Timeline for Intersection of three quadrics: associating something geometric to these analogously to intersection of two quadrics
Current License: CC BY-SA 4.0
9 events
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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 |