Fano Varieties and Extremal Laurent Polynomials

Minkowski D3 Forms

PF operator:
-28672*t^4*D^3 - 172032*t^4*D^2 - 10240*t^3*D^3 - 315392*t^4*D - 46080*t^3*D^2 - 1152*t^2*D^3 - 172032*t^4 - 66560*t^3*D - 3456*t^2*D^2 - 32*t*D^3 - 30720*t^3 - 3520*t^2*D - 48*t*D^2 + D^3 - 1216*t^2 - 16*t*D

Connection matrix:
[     0    304   9984 121088]
[     1     16    800   9984]
[     0      1     16    304]
[     0      0      1      0]

This is the genus-5 Fano with b_2=1.

This occurs for the following polytopes (PALP id, grdb id):
[(3313, 547429), (4004, 522681), (4166, 264812), (4193, 521215), (4202, 433519), (4204, 433517), (4216, 261645), (4230, 544534), (4237, 520915), (4243, 432471), (4249, 259467), (4250, 259468), (4266, 258004), (4268, 258029), (4274, 520538), (4279, 257048), (4289, 431045), (4297, 520329), (4298, 520332), (4303, 520261), (4312, 547389), (4313, 544405), (4314, 544406)]

The corresponding toric Fanos (for each polytope) have degrees: [8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8]

Each toric Fano has Hilbert series: 1 + 7*x + 25*x^2 + 63*x^3 + 129*x^4 + 231*x^5 + 377*x^6 + 575*x^7 + 833*x^8 + 1159*x^9 + 1561*x^10 + O(x^11)

The first 20 coefficients of the period are:
[1, 0, 152, 3840, 157656, 6428160, 280064960, 12618762240, 584579486680, 27660007173120, 1331176443653952, 64959370440222720, 3206739717096129984, 159854933855643586560, 8035635968276347571712, 406873893320932784947200, 20732408686051781974758360, 1062343600325163395688683520, 54705987739091805901411964480, 2829642239626465253780367452160]

==================================================

PF operator:
-128*t^4*D^3 - 768*t^4*D^2 - 1408*t^4*D - 28*t^2*D^3 - 768*t^4 - 84*t^2*D^2 - 88*t^2*D + D^3 - 32*t^2

Connection matrix:
[  0   8   0 192]
[  1   0  12   0]
[  0   1   0   8]
[  0   0   1   0]

This is a piece of the quantum cohomology of a (1,1) hypersurface in .
(This can be calculated using quantum Lefschetz.)  The piece is the span of 1, K, K^2, K^3; put differently
it is the -invariant part of the cohomology under the obvious -action.
So this is a G-Fano.
 
This occurs for the following polytopes (PALP id, grdb id):
[(12, 544356), (21, 520158), (103, 544063), (121, 519664), (155, 430096)]

The corresponding toric Fanos (for each polytope) have degrees: [48, 48, 48, 48, 48]

Each toric Fano has Hilbert series: 1 + 27*x + 125*x^2 + 343*x^3 + 729*x^4 + 1331*x^5 + 2197*x^6 + 3375*x^7 + 4913*x^8 + 6859*x^9 + 9261*x^10 + O(x^11)

The first 20 coefficients of the period are:
[1, 0, 4, 0, 60, 0, 1120, 0, 24220, 0, 567504, 0, 14030016, 0, 360222720, 0, 9513014940, 0, 256758913840, 0]

==================================================

PF operator:
-3600*t^4*D^3 - 21600*t^4*D^2 - 2040*t^3*D^3 - 39600*t^4*D - 9180*t^3*D^2 - 359*t^2*D^3 - 21600*t^4 - 13260*t^3*D - 1077*t^2*D^2 - 14*t*D^3 - 6120*t^3 - 1102*t^2*D - 21*t*D^2 + D^3 - 384*t^2 - 7*t*D

Connection matrix:
[    0    96  1692 12816]
[    1     7   216  1692]
[    0     1     7    96]
[    0     0     1     0]

This is the genus-7 Fano with b_2=1.

This occurs for the following polytopes (PALP id, grdb id):

[(2755, 474429), (2816, 355616), (2816, 355616), (3405, 321303), (3446, 147470), (3446, 147470), (3451, 147467), (3504, 610803), (3504, 610803), (3624, 305807), (3625, 306072), (3666, 129112), (3682, 127896), (3701, 597737), (3730, 544886), (3761, 446913), (3790, 294031), (3790, 294031), (3794, 292940), (3795, 292458), (3843, 585895), (3843, 585895), (3844, 585890), (3844, 585890), (3845, 585897), (3845, 585897), (3847, 585514), (3852, 585548), (3856, 585686), (3867, 29624), (3867, 29624), (3868, 29628), (3873, 5953), (3873, 5953), (3874, 5954), (3874, 5954), (3932, 281846), (3935, 281906), (3936, 281910), (3937, 282088), (3945, 281909), (3961, 98314), (3965, 93823), (3966, 95245), (3980, 574886), (3982, 573895), (3983, 574977), (3984, 574842), (3990, 25067), (4026, 439663), (4042, 274128), (4057, 87167), (4057, 87167), (4058, 86668), (4059, 86880), (4069, 83883), (4074, 566716), (4074, 566716), (4075, 566695), (4075, 566695), (4079, 21153), (4101, 437078), (4103, 436976), (4118, 268997), (4121, 268912), (4123, 268019), (4132, 78482), (4133, 78269), (4143, 558361), (4144, 558688), (4148, 521504), (4168, 264855), (4168, 264855), (4169, 263867), (4178, 72123), (4179, 72493), (4181, 72684), (4181, 72684), (4182, 72202), (4182, 72202), (4183, 72680), (4217, 261497), (4219, 260624), (4240, 520890), (4246, 432558), (4248, 432464), (4253, 259203), (4262, 431807), (4269, 257760), (4271, 257945), (4272, 257862), (4292, 431027), (4293, 430976)]
The corresponding toric Fanos (for each polytope) have degrees: [12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12]
Each toric Fano has Hilbert series: 1 + 9*x + 35*x^2 + 91*x^3 + 189*x^4 + 341*x^5 + 559*x^6 + 855*x^7 + 1241*x^8 + 1729*x^9 + 2331*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 48, 600, 13176, 276480, 6259800, 146064240, 3505282200, 85882130880, 2139884768448, 54055299735360, 1381203124913304, 35635462470447840, 927068343971532048, 24292067745404178720, 640545056435479080600, 16984212847751657439360, 452567591447122633310400, 12112614406551107374960320]
==================================================
PF operator:

-80*t^4*D^3 - 480*t^4*D^2 - 136*t^3*D^3 - 880*t^4*D - 612*t^3*D^2 - 59*t^2*D^3 - 480*t^4 - 884*t^3*D - 177*t^2*D^2 - 2*t*D^3 - 408*t^3 - 182*t^2*D - 3*t*D^2 + D^3 - 64*t^2 - t*D
Connection matrix:

[  0  16  84 336]
[  1   1  28  84]
[  0   1   1  16]
[  0   0   1   0]

This is probably a G-Fano.  See Galkin's example  here.
 is the blow-up of a 3-dimensional quadric along a
rational normal curve of degree 4. 

