Fano fourfold literature

an overview of the literature listing Fano fourfolds

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Bat99 Classification of toric Fano fourfolds

Batyrev, V. V. (1999). On the classification of toric Fano 4-folds. In J. Math. Sci. (New York) (Vol. 94, pp. 1021–1050).

count123 families
constructiontoric
entry numberingBatyrev list number
invariantsconstruction, fan, numerical
links10.1007/BF02367245 MR1703904
notes123 smooth toric Fano fourfolds with numerical invariants and geometric models; later completed to 124 (Sato). A machine-readable list of all 124 is Paffenholz's smooth reflexive 4-polytopes (identifiers F.4D.0000 to F.4D.0123); per-variety invariants (Picard rank = #rays − 4, degree, Betti/Hodge) are computed from the fan rather than stored, so they are not transcribed here.

Varieties

Each row is a family from this source. The description column records the construction as given in the paper (ambient and defining bundle, Cox-ring data, base variety, or birational model, depending on the source); the invariant columns are those the paper tabulates. Provenance and conventions are noted below the table.

entry$\rho_X$$(-\mathrm{K}_X)^4$$\mathrm{h}^0(-\mathrm{K}_X)$$\chi(T_X)$$\mathrm{h}^{1,1}$$\mathrm{h}^{1,2}$$\mathrm{h}^{1,3}$$\mathrm{h}^{2,2}$description
Bat99:01625126241001smooth toric Fano fourfold, Picard rank 1
Bat99:12800159362002smooth toric Fano fourfold, Picard rank 2
Bat99:22640129262002smooth toric Fano fourfold, Picard rank 2
Bat99:32544111202002smooth toric Fano fourfold, Picard rank 2
Bat99:42512105182002smooth toric Fano fourfold, Picard rank 2
Bat99:52512105182002smooth toric Fano fourfold, Picard rank 2
Bat99:62594120242003smooth toric Fano fourfold, Picard rank 2
Bat99:72513105182003smooth toric Fano fourfold, Picard rank 2
Bat99:82513105182003smooth toric Fano fourfold, Picard rank 2
Bat99:92486100162003smooth toric Fano fourfold, Picard rank 2
Bat99:103605123233003smooth toric Fano fourfold, Picard rank 3
Bat99:113489101173003smooth toric Fano fourfold, Picard rank 3
Bat99:12343190143003smooth toric Fano fourfold, Picard rank 3
Bat99:133592120243004smooth toric Fano fourfold, Picard rank 3
Bat99:143576117233004smooth toric Fano fourfold, Picard rank 3
Bat99:153560114223004smooth toric Fano fourfold, Picard rank 3
Bat99:163560114223004smooth toric Fano fourfold, Picard rank 3
Bat99:173496102183004smooth toric Fano fourfold, Picard rank 3
Bat99:183496102183004smooth toric Fano fourfold, Picard rank 3
Bat99:193486100183004smooth toric Fano fourfold, Picard rank 3
Bat99:20348099173004smooth toric Fano fourfold, Picard rank 3
Bat99:21346496163004smooth toric Fano fourfold, Picard rank 3
Bat99:22346496163004smooth toric Fano fourfold, Picard rank 3
Bat99:23345995163004smooth toric Fano fourfold, Picard rank 3
Bat99:24344893153004smooth toric Fano fourfold, Picard rank 3
Bat99:25343290143004smooth toric Fano fourfold, Picard rank 3
Bat99:26343290143004smooth toric Fano fourfold, Picard rank 3
Bat99:27343290143004smooth toric Fano fourfold, Picard rank 3
Bat99:28343290143004smooth toric Fano fourfold, Picard rank 3
Bat99:29340585123004smooth toric Fano fourfold, Picard rank 3
Bat99:30340084123004smooth toric Fano fourfold, Picard rank 3
Bat99:31340084123004smooth toric Fano fourfold, Picard rank 3
Bat99:323529108203005smooth toric Fano fourfold, Picard rank 3
Bat99:33345093163005smooth toric Fano fourfold, Picard rank 3
Bat99:34343390143005smooth toric Fano fourfold, Picard rank 3
Bat99:35341787133005smooth toric Fano fourfold, Picard rank 3
Bat99:36340685123005smooth toric Fano fourfold, Picard rank 3
Bat99:37340184123005smooth toric Fano fourfold, Picard rank 3
Bat99:384558114224005smooth toric Fano fourfold, Picard rank 4
Bat99:394505104194005smooth toric Fano fourfold, Picard rank 4
Bat99:40447899174005smooth toric Fano fourfold, Picard rank 4
Bat99:41444793164005smooth toric Fano fourfold, Picard rank 4
Bat99:42441587144005smooth toric Fano fourfold, Picard rank 4
Bat99:43440986134005smooth toric Fano fourfold, Picard rank 4
Bat99:44438281114005smooth toric Fano fourfold, Picard rank 4
Bat99:45437880124005smooth toric Fano fourfold, Picard rank 4
Bat99:46436778114005smooth toric Fano fourfold, Picard rank 4
Bat99:47435175104005smooth toric Fano fourfold, Picard rank 4
Bat99:48448099184006smooth toric Fano fourfold, Picard rank 4
Bat99:49446496174006smooth toric Fano fourfold, Picard rank 4
Bat99:50444893164006smooth toric Fano fourfold, Picard rank 4
Bat99:51443290154006smooth toric Fano fourfold, Picard rank 4
Bat99:52441687144006smooth toric Fano fourfold, Picard rank 4
