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