an overview of the literature listing Fano fourfolds
Küchle, O. (1995). On Fano 4-fold of index 1 and homogeneous vector bundles over Grassmannians. Math. Z., 218(4), 563–575.
| count | 21 families |
|---|---|
| construction | zero-locus-grassmannian |
| Picard rank $\rho_X$ | 1–8 |
| Fano index $r_X$ | 1 |
| entry numbering | Küchle label (b1–d3) |
| invariants | construction, hodge, anticanonical, index |
| links | 10.1007/BF02571923 MR1326986 |
| notes | Küchle's Theorem 3.1 enumerates 21 Fano-index-1 families of zero loci of homogeneous vector bundles on Grassmannians, labelled b1–d3. We tabulate 20 of them (all but d1) with invariants computed with PartialFlagVarieties.jl; c4, d2, d3 are recovered as the Fano fourfold whose anticanonical divisor is the corresponding Küchle Calabi–Yau threefold. The classically-cited count of 9 (8 up to deformation) refers to the genuinely new prime fourfolds among these. |
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 | $\rho_X$ | index | $\mathrm{h}^0(-\mathrm{K}_X)$ | $e(X)$ | $\mathrm{h}^{1,1}$ | $\mathrm{h}^{1,2}$ | $\mathrm{h}^{1,3}$ | $\mathrm{h}^{2,2}$ | description |
|---|---|---|---|---|---|---|---|---|---|
| Kuc95:b1 | 1 | 1 | 9 | 318 | 1 | 0 | 41 | 232 | $\mathcal{O}(3)\oplus\mathcal{O}(1)$ on $\mathrm{Gr}(2,5)$ |
| Kuc95:b2 | 1 | 1 | 10 | 176 | 1 | 0 | 20 | 132 | $\mathcal{O}(2)^{\oplus 2}$ on $\mathrm{Gr}(2,5)$ |
| Kuc95:b3 | 1 | 1 | 15 | 73 | 1 | 0 | 6 | 57 | $\wedge^3\mathcal{Q}^\vee\otimes\mathcal{O}(2)$ on $\mathrm{Gr}(2,6)$ |
| Kuc95:b4 | 2 | 1 | 15 | 152 | 2 | 0 | 16 | 114 | $\mathrm{Sym}^2\mathcal{S}^\vee\oplus\mathcal{O}(2)$ on $\mathrm{Gr}(2,6)$ |
| Kuc95:b5 | 1 | 1 | 13 | 90 | 1 | 0 | 8 | 70 | $\mathcal{O}(1)^{\oplus 2}\oplus\mathcal{S}^\vee(1)$ on $\mathrm{Gr}(2,6)$ |
| Kuc95:b6 | 1 | 1 | 12 | 140 | 1 | 0 | 15 | 106 | $\mathcal{O}(1)^{\oplus 3}\oplus\mathcal{O}(2)$ on $\mathrm{Gr}(2,6)$ |
| Kuc95:b7 | 1 | 1 | 15 | 73 | 1 | 0 | 6 | 57 | $\mathcal{O}(1)^{\oplus 6}$ on $\mathrm{Gr}(2,7)$ |
| Kuc95:b8 | 1 | 1 | 18 | 48 | 1 | 0 | 3 | 38 | $\mathcal{O}(1)^{\oplus 3}\oplus\mathrm{Sym}^2\mathcal{S}^\vee$ on $\mathrm{Gr}(2,7)$ |
| Kuc95:b9 | 8 | 1 | 21 | 48 | 8 | 0 | 0 | 30 | $(\mathrm{Sym}^2\mathcal{S}^\vee)^{\oplus 2}$ on $\mathrm{Gr}(2,7)$ |
| Kuc95:b10 | 1 | 1 | 14 | 132 | 1 | 0 | 14 | 100 | $\mathcal{O}(2)\oplus\mathcal{Q}^\vee(1)$ on $\mathrm{Gr}(2,7)$ |
| Kuc95:b11 | 1 | 1 | 18 | 39 | 1 | 0 | 2 | 31 | $\mathcal{O}(1)^{\oplus 2}\oplus\mathcal{Q}^\vee(1)$ on $\mathrm{Gr}(2,8)$ |
| Kuc95:c1 | 1 | 1 | 15 | 66 | 1 | 0 | 5 | 52 | $\mathcal{O}(1)^{\oplus 5}$ on $\mathrm{Gr}(3,6)$ |
| Kuc95:c2 | 1 | 1 | 13 | 132 | 1 | 0 | 14 | 100 | $\wedge^2\mathcal{S}^\vee\oplus\mathcal{O}(1)\oplus\mathcal{O}(2)$ on $\mathrm{Gr}(3,6)$ |
| Kuc95:c3 | 1 | 1 | 21 | 19 | 1 | 0 | 0 | 15 | $\mathcal{Q}^\vee(1)^{\oplus 2}$ on $\mathrm{Gr}(3,7)$ |
| Kuc95:c4 | 1 | 1 | 33 | 48 | 1 | 0 | 3 | 38 | $\mathrm{Sym}^2\mathcal{S}^\vee\oplus\mathcal{O}(1)^{\oplus 2}$ on $\mathrm{Gr}(3,7)$ |
| Kuc95:c5 | 1 | 1 | 20 | 30 | 1 | 0 | 1 | 24 | $\wedge^2\mathcal{S}^\vee\oplus\mathcal{O}(1)\oplus\mathcal{Q}^\vee(1)$ on $\mathrm{Gr}(3,7)$; K3 type |
| Kuc95:c6 | 1 | 1 | 19 | 38 | 1 | 0 | 2 | 30 | $(\wedge^2\mathcal{S}^\vee)^{\oplus 2}\oplus\mathcal{O}(1)^{\oplus 2}$ on $\mathrm{Gr}(3,7)$ |
| Kuc95:c7 | 2 | 1 | 27 | 30 | 2 | 0 | 1 | 22 | $\mathcal{O}(1)\oplus(\wedge^2\mathcal{Q}^\vee\otimes\mathcal{O}(1))$ on $\mathrm{Gr}(3,8)$; K3 type |
| Kuc95:d2 | 4 | 1 | 81 | 16 | 4 | 0 | 0 | 6 | $\mathrm{Sym}^2\mathcal{S}^\vee\oplus\wedge^2\mathcal{S}^\vee$ on $\mathrm{Gr}(4,9)$ |
| Kuc95:d3 | 5 | 1 | 31 | 40 | 5 | 0 | 1 | 26 | $(\wedge^2\mathcal{S}^\vee)^{\oplus 2}\oplus\mathcal{O}(1)$ on $\mathrm{Gr}(5,10)$ |
| Kuc95:d1 | 1 | family d1 (listed by Küchle, Theorem 3.1); bundle data unavailable in the tooling, not computed here |
Numerical invariants computed with PartialFlagVarieties.jl, reproducing Küchle (Math. Z. 218, 1995), Theorem 3.1. anticanonical = h^0(-K); euler = topological Euler characteristic. 20 of the 21 index-1 families are tabulated; d1 is listed but not computed (no bundle data). The classically-cited 9 (8 up to deformation) are the genuinely new prime fourfolds among these.