Bat99:53440084134006smooth toric Fano fourfold, Picard rank 4
Bat99:54438481124006smooth toric Fano fourfold, Picard rank 4
Bat99:55438481124006smooth toric Fano fourfold, Picard rank 4
Bat99:56438481124006smooth toric Fano fourfold, Picard rank 4
Bat99:57438481124006smooth toric Fano fourfold, Picard rank 4
Bat99:58435275104006smooth toric Fano fourfold, Picard rank 4
Bat99:59435275104006smooth toric Fano fourfold, Picard rank 4
Bat99:60435275104006smooth toric Fano fourfold, Picard rank 4
Bat99:614496102194006smooth toric Fano fourfold, Picard rank 4
Bat99:62446396164006smooth toric Fano fourfold, Picard rank 4
Bat99:63444292154006smooth toric Fano fourfold, Picard rank 4
Bat99:64443390164006smooth toric Fano fourfold, Picard rank 4
Bat99:65441587134006smooth toric Fano fourfold, Picard rank 4
Bat99:66441186144006smooth toric Fano fourfold, Picard rank 4
Bat99:67440084134006smooth toric Fano fourfold, Picard rank 4
Bat99:68438481124006smooth toric Fano fourfold, Picard rank 4
Bat99:69439082134006smooth toric Fano fourfold, Picard rank 4
Bat99:70438982124006smooth toric Fano fourfold, Picard rank 4
Bat99:71438481124006smooth toric Fano fourfold, Picard rank 4
Bat99:72434774104006smooth toric Fano fourfold, Picard rank 4
Bat99:73436878114006smooth toric Fano fourfold, Picard rank 4
Bat99:74435776104006smooth toric Fano fourfold, Picard rank 4
Bat99:75433772104006smooth toric Fano fourfold, Picard rank 4
Bat99:76438581124007smooth toric Fano fourfold, Picard rank 4
Bat99:77441787144007smooth toric Fano fourfold, Picard rank 4
Bat99:78436978114007smooth toric Fano fourfold, Picard rank 4
Bat99:79436978114007smooth toric Fano fourfold, Picard rank 4
Bat99:80436477114007smooth toric Fano fourfold, Picard rank 4
Bat99:81436878104007smooth toric Fano fourfold, Picard rank 4
Bat99:8243267084007smooth toric Fano fourfold, Picard rank 4
Bat99:83544292165008smooth toric Fano fourfold, Picard rank 5
Bat99:84540585145008smooth toric Fano fourfold, Picard rank 5
Bat99:85539483135008smooth toric Fano fourfold, Picard rank 5
Bat99:86540585145008smooth toric Fano fourfold, Picard rank 5
Bat99:87537379125008smooth toric Fano fourfold, Picard rank 5
Bat99:88536878125008smooth toric Fano fourfold, Picard rank 5
Bat99:89536377125008smooth toric Fano fourfold, Picard rank 5
Bat99:90535275115008smooth toric Fano fourfold, Picard rank 5
Bat99:91534173105008smooth toric Fano fourfold, Picard rank 5
Bat99:92533672105008smooth toric Fano fourfold, Picard rank 5
Bat99:93533672105008smooth toric Fano fourfold, Picard rank 5
Bat99:94533171105008smooth toric Fano fourfold, Picard rank 5
Bat99:9553307195008smooth toric Fano fourfold, Picard rank 5
Bat99:9653257095008smooth toric Fano fourfold, Picard rank 5
Bat99:9753206995008smooth toric Fano fourfold, Picard rank 5
Bat99:9853106795008smooth toric Fano fourfold, Picard rank 5
Bat99:9952996585008smooth toric Fano fourfold, Picard rank 5
Bat99:100536478105006smooth toric Fano fourfold, Picard rank 5
Bat99:101535476105006smooth toric Fano fourfold, Picard rank 5
Bat99:102533472105006smooth toric Fano fourfold, Picard rank 5
Bat99:103532470105006smooth toric Fano fourfold, Picard rank 5
Bat99:104533271105009smooth toric Fano fourfold, Picard rank 5
Bat99:10553216995009smooth toric Fano fourfold, Picard rank 5
Bat99:10653056685009smooth toric Fano fourfold, Picard rank 5
Bat99:10753106785009smooth toric Fano fourfold, Picard rank 5
Bat99:108630867860010smooth toric Fano fourfold, Picard rank 6
Bat99:109629865860010smooth toric Fano fourfold, Picard rank 6
Bat99:110629865860010smooth toric Fano fourfold, Picard rank 6
Bat99:111628863860010smooth toric Fano fourfold, Picard rank 6
Bat99:112628863860010smooth toric Fano fourfold, Picard rank 6
Bat99:113628863860010smooth toric Fano fourfold, Picard rank 6
Bat99:114627861860010smooth toric Fano fourfold, Picard rank 6
Bat99:115626859860010smooth toric Fano fourfold, Picard rank 6
Bat99:116623051460016smooth toric Fano fourfold, Picard rank 6
Bat99:117530766850011smooth toric Fano fourfold, Picard rank 5
Bat99:118629464860011smooth toric Fano fourfold, Picard rank 6
Bat99:119725256670014smooth toric Fano fourfold, Picard rank 7
Bat99:120821649480018smooth toric Fano fourfold, Picard rank 8
Bat99:12143226984008smooth toric Fano fourfold, Picard rank 4
Bat99:12243277084008smooth toric Fano fourfold, Picard rank 4
Bat99:123524955450012smooth toric Fano fourfold, Picard rank 5

The 124 smooth toric Fano fourfolds (Batyrev + Sato). Invariants computed with Macaulay2 from the fan; ref numbering is the fanobase toric-dataset index (0-123), not Batyrev's original numbering. Toric Fano fourfolds have h12 = h13 = 0.