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FRG Workshop on Moduli Spaces and Stability - Shared screen with speaker view - Recording 1/3
AARON BERTRAM
30:36
Have we lost Sasha?
John Kopper
30:46
oh no
Dmitrii Pedchenko
41:52
Is there some condition that could be imposed either on the collection or on X_H to guarantee that after restricting to X_H one still gets a full collection?
Pieter Belmans
42:41
it can be read off from the geometry of the homological projective dual, but that's a rather tautological answer
Pieter Belmans
43:45
the question is somewhat similar to applying the usual Lefschetz theorem: can one find conditions that guarantee that there is no new middle cohomology?
Pieter Belmans
44:04
I'm not aware of such conditions in this classical setting, and also not in the exceptional collections setting :)
Dmitrii Pedchenko
44:08
Ok, makes sense
Dmitrii Pedchenko
47:02
But does the count work at least on the level of K groups of X_H and X (if say X_H) is smooth)?
Pieter Belmans
49:14
I think it goes the other way? if you can compute something on X_H (say Hodge numbers, or rank of K_0) you know whether there's any chance of the Lefschetz collection for X restricting to X_H without additional components
Svetlana Makarova
49:35
I believe no, because take e.g. X=P^3, the ample line bundle O(2), so H=quadric=P^1 x P^1. Then restriction of the part of FEC from P^3 has length 2, but a FEC on P^1 x P^1 has length 4
Dmitrii Pedchenko
49:43
Ok, that is what I thought =(
Dmitrii Pedchenko
50:04
Sveta, sure, thanks