an overview of the literature listing Fano fourfolds
Secci, S. A. (2023). Fano fourfolds having a prime divisor of Picard number 1. Adv. Geom., 23(2), 267–280.
| count | 28 families |
|---|---|
| construction | blowup |
| Picard rank $\rho_X$ | 3 |
| entry numbering | family number |
| invariants | hodge, volume, numerical, construction |
| links | 10.1515/advgeom-2023-0002 MR4596223 |
| notes | 28 families of Fano fourfolds of Picard number 3 with a prime divisor of Picard number 1, namely blow-ups of a ℙ¹-bundle over a smooth Fano of Picard number 1 along a codimension-2 subvariety. Numerical invariants and Hodge numbers provided. |
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 | $\rho_X$ | $(-\mathrm{K}_X)^4$ | $\mathrm{h}^0(-\mathrm{K}_X)$ | $\mathrm{h}^{1,1}$ | $\mathrm{h}^{1,2}$ | $\mathrm{h}^{1,3}$ | $\mathrm{h}^{2,2}$ | description |
|---|---|---|---|---|---|---|---|---|
| Sec23:X^1_{0,1} | 3 | 47 | 17 | 3 | 21 | 0 | 11 | conic bundle over Z_1 (sextic in P(1,1,1,2,3)), (a,d)=(0,1) |
| Sec23:X^1_{1,2} | 3 | 30 | 13 | 3 | 21 | 1 | 22 | over Z_1, (a,d)=(1,2) |
| Sec23:X^2_{0,1} | 3 | 94 | 26 | 3 | 10 | 0 | 10 | over Z_2 (double cover of P^3 branched in a quartic), (a,d)=(0,1) |
| Sec23:X^2_{1,2} | 3 | 60 | 19 | 3 | 10 | 1 | 22 | over Z_2, (a,d)=(1,2) |
| Sec23:X^3_{0,1} | 3 | 141 | 35 | 3 | 5 | 0 | 9 | over Z_3 (cubic in P^4), (a,d)=(0,1) |
| Sec23:X^3_{1,2} | 3 | 90 | 25 | 3 | 5 | 1 | 22 | over Z_3, (a,d)=(1,2) |
| Sec23:X^4_{0,1} | 3 | 188 | 44 | 3 | 2 | 0 | 8 | over Z_4 (intersection of two quadrics in P^5), (a,d)=(0,1) |
| Sec23:X^4_{1,2} | 3 | 120 | 31 | 3 | 2 | 1 | 22 | over Z_4, (a,d)=(1,2) |
| Sec23:X^5_{0,1} | 3 | 235 | 53 | 3 | 0 | 0 | 7 | over Z_5 (codim-3 linear section of Gr(2,5)), (a,d)=(0,1) |
| Sec23:X^5_{1,2} | 3 | 150 | 37 | 3 | 0 | 1 | 22 | over Z_5, (a,d)=(1,2) |
| Sec23:X^6_{0,1} | 3 | 346 | 74 | 3 | 0 | 0 | 4 | over Z_6 (quadric in P^4), (a,d)=(0,1) |
| Sec23:X^6_{0,2} | 3 | 296 | 65 | 3 | 0 | 0 | 8 | over Z_6, (a,d)=(0,2) |
| Sec23:X^6_{1,2} | 3 | 260 | 58 | 3 | 0 | 0 | 8 | over Z_6, (a,d)=(1,2) |
| Sec23:X^6_{1,3} | 3 | 210 | 49 | 3 | 0 | 1 | 22 | over Z_6, (a,d)=(1,3) |
| Sec23:X^6_{2,1} | 3 | 430 | 90 | 3 | 0 | 0 | 4 | over Z_6, (a,d)=(2,1) |
| Sec23:X^6_{2,4} | 3 | 160 | 40 | 3 | 0 | 5 | 54 | over Z_6, (a,d)=(2,4) |
| Sec23:X^7_{0,1} | 3 | 431 | 90 | 3 | 0 | 0 | 3 | over Z_7 = P^3, (a,d)=(0,1); toric, = Batyrev E_3 |
| Sec23:X^7_{0,2} | 3 | 376 | 80 | 3 | 0 | 0 | 4 | over P^3, (a,d)=(0,2) |
| Sec23:X^7_{0,3} | 3 | 341 | 74 | 3 | 0 | 0 | 9 | over P^3, (a,d)=(0,3) |
| Sec23:X^7_{1,2} | 3 | 350 | 75 | 3 | 0 | 0 | 4 | over P^3, (a,d)=(1,2) |
| Sec23:X^7_{1,3} | 3 | 295 | 65 | 3 | 0 | 0 | 9 | over P^3, (a,d)=(1,3) |
| Sec23:X^7_{1,4} | 3 | 260 | 59 | 3 | 0 | 1 | 22 | over P^3, (a,d)=(1,4) |
| Sec23:X^7_{2,1} | 3 | 489 | 101 | 3 | 0 | 0 | 3 | over P^3, (a,d)=(2,1); toric, = Batyrev E_2 |
| Sec23:X^7_{2,4} | 3 | 240 | 55 | 3 | 0 | 1 | 22 | over P^3, (a,d)=(2,4) |
| Sec23:X^7_{2,5} | 3 | 205 | 49 | 3 | 0 | 4 | 47 | over P^3, (a,d)=(2,5) |
| Sec23:X^7_{3,1} | 3 | 605 | 123 | 3 | 0 | 0 | 3 | over P^3, (a,d)=(3,1); toric, = Batyrev E_1 |
| Sec23:X^7_{3,2} | 3 | 454 | 95 | 3 | 0 | 0 | 4 | over P^3, (a,d)=(3,2) |
| Sec23:X^7_{3,6} | 3 | 170 | 43 | 3 | 0 | 10 | 88 | over P^3, (a,d)=(3,6) |
Invariants from Table 2 of arXiv:2103.16140. Base threefolds Z_1..Z_7 as in the paper's Table 1. Families X^7_{0,1}, X^7_{2,1}, X^7_{3,1} coincide with Batyrev's toric E_3, E_2, E_1.