## New data

Here are all the Minkowski period sequences, cross-linked with useful data.

Here are all the 3D reflexive polytopes, with face decompositions and other useful data.

1. Sergey says:

On page with polytopes:
it is too heavy – just hangs the browser.

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On page with Minkowski sequences:

a. “Hilbert function” – this field can be safely removed as long as we restrict ourselves to reflexive polytopes,
it contains same info as just the degree.

b. some stratification of “polytopes” with respect to ‘smoothness’ data (first “smooth”, then “terminal”,
then the rest) and Picard numbers (from higher to lower) as it was in MinkowskiSliceforSergei2.txt)
would be nice.

c. Please include highest Picard number in the data.

d. Specify the index.

e. If possible, after ranks – discriminants of Picard lattices.

f. ‘expected values of N along with the values achieved’
(from post “expected distribution”)

2. Sergey says:

In description of PF for period sequences
87, 93, 147
there are few strange remarks that degree N is unknown but N<=7 and most likely N=7. About PS(87) - it is I-series of 5.2, so indeed should have N <=7. But according to Matsuki it has an involution, so should have N=6. About PS(93) - it is I-series of 4.11, so apriori N <= 6, and expected N is also 6. About PS(147) - it is associated with STD of 4.5, so conjecturally should have N <= 6. How similar remarks for PS 83, 99, 125, 156 were obtained? Just found some annihilating operator of high degree?

3. Sergey says:

I assume further PF operator for all sequences computed indeed have the minimal degree in D (once one has arbitrary operator it can be used to produce a lot of next coefficients and to find operator of really minimal degree).

164, 161, 159, 157, 155, 154, 153, 149, 145, 143, 141-131, 130, 128, 127, 126, 124, 123, 121, 120, 118, 117, 116, 115, 111, 108, 106, 104, 101, 100, 96, 95, 94, 91, 89, 82-72, 27, 25, 24, 23

are easilly excluded from candidates for smooth Fano threefolds for trivial reason:
they have wrong restriction to t=0 – there are some factors not of the form (D-n), n integer.

Exclusion of 23, 24, 25, 27 is in accordance with
this comment – this means the only remaining N=4 sequence number 68 corresponds to the only remaining N=4 threefold 3.9. And the rest 4 remaining threefolds really have N=5 and correspond to some sequences with N=5.

The consideration above excludes 59 sequences out of 165, so 106 is left, however there should be 8 more parasitic among those that we haven’t found Picard-Fuchs equation yet.

4. Sergey says:

Now it seems we deciphered the correspondence between period sequences and Fano threefolds
(but some details could be missing, and of course we need approx. 20 proofs for guessed 3fold< ->sequence correspondence).

Altogether there are 165 period sequences.

7 Fano threefolds do not seem to correspond to any of them,
these threefolds has degrees 2 (V2), 8 (B1), 4 (2.1), 6 (2.2), 8 (2.3), 12 (9.1), 6 (10.1).

It seems case 3.5 (degree 20) is actually miscalculated and should correspond to sequence 109.

So 165-(105-7)=165-98=67 period sequences should be parasitic.

11 sequences have unknown Picard-Fuchs, among them:
3 correspond to Fanos (87, 93, 147),
8 are probably parasitic (83, 92, 99, 110, 125, 129, 156, 162).

59 more parasitic sequences are listed in the previous post.
59+8 = 67.

5. Sergey says:

Important fix – period sequence 130 is also bad!

6. Sergey says:

23, 24, 25, 27, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 89, 91, 94, 95, 96, 100, 101, 104, 106, 108, 111, 115, 116, 117, 118, 120, 121, 123, 124, 126, 127, 128, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 143, 145, 149, 153, 154, 155, 157, 159, 161, 164

83, 92, 99, 110, 125, 129, 156, 162

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2 GOOD period sequences with uncomputed Picard-Fuchs operator:

87, 93

1 conjecturally GOOD period sequence with uncomputed Picard-Fuchs operator:

147

95 GOOD period sequences with computed Picard-Fuchs operator:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 84, 85, 86, __, 88, 90, __, 97, 98, 102, 103, 105, 107, 109, 112, 113, 114, 119, 122, 142, 144, 146, ___, 148, 150, 151, 152, 158, 160, 163