This occurs for the following polytopes (PALP id, grdb id):
[(122, 519649), (237, 518830), (294, 429082), (463, 543553), (512, 516974), (625, 425410), (699, 61968), (702, 61981), (730, 674688), (732, 674685), (828, 513257), (909, 420832), (917, 420911), (963, 419969), (1094, 674577), (1095, 674598), (1101, 674607), (1102, 674578), (1110, 61963), (1119, 546609), (1132, 539540), (1133, 539453), (1146, 539478), (1183, 507651), (1187, 507587), (1189, 507636), (1227, 506691), (1360, 412197), (1374, 412119), (1388, 246297), (1399, 246050), (1404, 246108), (1435, 251552), (1445, 251086), (1509, 674087), (1517, 61936), (1653, 498796), (1705, 402373), (1722, 402266), (1752, 397711), (1846, 236262), (1872, 669437), (1892, 672913), (1893, 672826), (1914, 671908), (1919, 671936), (1934, 61770), (2038, 491281), (2066, 486941), (2086, 388962), (2105, 388129), (2143, 388557), (2146, 388498), (2164, 385229), (2193, 223127), (2244, 222612), (2253, 231089), (2319, 669046), (2333, 61316), (2414, 482045), (2431, 482353), (2490, 371033), (2521, 370589), (2531, 370638), (2598, 204086), (2674, 662260), (2728, 529927), (2729, 529925), (2783, 473103), (2784, 473149), (2867, 353915), (2870, 353914), (2873, 352836), (2948, 181701), (2955, 181632), (3002, 649478), (3005, 649168), (3116, 464073), (3180, 332292), (3192, 334858), (3339, 526756), (3383, 456760), (3428, 314752), (3609, 451932), (3611, 450146), (3658, 303278), (3746, 524263)]
The corresponding toric Fanos (for each polytope) have degrees: [28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28]
Each toric Fano has Hilbert series: 1 + 17*x + 75*x^2 + 203*x^3 + 429*x^4 + 781*x^5 + 1287*x^6 + 1975*x^7 + 2873*x^8 + 4009*x^9 + 5411*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 8, 24, 240, 1440, 11960, 89040, 731920, 5913600, 49519008, 416095680, 3554134584, 30566888352, 265408469040, 2319246113184, 20400180886800, 180392364801792, 1603103294139776, 14307945541718400]
==================================================
PF operator:

-8784*t^4*D^3 - 52704*t^4*D^2 - 4080*t^3*D^3 - 96624*t^4*D - 18360*t^3*D^2 - 592*t^2*D^3 - 52704*t^4 - 26520*t^3*D - 1776*t^2*D^2 - 20*t*D^3 - 12240*t^3 - 1808*t^2*D - 30*t*D^2 + D^3 - 624*t^2 - 10*t*D
Connection matrix:

[    0   156  3600 33120]
[    1    10   380  3600]
[    0     1    10   156]
[    0     0     1     0]

This is the genus-6 Fano with b_2=1.

This occurs for the following polytopes (PALP id, grdb id):

[(3050, 528558), (3791, 294041), (3902, 442714), (3921, 283511), (3926, 283519), (3927, 282264), (3963, 98325), (3964, 98326), (4006, 522324), (4022, 439394), (4023, 439399), (4031, 438025), (4041, 275510), (4043, 274167), (4055, 86711), (4073, 566718), (4117, 269340), (4130, 78248), (4131, 78175), (4134, 78330), (4142, 560035), (4160, 435180), (4167, 264850), (4180, 72114), (4185, 555254), (4189, 521210), (4190, 521212), (4199, 433689), (4201, 433642), (4205, 433633), (4213, 261697), (4215, 261648), (4218, 260631), (4224, 68371), (4227, 551994), (4244, 432671), (4251, 259464), (4254, 65832), (4257, 520706), (4260, 431891), (4267, 258031), (4280, 257095), (4290, 431005), (4291, 431051), (4294, 544439), (4300, 520319), (4302, 430778), (4306, 430676), (4310, 520191)]
The corresponding toric Fanos (for each polytope) have degrees: [10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10]
Each toric Fano has Hilbert series: 1 + 8*x + 30*x^2 + 77*x^3 + 159*x^4 + 286*x^5 + 468*x^6 + 715*x^7 + 1037*x^8 + 1444*x^9 + 1946*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 78, 1320, 37746, 1051920, 31464780, 971757360, 30859805970, 1000739433120, 33005374791228, 1103665924746000, 37332067569231204, 1275110813852235360, 43916809784375982168, 1523516070448697925600, 53186807444533509802770, 1867145144587345769889600, 65871893221799542211085180, 2334228501643430357642648400]
==================================================
PF operator:

-648*t^4*D^3 - 3888*t^4*D^2 - 540*t^3*D^3 - 7128*t^4*D - 2430*t^3*D^2 - 135*t^2*D^3 - 3888*t^4 - 3510*t^3*D - 405*t^2*D^2 - 6*t*D^3 - 1620*t^3 - 414*t^2*D - 9*t*D^2 + D^3 - 144*t^2 - 3*t*D
Connection matrix:

[   0   36  378 1944]
[   1    3   72  378]
[   0    1    3   36]
[   0    0    1    0]

This is the genus-10 Fano with b_2=1.

