an overview of the literature listing Fano fourfolds
Höring, A., & Secci, S. A. (2025). Fano fourfolds with large anticanonical base locus. J. Inst. Math. Jussieu, 24(3), 1021–1051.
| count | 22 families |
|---|---|
| construction | anticanonical-base-locus |
| entry numbering | family number |
| invariants | numerical, base_locus, construction |
| links | 10.1017/S1474748024000604 MR4891410 |
| notes | 22 families with |−K_X| of dimension ≥ 3 and base locus of dimension two; numerical invariants provided. Such manifolds are never strictly Fano. First announced 2023, full classification 2025 (MR4891410). |
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 | $\rho_X$ | $(-\mathrm{K}_X)^4$ | $\mathrm{h}^0(-\mathrm{K}_X)$ | description |
|---|---|---|---|---|
| HS23:1 | ≤4 | 18 | 13 | B=P^2, N*=O+O(2) |
| HS23:2 | 2 | 33 | 16 | B=P^2, N*=O+O(1) |
| HS23:3 | 10 | 54 | 20 | B=P^2, N*=O^2 |
| HS23:4 | ? | 17 | 12 | B=P^2, N*=O(1)^2 |
| HS23:5 | 2 | 32 | 15 | B=P^2, N*=T_{P^2}(-1) |
| HS23:6 | 2 | 16 | 11 | B=P^2, N*=F with 0->O->T_{P^2}(-1)+O(1)->F->0 |
| HS23:7 | 2 | 15 | 10 | B=P^2, N*=F with 0->O(-1)^2->O^4->F->0 |
| HS23:8 | 2 | 14 | 9 | B=P^2, N*=F with 0->O(-2)->O^3->F->0 |
| HS23:9 | 3 | 22 | 13 | B=P^1xP^1, N*=O+O(1,1) |
| HS23:10 | 11 | 48 | 18 | B=P^1xP^1, N*=O^2 |
| HS23:11 | 3 | 32 | 15 | B=P^1xP^1, N*=O+O(1,0) |
| HS23:12 | 3 | 21 | 12 | B=P^1xP^1, N*=O(1,0)+O(0,1) |
| HS23:13 | 3 | 20 | 11 | B=P^1xP^1, N*=F with 0->O(-1,-1)->O^3->F->0 |
| HS23:14 | 3 | 27 | 14 | B=F_1, N*=g*(O+O(1)), g:F_1->P^2 |
| HS23:15 | 11 | 48 | 18 | B=F_1, N*=O^2 |
| HS23:16 | 3 | 26 | 13 | B=F_1, N*=g*T_{P^2}(-1) |
| HS23:17 | 12 | 42 | 16 | B=S_7 (del Pezzo degree 7), N*=O^2 |
| HS23:18 | 13 | 36 | 14 | B=S_6, N*=O^2 |
| HS23:19 | 14 | 30 | 12 | B=S_5, N*=O^2 |
| HS23:20 | 15 | 24 | 10 | B=S_4, N*=O^2 |
| HS23:21 | 16 | 18 | 8 | B=S_3, N*=O^2 |
| HS23:22 | 17 | 12 | 6 | B=S_2, N*=O^2 |
Invariants from Table 1 of arXiv:2510.21216. volume = (-K)^4, anticanonical = h^0(-K). B = the surface blown up, N* its conormal data. No Hodge numbers tabulated.