Cone F over Segre variety

has index 3, and its mirror is isogeneous to

So its regularized I-series are

Then

Normalizing gives period sequence:

It is intersection of divisors of polydegrees

(1,1,0),(1,0,1) and (0,1,1) on .

Up to Givental’s constant it’s $I$-series are

after the shift of Givental’s constant to it becomes

]]>It has a terminal representative

[1 0 0 -1 2 1 1 -1 -1 -2 1 0 0 -1]

[0 1 0 1 -1 0 -1 1 0 1 -1 0 -1 0]

[0 0 1 1 -1 -1 0 0 1 1 -1 -1 0 0]

so it is a degeneration of smooth Fano variety

with and .

There are exactly 8 families containing G-Fano threefolds. 7 were already listed. The last one is – it is a blowup of with the center in the curve of genus 3 and degree 6 (which is intersection of cubics).

It is an intersection of three divisors of bi-degree on , so also can be done by quantum Lefschetz.

,

after normalizing (shift of Givental’s constant to 0) it becomes

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?? currently unknown: conjecturally a G-Fano, arising as a piece of the quantum cohomology of a 2:1 cover of branched over a (2,2,2) hypersurface.

This can be done by quantum Lefschetz since this variety is a hypersurface in smooth toric Fano fourfold

Some other notes.

Denote by UU,

double cover of UU branched in divisor of degree (2,,2,2) by U,

divisor of bidegree (1,1) in by W,

divisor of bidegree (2,2) in by V.

QDE for U is derived from UU in the same way as V is derived from W:

first replace t^2 with t, then shift Givental’s costant to zero.

Hyperplane section of W and UU conicide – it is a del Pezzo surface of degree 6, so their irreducible parts of QDE’s are also related by a “house”.

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