This occurs for the following polytopes (PALP id, grdb id):

[(1558, 537117), (1591, 500518), (1615, 500364), (1694, 402564), (1783, 236732), (1791, 236722), (1826, 236600), (2048, 491706), (2107, 387279), (2115, 388505), (2186, 223100), (2187, 223102), (2188, 223154), (2222, 222475), (2270, 662848), (2288, 662834), (2301, 662832), (2336, 61365), (2468, 372892), (2479, 370754), (2493, 370852), (2499, 370782), (2519, 372147), (2538, 372148), (2564, 205912), (2570, 205938), (2577, 205973), (2594, 204885), (2604, 204865), (2632, 652803), (2645, 652691), (2647, 652669), (2657, 662250), (2679, 57386), (2682, 60038), (2683, 60041), (2690, 59974), (2698, 12465), (2700, 12510), (2702, 1515), (2762, 473300), (2831, 355341), (2839, 353935), (2846, 353942), (2857, 352810), (2897, 186448), (2901, 186252), (2910, 186253), (2920, 186479), (2922, 186663), (2962, 639854), (2979, 639732), (2981, 639027), (2984, 639695), (2988, 639710), (3010, 53085), (3014, 57336), (3019, 57256), (3022, 57226), (3023, 57290), (3024, 57272), (3034, 12193), (3036, 12161), (3037, 1505), (3054, 528551), (3060, 527955), (3098, 464419), (3143, 338018), (3150, 338244), (3154, 335142), (3159, 333359), (3160, 335731), (3179, 335733), (3220, 165806), (3225, 165835), (3229, 165869), (3232, 161596), (3234, 163370), (3236, 161572), (3268, 624171), (3273, 624115), (3275, 624073), (3277, 624451), (3278, 624132), (3280, 624343), (3286, 637790), (3289, 635469), (3290, 630597), (3292, 47445), (3296, 47429), (3298, 52838), (3304, 52997), (3306, 11567), (3307, 11568), (3360, 456303), (3393, 320569), (3394, 320563), (3414, 314193), (3422, 317467), (3455, 144137), (3456, 146567), (3462, 146358), (3465, 146307), (3469, 146566), (3485, 140871), (3495, 140548), (3507, 610446), (3513, 608684), (3515, 608813), (3516, 608627), (3519, 610102), (3521, 606825), (3525, 610103), (3533, 40829), (3540, 45832), (3544, 10527), (3585, 450161), (3587, 449907), (3595, 450327), (3602, 451427), (3604, 450160), (3626, 305770), (3645, 300021), (3670, 125848), (3677, 125176), (3678, 126080), (3680, 126185), (3681, 127686), (3691, 120787), (3692, 120954), (3698, 597560), (3706, 595148), (3712, 592451), (3713, 595151), (3715, 595061), (3716, 595051), (3721, 37682), (3722, 39241), (3763, 446171), (3767, 446903), (3768, 445062), (3769, 444829), (3811, 287942), (3822, 287241), (3828, 108694), (3830, 107924), (3833, 108116), (3838, 103584), (3858, 579032), (3861, 580364), (3865, 580302), (3911, 442307), (3929, 283324), (3939, 282905), (3941, 282805), (3950, 278148), (3954, 281543), (3957, 280321), (3958, 277552), (3968, 96299), (3970, 96497), (3978, 89065), (3986, 568804), (3987, 568869), (4052, 272829), (4067, 86200), (4072, 82865), (4084, 544685), (4095, 521906), (4109, 436777), (4111, 436667), (4113, 436698), (4122, 268454), (4129, 267792), (4141, 77057), (4173, 264169), (4177, 262832), (4241, 520868), (4264, 431504)]
The corresponding toric Fanos (for each polytope) have degrees: [18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18]
Each toric Fano has Hilbert series: 1 + 12*x + 50*x^2 + 133*x^3 + 279*x^4 + 506*x^5 + 832*x^6 + 1275*x^7 + 1853*x^8 + 2584*x^9 + 3486*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 18, 120, 1566, 18360, 237060, 3129840, 42576030, 590756880, 8335922868, 119226824640, 1724740692444, 25190718593040, 370964999876328, 5502056132532960, 82116488056364190, 1232339463675983520, 18584862355966240620, 281509812816269534640]
==================================================
PF operator:

-108*t^2*D^3 - 324*t^2*D^2 - 312*t^2*D + D^3 - 96*t^2
Connection matrix:

[  0  24   0 576]
[  1   0  60   0]
[  0   1   0  24]
[  0   0   1   0]

This is the cubic 3-fold .

This occurs for the following polytopes (PALP id, grdb id):

[(231, 518837), (741, 547501)]
The corresponding toric Fanos (for each polytope) have degrees: [24, 24]
Each toric Fano has Hilbert series: 1 + 15*x + 65*x^2 + 175*x^3 + 369*x^4 + 671*x^5 + 1105*x^6 + 1695*x^7 + 2465*x^8 + 3439*x^9 + 4641*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 12, 0, 540, 0, 33600, 0, 2425500, 0, 190702512, 0, 15849497664, 0, 1369618398720, 0, 121821136479900, 0, 11079206239530000, 0]
==================================================
PF operator:

-3584*t^4*D^3 - 21504*t^4*D^2 - 2112*t^3*D^3 - 39424*t^4*D - 9504*t^3*D^2 - 368*t^2*D^3 - 21504*t^4 - 13728*t^3*D - 1104*t^2*D^2 - 12*t*D^3 - 6336*t^3 - 1088*t^2*D - 18*t*D^2 + D^3 - 352*t^2 - 6*t*D
Connection matrix:

[    0    88  1584 11328]
[    1     6   228  1584]
[    0     1     6    88]
[    0     0     1     0]

This occurs as a piece (the block 1, K, K^2, K^3) in the quantum cohomology of
a hypersurface of bidegree (2,2) in ; as before this can be calculated using
quantum Lefschetz.  In particular, this is a G-Fano with  and the obvious G-action.

This occurs for the following polytopes (PALP id, grdb id):

[(2710, 530438), (2816, 355616), (3078, 466014), (3318, 545139), (3330, 526891), (3348, 525745), (3389, 321877), (3415, 317924), (3446, 147470), (3504, 610803), (3572, 452175), (3619, 306960), (3755, 446982), (3759, 446933), (3789, 294043), (3790, 294031), (3843, 585895), (3844, 585890), (3845, 585897), (3856, 585686), (3867, 29624), (3873, 5953), (3874, 5954), (3900, 442762), (3922, 283523), (3932, 281846), (3982, 573895), (4002, 522683), (4003, 522703), (4040, 275527), (4042, 274128), (4057, 87167), (4074, 566716), (4075, 566695), (4116, 269333), (4158, 435216), (4159, 434956), (4168, 264855), (4169, 263867), (4181, 72684), (4182, 72202), (4214, 261714), (4235, 544536), (4240, 520890), (4245, 432661), (4248, 432464), (4259, 431910), (4277, 431397), (4299, 520330)]
The corresponding toric Fanos (for each polytope) have degrees: [12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12]
Each toric Fano has Hilbert series: 1 + 9*x + 35*x^2 + 91*x^3 + 189*x^4 + 341*x^5 + 559*x^6 + 855*x^7 + 1241*x^8 + 1729*x^9 + 2331*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 44, 528, 11292, 228000, 4999040, 112654080, 2613620380, 61885803840, 1490373453744, 36386869189440, 898607084375616, 22407788170639872, 563420642442797568, 14268868232534243328, 363645785196486035100, 9319184268623160277632, 240004469570133005573264, 6208361451073231685966016]
==================================================
PF operator:

-256*t^4*D^3 - 1536*t^4*D^2 - 2816*t^4*D - 1536*t^4 + D^3
Connection matrix:

[  0   0   0 256]
[  1   0   0   0]
[  0   1   0   0]
[  0   0   1   0]

This is .

