an overview of the literature listing Fano fourfolds
Casagrande, C., Romano, E. A., & Secci, S. A. (2022). Fano manifolds with Lefschetz defect 3. J. Math. Pures Appl. (9), 163, 625–653.
| count | 19 families |
|---|---|
| construction | lefschetz-defect-classification |
| Picard rank $\rho_X$ | 5–8 |
| Lefschetz defect $\delta_X$ | 3 |
| entry numbering | case number |
| invariants | lefschetz_defect, picard_rank, volume, anticanonical, hodge, chi_T, construction |
| links | 10.1016/j.matpur.2022.05.016 MR4438911 |
| notes | Completes the classification. The final (post-corrigendum) count is 19 families with 5 <= Picard number <= 8, of which 5 are non-toric fourfolds: 2 at Picard number 5 and 3 at Picard number 6 (the original published version had 18 families / 4 non-toric; the corrigendum, MR4515254, added one). The Picard-5 families are tabulated in the companion paper arXiv:2007.11229. |
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 | $\rho_X$ | $\delta_X$ | $(-\mathrm{K}_X)^4$ | $\mathrm{h}^0(-\mathrm{K}_X)$ | $\chi(T_X)$ | $\mathrm{h}^{1,1}$ | $\mathrm{h}^{1,2}$ | $\mathrm{h}^{1,3}$ | $\mathrm{h}^{2,2}$ | description |
|---|---|---|---|---|---|---|---|---|---|---|
| CRS22:X_B0 | 6 | 3 | 224 | 51 | 4 | 6 | 0 | 0 | 10 | non-toric, rho=6: P^1 x Y, Y = blow-up of P^3 along a line, a disjoint conic, and two fibres over the line |
| CRS22:X_B1 | 6 | 3 | 222 | 51 | -2 | 6 | 0 | 0 | 14 | non-toric, rho=6: blow-up of P_{P1xP1}(O(1,1)+O+O) along three disjoint surfaces (Construction B, T=P1xP1, N in |O(1,1)|) |
| CRS22:X_B2 | 6 | 3 | 223 | 51 | 1 | 6 | 0 | 0 | 12 | non-toric, rho=6 (added by the corrigendum): Construction B from T=F_1, N = pullback of a general line in P^2 |
| CRS22:Ex1 | 5 | 3 | 253 | 57 | 3 | 5 | 0 | 0 | 7 | non-toric, rho=5: blow-up of P^4 along a line pq, the fibres over p,q, and the transform of a disjoint quadric surface |
| CRS22:Ex2 | 5 | 3 | 250 | 57 | -6 | 5 | 0 | 0 | 13 | non-toric, rho=5: blow-up of P_{P2}(O+O+O(2)) along a degree-2 del Pezzo surface and two fibres of the exceptional divisor |
| CRS22:K1 | 5 | 3 | 364 | 78 | 10 | 5 | 0 | 0 | 6 | toric, rho=5 (companion paper) |
| CRS22:K2 | 5 | 3 | 354 | 76 | 10 | 5 | 0 | 0 | 6 | toric, rho=5 (companion paper) |
| CRS22:K3 | 5 | 3 | 334 | 72 | 10 | 5 | 0 | 0 | 6 | toric, rho=5 (companion paper) |
| CRS22:K4 | 5 | 3 | 324 | 70 | 10 | 5 | 0 | 0 | 6 | toric, rho=5: P^2 x S_4, S_4 = blow-up of P^2 at 3 non-collinear points |
Non-toric rho=6 from arXiv:2201.02413 (the corrigendum-included v2); rho=5 families from the companion arXiv:2007.11229. Toric rho=6 (U_1..U_8) and the rho=7,8 products are not tabulated with invariants.