## 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.