This occurs for the following polytopes (PALP id, grdb id):

[(0, 547386)]
The corresponding toric Fanos (for each polytope) have degrees: [64]
Each toric Fano has Hilbert series: 1 + 35*x + 165*x^2 + 455*x^3 + 969*x^4 + 1771*x^5 + 2925*x^6 + 4495*x^7 + 6545*x^8 + 9139*x^9 + 12341*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 0, 0, 24, 0, 0, 0, 2520, 0, 0, 0, 369600, 0, 0, 0, 63063000, 0, 0, 0]
==================================================
PF operator:

-192*t^3*D^3 - 864*t^3*D^2 - 80*t^2*D^3 - 1248*t^3*D - 240*t^2*D^2 - 4*t*D^3 - 576*t^3 - 256*t^2*D - 6*t*D^2 + D^3 - 96*t^2 - 2*t*D
Connection matrix:

[  0  24 144 576]
[  1   2  36 144]
[  0   1   2  24]
[  0   0   1   0]

This is a G-Fano.  It occurs as the G-invariant piece (=span of 1, K, K^2, K^3) inside the quantum
cohomology of the degree (1,1,1,1) hypersurface in .
The group G here is the symmetric group  acting in the obvious way ( would suffice).

This occurs for the following polytopes (PALP id, grdb id):

[(488, 543412), (577, 516786), (609, 425459), (1190, 507649), (1275, 413301), (1460, 250869), (1502, 674139), (1529, 12645), (1774, 408626), (1986, 534504), (1990, 534507), (2022, 491583), (2040, 491586), (2152, 387007), (2200, 222775), (2257, 234713), (2337, 61130), (2347, 60878), (2354, 1517), (2618, 204551), (2809, 468913), (2960, 195976), (3006, 646162), (3118, 463650), (3256, 160162), (3489, 135822), (3813, 290693)]
The corresponding toric Fanos (for each polytope) have degrees: [24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24]
Each toric Fano has Hilbert series: 1 + 15*x + 65*x^2 + 175*x^3 + 369*x^4 + 671*x^5 + 1105*x^6 + 1695*x^7 + 2465*x^8 + 3439*x^9 + 4641*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 12, 48, 540, 4320, 42240, 403200, 4038300, 40958400, 423550512, 4434978240, 46982827584, 502437551616, 5417597053440, 58831951546368, 642874989479580, 7063600894137216, 77991775777488144, 864910651813116480]
==================================================
PF operator:

-336*t^4*D^3 - 2016*t^4*D^2 - 368*t^3*D^3 - 3696*t^4*D - 1656*t^3*D^2 - 112*t^2*D^3 - 2016*t^4 - 2392*t^3*D - 336*t^2*D^2 - 4*t*D^3 - 1104*t^3 - 336*t^2*D - 6*t*D^2 + D^3 - 112*t^2 - 2*t*D
Connection matrix:

[   0   28  240 1120]
[   1    2   60  240]
[   0    1    2   28]
[   0    0    1    0]

This is G-Fano , see the comment below.
 is a section of Segre variety .

This occurs for the following polytopes (PALP id, grdb id):

[(1135, 539569), (1193, 507586), (1280, 413310), (1347, 413085), (1496, 674248), (1545, 537133), (1547, 537113), (1583, 500497), (1683, 402540), (1821, 236574), (1884, 672737), (1887, 672861), (2081, 388954), (2091, 388966), (2151, 388570), (2224, 222624), (2277, 662786), (2278, 662796), (2299, 662797), (2310, 668983), (2327, 61360), (2349, 12608), (2355, 10), (2411, 482428), (2480, 372136), (2539, 359406), (2548, 206001), (2579, 204760), (2643, 652681), (2653, 652715), (2665, 662262), (2681, 60032), (2688, 60031), (2697, 12519), (2721, 529907), (2742, 529800), (2748, 474431), (2761, 473110), (2780, 473340), (2819, 354967), (2820, 355397), (2889, 186748), (2915, 186256), (2954, 181697), (2974, 639652), (2975, 639467), (2995, 649245), (2999, 651604), (3009, 53121), (3028, 56368), (3076, 465966), (3089, 464399), (3133, 338175), (3134, 338158), (3200, 166803), (3253, 163000), (3270, 623294), (3302, 52267), (3336, 526335), (3354, 458626), (3401, 321330), (3420, 314336), (3439, 315731), (3477, 145166), (3494, 134327), (3500, 140258), (3524, 608931), (3539, 45839), (3551, 525020), (3590, 450075), (3603, 450158), (3633, 306075), (3668, 125301), (3704, 592333), (3774, 446837), (3805, 292773), (3860, 580376), (3892, 442983), (3979, 91442), (4014, 522631), (4016, 522644), (4027, 439692), (4176, 262193)]
The corresponding toric Fanos (for each polytope) have degrees: [20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20]
Each toric Fano has Hilbert series: 1 + 13*x + 55*x^2 + 147*x^3 + 309*x^4 + 561*x^5 + 923*x^6 + 1415*x^7 + 2057*x^8 + 2869*x^9 + 3871*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 14, 72, 882, 8400, 95180, 1060080, 12389650, 146472480, 1767391164, 21581516880, 266718438756, 3327025429728, 41849031952728, 530135326392672, 6757845419895570, 86619827323917888, 1115719258312182524, 14434274832755201424]
==================================================
PF operator:

-108*t^3*D^3 - 486*t^3*D^2 - 702*t^3*D - 324*t^3 + D^3
Connection matrix:

[ 0  0 54  0]
[ 1  0  0 54]
[ 0  1  0  0]
[ 0  0  1  0]

This is the quadric 3-fold.

This occurs for the following polytopes (PALP id, grdb id):

[(1, 547378), (3, 544395)]
The corresponding toric Fanos (for each polytope) have degrees: [54, 54]
Each toric Fano has Hilbert series: 1 + 30*x + 140*x^2 + 385*x^3 + 819*x^4 + 1496*x^5 + 2470*x^6 + 3795*x^7 + 5525*x^8 + 7714*x^9 + 10416*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 0, 12, 0, 0, 540, 0, 0, 33600, 0, 0, 2425500, 0, 0, 190702512, 0, 0, 15849497664, 0]
==================================================
PF operator:

-16*t^4*D^3 - 96*t^4*D^2 - 176*t^4*D - 44*t^2*D^3 - 96*t^4 - 132*t^2*D^2 - 136*t^2*D + D^3 - 48*t^2
Connection matrix:

[  0  12   0 160]
[  1   0  20   0]
[  0   1   0  12]
[  0   0   1   0]

This is .

