 14:46

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