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Oct 15, 2023 at 7:55 comment added Sasha I strongly suggest you to look into the Tjurin's paper; I guess you will find there the answers to all your questions.
Oct 14, 2023 at 18:40 comment added alg_et_geom How much does can the ramification locus vary as we vary the smooth $X$ within its moduli space? If my understanding is correct, for smooth $Y$ the ramification locus is always (different configurations of) $6$ points which gives $M_Y \cong M_{C_2}$. I'm guessing if the ramification can vary more, and even be empty, then there is no such nice isomorphism $M_X \cong ...$ of moduli spaces here?
Oct 14, 2023 at 18:13 comment added Sasha It is ramified over those points of $C_7$ (if any) where the rank of the quadric drops further.
Oct 14, 2023 at 14:46 comment added alg_et_geom In this case, what is the covering of $C_7$ ramified in? Or is it unramified?
Oct 13, 2023 at 20:28 comment added Sasha 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.
Oct 13, 2023 at 9:09 comment added alg_et_geom 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?
Oct 13, 2023 at 4:51 comment added Sasha 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.
Oct 12, 2023 at 22:24 comment added Will Sawin 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.
Oct 12, 2023 at 20:17 history asked alg_et_geom CC BY-SA 4.0