This occurs for the following polytopes (PALP id, grdb id):

[(42, 520063), (67, 430442), (220, 543857), (245, 518819)]
The corresponding toric Fanos (for each polytope) have degrees: [40, 40, 40, 40]
Each toric Fano has Hilbert series: 1 + 23*x + 105*x^2 + 287*x^3 + 609*x^4 + 1111*x^5 + 1833*x^6 + 2815*x^7 + 4097*x^8 + 5719*x^9 + 7721*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 6, 0, 114, 0, 2940, 0, 87570, 0, 2835756, 0, 96982116, 0, 3446781624, 0, 126047377170, 0, 4712189770860, 0]
==================================================
PF operator:

-3207168*t^4*D^3 - 19243008*t^4*D^2 - 387072*t^3*D^3 - 35278848*t^4*D - 1741824*t^3*D^2 - 14976*t^2*D^3 - 19243008*t^4 - 2515968*t^3*D - 44928*t^2*D^2 - 160*t*D^3 - 1161216*t^3 - 45504*t^2*D - 240*t*D^2 + D^3 - 15552*t^2 - 80*t*D
Connection matrix:

[       0     3888   504576 18323712]
[       1       80    13600   504576]
[       0        1       80     3888]
[       0        0        1        0]

This is the genus-3 Fano with b_2=1.

This occurs for the following polytopes (PALP id, grdb id):

[(4311, 547390)]
The corresponding toric Fanos (for each polytope) have degrees: [4]
Each toric Fano has Hilbert series: 1 + 5*x + 15*x^2 + 35*x^3 + 69*x^4 + 121*x^5 + 195*x^6 + 295*x^7 + 425*x^8 + 589*x^9 + 791*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 1944, 215808, 35295192, 5977566720, 1073491139520, 199954313717760, 38302652395770840, 7497487505353251840, 1493126207370030913344, 301572606517249894041600, 61627017349214878082106816, 12718722541880617724729659392, 2647214316675322606900471500288, 555025852555692350954771059458048, 117115566711982792889808392397460440, 24852125190258474673660570907685107712, 5300118106455156542190578158338402598464, 1135402757829368078738986314513572479911936]
==================================================
PF operator:

-216*t^4*D^3 - 1296*t^4*D^2 - 156*t^3*D^3 - 2376*t^4*D - 702*t^3*D^2 - 43*t^2*D^3 - 1296*t^4 - 1014*t^3*D - 129*t^2*D^2 - 2*t*D^3 - 468*t^3 - 134*t^2*D - 3*t*D^2 + D^3 - 48*t^2 - t*D
Connection matrix:

[  0  12  90 360]
[  1   1  20  90]
[  0   1   1  12]
[  0   0   1   0]

This is G-Fano: see example  in Galkin's paper here and a comment below.


This occurs for the following polytopes (PALP id, grdb id):

[(92, 544075), (128, 519656), (369, 254882), (420, 62083), (555, 516755), (635, 425353), (736, 674673), (738, 674678), (756, 546977), (776, 541681), (801, 541398), (830, 513201), (973, 420086), (976, 420478), (981, 419971), (1042, 253678), (1062, 253974), (1084, 674626), (1104, 674529), (1234, 506708), (1295, 413098), (1497, 673893), (1507, 674094), (1523, 61915), (1764, 400494), (1766, 400145), (1767, 400227), (1811, 236421), (1851, 236231), (1853, 236555), (1858, 245022), (1913, 672076), (1933, 61769), (2030, 490916), (2175, 385001), (2240, 221492), (2321, 667186), (2442, 481366), (2450, 479732), (2504, 370044), (2528, 372132), (2529, 371259), (2610, 202555), (2797, 472610), (2882, 346002), (3068, 527903), (3176, 331526), (3188, 332295), (3440, 314609), (3787, 446396)]
The corresponding toric Fanos (for each polytope) have degrees: [30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30]
Each toric Fano has Hilbert series: 1 + 18*x + 80*x^2 + 217*x^3 + 459*x^4 + 836*x^5 + 1378*x^6 + 2115*x^7 + 3077*x^8 + 4294*x^9 + 5796*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 6, 24, 162, 1080, 7620, 55440, 415170, 3166800, 24570756, 193152960, 1535529996, 12323014704, 99702108792, 812367620064, 6660134536770, 54901225345824, 454766527525596, 3783396651070608]
==================================================
PF operator:

-304*t^4*D^3 - 1824*t^4*D^2 - 300*t^3*D^3 - 3344*t^4*D - 1350*t^3*D^2 - 88*t^2*D^3 - 1824*t^4 - 1950*t^3*D - 264*t^2*D^2 - 4*t*D^3 - 900*t^3 - 272*t^2*D - 6*t*D^2 + D^3 - 96*t^2 - 2*t*D
Connection matrix:

[  0  24 198 880]
[  1   2  44 198]
[  0   1   2  24]
[  0   0   1   0]

This is the genus-12 Fano with b_2=1.

This occurs for the following polytopes (PALP id, grdb id):

