there is a typo in the very end (!)

the period **is _not_ ps[130]**: [1, 0, 4, 6, 60, 180, 1210, 5040, 30940, 150360]

but ps[86]: [1, 0, 4, 6, 60, 180, 1210, 5460, 30940, 165480]

They have same first 7 entries, so it was easy to mistake.

Moreover, period sequence 130 is bad – it has (2D-1) as a multiple. So this resolves the ambiguity we had before.

Also – 3.24 has 3 terminal Gorenstein degenerations,

and TGTD ansatz applied to them is exactly ps[86].

3.24 corresponds to ps[86]

3.23 should correspond to ps[158]

And there will be exactly 1-1 correspondence between 98 Fano threefolds and 98 good period sequences.

Following is the computation of first 10 terms:

> sum(l=0,10,sum(m=0,10,sum(n=m,10, t^(2*l+m+n) * (l+n)! / l!^3 /m!^2 /n! /(n-m)!))) +O(t^11)

1 + t + 5/2*t^2 + 19/6*t^3 + 109/24*t^4 + 581/120*t^5 + 3371/720*t^6 + 4021/1008*t^7 + 123229/40320*t^8 + 773029/362880*t^9 + 983333/725760*t^10 + O(t^11)

> reg(%)

1 + t + 5*t^2 + 19*t^3 + 109*t^4 + 581*t^5 + 3371*t^6 + 20105*t^7 + 123229*t^8 + 773029*t^9 + O(t^10)

> nor(%)

1 + 4*t^2 + 6*t^3 + 60*t^4 + 180*t^5 + 1210*t^6 + 5460*t^7 + 30940*t^8 + 165480*t^9 + O(t^10)

> Vec(%)

[1, 0, 4, 6, 60, 180, 1210, 5460, 30940, 165480]

Moreover, the description for 3.24 seems too complicated.

After on the third line X is described as the complete intersection of degrees (1,1,0) and (0,1,1) in we can already apply quantum Lefschetz:

,

regularizing and normalizing we get

i.e. period sequence 86.

3.6 has period sequence 146

It is represented by complete intersection of degrees (1,0,2) and (0,1,1) in .

3.10 has period sequence 67.

It is represented by complete intersection of degrees (1,0,1), (0,1,1) and (0,0,2) in .

3.23 has period sequence 86.

It is represented by complete intersection of degrees (1,1,0) and (0,1,1) in .

Note that this is a D4 form (i.e. the Picardâ€“Fuchs equation has unexpectedly low degree in D). It is almost certainly a G-Fano, as there is an obvious -action.

It is not a G-Fano, but has a little of symmetry (), so it looks like

Fano with Picard number 2.

(According to Matsuki’s data) other Fanos with this property in this list should have the following numbers:

3, 9, 10, 17, 19, 20, 25, 31

I wrote a wider review of these issues in the post

expected distribution of equations

—-

Also I propose not to call this type of equation D4, since D4 is already reserved for equations that look like RQDE of P^4 (or 4-dimensional quadric). Better name for RQDE

of general Fano 3-fold with Picard number 2 is D3+1.

In general, type of RQDE for generic Fano variety is classified by its Lefschetz decomposition i.e. partition or Young tableux.

For Fano threefolds we will probably have just D3, D3+1, D3+2 and four D3+3’s.

(3+2 is a nickname for 3+2×1,

3+3 is a nickname for 3+3×1).

They correspond to period sequences 41, 53, 84 and 163.

To be more precise:

25 is unique smooth toric Fano with P=3 and degree 44,

it corresponds to period sequence 41.

31 is unique smooth toric Fano with P=3 and degree 52,

it corresponds to period sequence 53.

29 and 30 has same degree 50 and correspond to p.s. 84 (grdb – 520136) and 163 (grdb – 520127).

Need an extra computation to separate these two.

Here is it.

Mirror of is

is a blowup of point, its mirror is

.

2.29 is blowup of line on exceptional divisor, so its mirror is

This Laurent polynomial has period sequence 163

[1, 0, 2, 0, 30, 60, 380, 840, 5950, 22680]

2.30 is blowup of line that strict transform of line passing the center of blowup ,

so its mirror is

This Laurent polynomial has period sequence 84

[1, 0, 2, 6, 30, 60, 470, 1680, 7630, 34440]

[Sorry, I had a typo here (!), 84 is correct]

Also, for 2.25 mirror is

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