A riddle wrapped in a mystery inside a balls-up
Consider #2 on the Mori-Mukai list of rank-3 Fano 3-folds. This has been giving us some difficulty, which I have now resolved. We were making a combination of mistakes. Mori and Mukai describe the variety as follows.
A member of on the
-bundle
over
such that
is irreducible, where
is the tautological line bundle and
is a member of
.
Our first mistake, as Mukai-sensei pointed out in an email to Corti, was using the wrong weight convention for projective bundles. Mori and Mukai use negative weights, so the ambient -bundle
is the toric variety with weight data:
For later convenience we change basis, expressing as the toric variety with weight data:
is a section of
.
Our second mistake was failing to accurately account for the fact that although is Fano, the bundle
(which restricts to
on
) is only semi-positive on
. Thus we are in the situation described in this post and, in the notation defined there, we have:
and hence:
Thus the cohomological-degree-zero part of the J-function is:
and setting ,
,
,
yields:
Regularizing this gives:
As Galkin conjectured, this is period sequence 97.