[(838, 513271), (929, 420909), (1021, 251679), (1027, 251680), (1140, 539490), (1271, 413299), (1312, 413150), (1427, 246241), (1481, 674247), (1483, 674241), (1603, 500036), (1699, 402281), (1715, 402297), (1721, 402434), (1785, 236692), (1815, 236477), (1854, 236595), (1879, 669463), (1885, 672886), (1886, 672906), (1888, 672877), (1897, 672700), (1898, 672887), (1900, 672881), (1928, 61779), (1929, 61780), (1936, 61803), (1941, 12639), (1942, 125), (2114, 388599), (2120, 387405), (2180, 223147), (2211, 222629), (2213, 222761), (2227, 222574), (2228, 222533), (2234, 222478), (2247, 222531), (2272, 662846), (2273, 662772), (2283, 662809), (2287, 662764), (2308, 668938), (2329, 61322), (2334, 61355), (2348, 12613), (2350, 12611), (2372, 531930), (2458, 372905), (2466, 372746), (2494, 370851), (2502, 370599), (2522, 370618), (2584, 202565), (2591, 202512), (2596, 204789), (2599, 203231), (2617, 204738), (2642, 652718), (2654, 662248), (2656, 660114), (2660, 662294), (2661, 662301), (2663, 662177), (2664, 660113), (2691, 60018), (2693, 59484), (2695, 59329), (2696, 12518), (2701, 12361), (2760, 472027), (2825, 354996), (2841, 353944), (2843, 353483), (2851, 352703), (2905, 186059), (2913, 186612), (2916, 186255), (2919, 186187), (2932, 181709), (2933, 181894), (2935, 181758), (2936, 181988), (2952, 181692), (2973, 639032), (2991, 638807), (2998, 651804), (3003, 651573), (3015, 56731), (3016, 57186), (3017, 57061), (3021, 57254), (3029, 56365), (3093, 464170), (3127, 337772), (3148, 337640), (3156, 333577), (3224, 165968), (3235, 163475), (3237, 161994), (3239, 160242), (3246, 163474), (3247, 161995), (3248, 161991), (3249, 163471), (3254, 160747), (3269, 622927), (3271, 624138), (3283, 624083), (3284, 624143), (3301, 51708), (3305, 50760), (3362, 456388), (3364, 456230), (3365, 456769), (3370, 456302), (3380, 456300), (3419, 315934), (3430, 315935), (3431, 314507), (3458, 144273), (3482, 140624), (3487, 136237), (3492, 140571), (3493, 140188), (3498, 140567), (3499, 140594), (3526, 619055), (3588, 451905), (3629, 305845), (3655, 300039), (3656, 301802), (3687, 125281), (3694, 116207), (3714, 592519), (3771, 446183), (3818, 291701), (3819, 287964), (3840, 100388), (3842, 100550), (3909, 442425), (3917, 442338), (3959, 277791), (4036, 438626), (4053, 271678), (4054, 272813), (4128, 266401), (4154, 521439)]

The corresponding toric Fanos (for each polytope) have degrees: [22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22]

Each toric Fano has Hilbert series: 1 + 14*x + 60*x^2 + 161*x^3 + 339*x^4 + 616*x^5 + 1014*x^6 + 1555*x^7 + 2261*x^8 + 3154*x^9 + 4256*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 12, 60, 636, 5760, 58620, 604800, 6447420, 70022400, 773578512, 8660892240, 98071697724, 1121159239200, 12922708539312, 150012340798320, 1752282573404220, 20581063744475520, 242914688315991120, 2879646849072124560]
==================================================
PF operator:

-256*t^2*D^3 - 768*t^2*D^2 - 704*t^2*D + D^3 - 192*t^2
Connection matrix:

[   0   48    0 2304]
[   1    0  160    0]
[   0    1    0   48]
[   0    0    1    0]

This is .

This occurs for the following polytopes (PALP id, grdb id):

[(427, 547520)]
The corresponding toric Fanos (for each polytope) have degrees: [16]
Each toric Fano has Hilbert series: 1 + 11*x + 45*x^2 + 119*x^3 + 249*x^4 + 451*x^5 + 741*x^6 + 1135*x^7 + 1649*x^8 + 2299*x^9 + 3101*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 24, 0, 2520, 0, 369600, 0, 63063000, 0, 11732745024, 0, 2308743493056, 0, 472518347558400, 0, 99561092450391000, 0, 21452752266265320000, 0]
==================================================
PF operator:

-1840*t^4*D^3 - 11040*t^4*D^2 - 1208*t^3*D^3 - 20240*t^4*D - 5436*t^3*D^2 - 243*t^2*D^3 - 11040*t^4 - 7852*t^3*D - 729*t^2*D^2 - 10*t*D^3 - 3624*t^3 - 742*t^2*D - 15*t*D^2 + D^3 - 256*t^2 - 5*t*D
Connection matrix:

[   0   64  924 5936]
[   1    5  140  924]
[   0    1    5   64]
[   0    0    1    0]

This is the genus-8 Fano with b_2=1.

This occurs for the following polytopes (PALP id, grdb id):

[(2365, 532085), (2400, 483089), (2463, 372785), (2471, 372935), (2552, 206003), (2752, 474446), (2764, 473336), (2823, 355511), (2965, 639875), (3055, 527980), (3081, 465958), (3129, 338179), (3201, 166804), (3213, 165796), (3215, 166501), (3218, 166517), (3228, 166494), (3265, 625365), (3267, 625031), (3282, 625034), (3297, 47449), (3334, 526350), (3350, 458642), (3396, 321233), (3398, 321301), (3399, 321302), (3447, 147436), (3460, 146429), (3470, 146472), (3505, 610621), (3508, 610744), (3510, 610784), (3529, 41168), (3534, 41131), (3535, 41126), (3536, 41127), (3542, 9094), (3543, 9098), (3558, 524993), (3573, 452201), (3586, 449943), (3623, 306022), (3628, 305762), (3632, 306089), (3635, 306132), (3636, 306131), (3639, 299475), (3665, 129152), (3669, 125267), (3674, 127678), (3675, 127722), (3697, 597527), (3700, 597556), (3703, 597738), (3711, 596349), (3717, 34551), (3718, 34356), (3723, 7560), (3764, 445117), (3765, 445154), (3793, 292448), (3798, 292446), (3799, 292443), (3823, 112305), (3824, 112517), (3831, 108038), (3836, 110185), (3846, 585232), (3848, 585544), (3851, 585519), (3854, 585566), (3855, 584997), (3866, 29610), (3869, 28871), (3870, 28930), (3875, 5946), (3886, 523387), (3890, 442981), (3895, 442754), (3899, 442953), (3924, 283515), (3928, 282712), (3930, 282243), (3931, 281904), (3934, 282251), (3943, 282871), (3946, 283322), (3962, 98196), (3967, 93948), (3969, 94721), (3972, 93843), (3975, 94624), (3976, 95280), (3981, 574884), (3985, 574262), (3991, 24223), (4008, 522622), (4033, 438004), (4045, 274849), (4048, 273916), (4056, 86778), (4060, 87169), (4061, 82704), (4065, 85225), (4068, 86228), (4076, 564955), (4077, 565284), (4078, 565273), (4097, 521909), (4110, 435745), (4119, 269275), (4125, 268804), (4126, 269009), (4135, 77741), (4137, 76994), (4138, 77044), (4139, 75065), (4165, 434943), (4170, 264166), (4172, 264772), (4184, 69823), (4191, 521209), (4197, 521161), (4200, 433641), (4206, 433730), (4208, 433608), (4220, 261600), (4222, 261134), (4223, 261536), (4225, 67721), (4239, 520906), (4252, 259188), (4261, 431894), (4270, 257604)]
The corresponding toric Fanos (for each polytope) have degrees: [14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14]
Each toric Fano has Hilbert series: 1 + 10*x + 40*x^2 + 105*x^3 + 219*x^4 + 396*x^5 + 650*x^6 + 995*x^7 + 1445*x^8 + 2014*x^9 + 2716*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 32, 312, 5520, 91680, 1651640, 30604560, 583436560, 11352768000, 224645958432, 4506386808000, 91434963225144, 1873214113234464, 38695098186167280, 805079741247573792, 16855708323075233040, 354862100179318727424, 7507744262279847981824, 159540694369215435619200]
==================================================
PF operator:

-165888*t^4*D^3 - 995328*t^4*D^2 - 39744*t^3*D^3 - 1824768*t^4*D - 178848*t^3*D^2 - 3024*t^2*D^3 - 995328*t^4 - 258336*t^3*D - 9072*t^2*D^2 - 60*t*D^3 - 119232*t^3 - 9216*t^2*D - 90*t*D^2 + D^3 - 3168*t^2 - 30*t*D
Connection matrix:

[     0    792  43632 793152]
[     1     30   2340  43632]
[     0      1     30    792]
[     0      0      1      0]

This is the genus-4 Fano with b_2=1.

This occurs for the following polytopes (PALP id, grdb id):

[(4281, 547396), (4283, 544447), (4285, 520428), (4286, 520416), (4296, 520331), (4309, 520190), (4317, 547387)]
The corresponding toric Fanos (for each polytope) have degrees: [6, 6, 6, 6, 6, 6, 6]
Each toric Fano has Hilbert series: 1 + 6*x + 20*x^2 + 49*x^3 + 99*x^4 + 176*x^5 + 286*x^6 + 435*x^7 + 629*x^8 + 874*x^9 + 1176*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 396, 17616, 1217052, 85220640, 6349812480, 490029523200, 38883641777820, 3152020367254080, 259917911904104496, 21734333812757966400, 1838661488140000620096, 157079889830458865926656, 13532851175260662410660352, 1174404851170539753963534336, 102567197002978827892527466140, 9008119389478435783966748566656, 795101313792044241851836759258896, 70492745075535384957296839185342912]
==================================================
PF operator:

-1024*t^4*D^3 - 6144*t^4*D^2 - 768*t^3*D^3 - 11264*t^4*D - 3456*t^3*D^2 - 176*t^2*D^3 - 6144*t^4 - 4992*t^3*D - 528*t^2*D^2 - 8*t*D^3 - 2304*t^3 - 544*t^2*D - 12*t*D^2 + D^3 - 192*t^2 - 4*t*D
Connection matrix:

[   0   48  576 3328]
[   1    4   96  576]
[   0    1    4   48]
[   0    0    1    0]

This is the genus-9 Fano with b_2=1.

This occurs for the following polytopes (PALP id, grdb id):

[(1953, 546004), (2009, 492045), (2023, 491381), (2092, 388964), (2196, 223136), (2290, 662829), (2413, 482431), (2436, 482420), (2481, 372120), (2497, 372105), (2649, 652773), (2723, 529956), (2772, 473102), (2887, 186783), (2890, 186772), (2893, 186627), (2895, 186614), (2896, 186640), (2898, 186626), (2902, 186306), (2966, 639881), (2968, 639889), (2986, 639750), (2987, 639688), (2989, 639751), (3007, 53123), (3011, 53112), (3012, 53111), (3025, 57234), (3026, 57359), (3031, 11582), (3033, 12197), (3111, 464458), (3128, 338106), (3130, 338225), (3131, 338235), (3161, 335117), (3208, 166302), (3209, 165850), (3226, 166025), (3276, 624084), (3279, 624186), (3293, 47436), (3295, 47424), (3299, 52868), (3308, 11546), (3309, 11569), (3311, 1454), (3338, 526791), (3356, 458619), (3372, 455988), (3373, 456763), (3374, 456807), (3375, 456805), (3395, 321327), (3400, 321353), (3408, 321054), (3454, 145662), (3461, 145189), (3463, 146791), (3472, 146294), (3476, 146798), (3511, 610605), (3512, 610156), (3514, 608808), (3517, 608909), (3518, 608870), (3528, 41156), (3530, 40830), (3531, 40767), (3532, 41052), (3537, 41139), (3541, 9091), (3545, 10501), (3600, 450189), (3601, 449940), (3613, 449941), (3641, 299999), (3647, 301745), (3663, 129141), (3673, 126173), (3684, 126035), (3685, 125153), (3686, 125313), (3699, 597548), (3702, 597551), (3708, 595052), (3709, 595268), (3710, 595226), (3719, 34364), (3720, 40408), (3724, 1223), (3741, 524265), (3758, 446950), (3782, 445152), (3796, 293345), (3797, 292762), (3801, 292681), (3802, 292451), (3815, 288486), (3827, 112423), (3829, 108209), (3832, 109121), (3835, 111932), (3850, 585038), (3853, 584967), (3857, 579004), (3859, 580257), (3863, 582522), (3864, 582645), (3871, 28869), (3872, 28187), (3903, 443014), (3913, 442583), (3933, 283208), (3938, 281831), (3944, 282143), (3953, 278160), (3971, 94626), (3974, 93844), (3988, 569725), (3989, 568994), (3992, 23122), (3997, 544752), (4015, 522325), (4044, 275383), (4049, 272798), (4062, 82771), (4064, 85219), (4066, 85150), (4071, 86196), (4108, 436835), (4120, 268442), (4136, 76930), (4145, 555757), (4149, 521466), (4151, 521506), (4162, 435181), (4163, 434689), (4164, 434819), (4171, 264649), (4174, 264553), (4226, 67006), (4263, 431653), (4301, 520292)]
The corresponding toric Fanos (for each polytope) have degrees: [16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16]
Each toric Fano has Hilbert series: 1 + 11*x + 45*x^2 + 119*x^3 + 249*x^4 + 451*x^5 + 741*x^6 + 1135*x^7 + 1649*x^8 + 2299*x^9 + 3101*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 24, 192, 2904, 40320, 611520, 9515520, 152412120, 2491104000, 41404233024, 697598415360, 11887922751936, 204543454123008, 3548536418059776, 62004013026988032, 1090207952792089560, 19275174299698656768, 342468223449089584704, 6111542317740696764928]
==================================================
PF operator:

144*t^4*D^3 + 864*t^4*D^2 + 1584*t^4*D - 40*t^2*D^3 + 864*t^4 - 120*t^2*D^2 - 128*t^2*D + D^3 - 48*t^2
Connection matrix:

[ 0 12  0  0]
[ 1  0 16  0]
[ 0  1  0 12]
[ 0  0  1  0]

This is .

This occurs for the following polytopes (PALP id, grdb id):

[(17, 544342), (30, 520140), (121, 519664), (155, 430096)]
The corresponding toric Fanos (for each polytope) have degrees: [48, 48, 48, 48]
Each toric Fano has Hilbert series: 1 + 27*x + 125*x^2 + 343*x^3 + 729*x^4 + 1331*x^5 + 2197*x^6 + 3375*x^7 + 4913*x^8 + 6859*x^9 + 9261*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 6, 0, 90, 0, 1860, 0, 44730, 0, 1172556, 0, 32496156, 0, 936369720, 0, 27770358330, 0, 842090474940, 0]
==================================================
PF operator:

-2160*t^4*D^3 - 12960*t^4*D^2 - 1728*t^3*D^3 - 23760*t^4*D - 7776*t^3*D^2 - 360*t^2*D^3 - 12960*t^4 - 11232*t^3*D - 1080*t^2*D^2 - 16*t*D^3 - 5184*t^3 - 1152*t^2*D - 24*t*D^2 + D^3 - 432*t^2 - 8*t*D
Connection matrix:

[    0   108  1728 13824]
[    1     8   208  1728]
[    0     1     8   108]
[    0     0     1     0]

??? 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 occurs for the following polytopes (PALP id, grdb id):

[(2816, 355616), (3328, 545072), (3349, 525553), (3392, 321879), (3446, 147470), (3504, 610803), (3739, 524375), (3790, 294031), (3794, 292940), (3843, 585895), (3844, 585890), (3845, 585897), (3867, 29624), (3873, 5953), (3874, 5954), (3966, 95245), (4026, 439663), (4057, 87167), (4074, 566716), (4075, 566695), (4168, 264855), (4181, 72684), (4182, 72202), (4240, 520890), (4248, 432464)]
The corresponding toric Fanos (for each polytope) have degrees: [12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12]
Each toric Fano has Hilbert series: 1 + 9*x + 35*x^2 + 91*x^3 + 189*x^4 + 341*x^5 + 559*x^6 + 855*x^7 + 1241*x^8 + 1729*x^9 + 2331*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 54, 672, 15642, 336960, 7919460, 191177280, 4751272890, 120527514240, 3109617573804, 81336932550720, 2152021034457756, 57492903382282368, 1548779408915134968, 42023349488330001792, 1147430010272379786810, 31504595489224640310528, 869291145239891056320924, 24092072648449534765168704]
==================================================
PF operator:

-64*t^2*D^3 - 192*t^2*D^2 - 192*t^2*D + D^3 - 64*t^2
Connection matrix:

[  0  16   0 256]
[  1   0  32   0]
[  0   1   0  16]
[  0   0   1   0]

This is .

This occurs for the following polytopes (PALP id, grdb id):

[(8, 547367), (91, 544065), (119, 519672), (153, 430103), (197, 255744), (428, 547516), (432, 547285), (433, 547298)]
The corresponding toric Fanos (for each polytope) have degrees: [32, 32, 32, 32, 32, 32, 32, 32]
Each toric Fano has Hilbert series: 1 + 19*x + 85*x^2 + 231*x^3 + 489*x^4 + 891*x^5 + 1469*x^6 + 2255*x^7 + 3281*x^8 + 4579*x^9 + 6181*x^10 + O(x^11)
The first 20 coefficients of the period are:

[1, 0, 8, 0, 216, 0, 8000, 0, 343000, 0, 16003008, 0, 788889024, 0, 40424237568, 0, 2131746903000, 0, 114933031928000, 0]
==================================================
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4 Comments

Sergey says:

May 17, 2010 at 8:53 pm

?? currently unknown: conjecturally a G-Fano, arising as a piece of the quantum cohomology of a 2:1 cover of $\PP^1 \times \PP^1 \times \PP^1$ 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
$\PP_{\PP^1 \times \PP^1 \times \PP^1} (O \oplus O(1,1,1))$

Some other notes.

Denote $\PP^1 \times \PP^1 \times \PP^1$ by UU,
double cover of UU branched in divisor of degree (2,,2,2) by U,
divisor of bidegree (1,1) in $\PP^2 \times \PP^2$ by W,
divisor of bidegree (2,2) in $\PP^2 \times \PP^2$ 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”.
Sergei says:

May 17, 2010 at 11:56 pm

On $Y_{20}$ .

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 $Y_{20} = V_{2.12}$
with $\rho=2, deg=20$ and $h^{1,2} = 3$ .

There are exactly 8 families containing G-Fano threefolds. 7 were already listed. The last one is $Y_{20}$ – it is a blowup of $\PP^3$ 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 $(1,1)$ on $\PP^3 \times \PP^3$ , so also can be done by quantum Lefschetz.

$I_{Y_{20}}^r(t) = \sum_{a,b \geq 0} \frac{(a+b)!^4}{a!^4 b!^4} t^{a+b} = 1 + 2 t + 18 t^2 + 164 t^3 + 1810 t^4 + 21252 t^5 + \cdots$ ,

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

$I_{Y_{20}}^{r,0}(t) = 1 + 14 t^2 + 72 t^3 + 882 t^4 + 8400 t^5 + \cdots$
Sergey says:

May 18, 2010 at 11:42 am

On $Y_{30}$ .

It is intersection of divisors of polydegrees
(1,1,0),(1,0,1) and (0,1,1) on $P^2 \times P^2 \times P^2$ .

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

$I_{Y_{30}}^r(t) = \sum_{a,b,c \geq 0} \frac{(a+b)!(a+c)!(b+c)!(a+b+c)!}{a!^3 b!^3 c!^3} t^{a+b+c} = 1 + 3 t + 15 t^2 + 105 t^3 + 855 t^4 + 7533 t^5 + \cdots$

after the shift of Givental’s constant to $0$ it becomes

$I_{Y_{30}}^{r,0}(t) = 1 + 6 t^2 + 24 t^3 + 162 t^4 + 1080 t^5 + \cdots$
Sergey says:

September 19, 2010 at 6:45 am

On $U_{12}$ :

Cone F over Segre variety $\PP^1 \times \PP^1 \times \PP^1 \subset \PP^7$
has index 3, and its mirror is isogeneous to
$w = u (x+y+z+1/x+1/y+1/z) + 1/u^2$

So its regularized I-series are
$I_F^r = \sum_n i_n t^{3n} = 1 + 18 t^3 + 1350 t^6 + 156240 t^9 + 22141350 t^{12} + 3521185668 t^{15} + 603258639984 t^{18} + 108881071041600 t^{21} + \dots$

Then $I_U^r = \sum i_n \cdot \frac{n! (2n)!}{(3n)!} = 1 + 6 t + 90 t^2 + 1860 t^3 + 44730 t^4 + 1172556 t^5 + 32496156 t^6 +\cdots$

Normalizing gives period sequence:
$I_U^{r,0} = 1 + 54 t^2 + 672 t^3 + 15642 t^4 + 336960 t^5 + 7919460 t^6 +$