an overview of the literature listing Fano fourfolds
Szurek, M., & Wiśniewski, J. A. (1990). On Fano manifolds, which are Pk-bundles over P2. Nagoya Math. J., 120, 89–101.
| count | 12 cases |
|---|---|
| construction | projective-bundle |
| Picard rank $\rho_X$ | 2 |
| entry numbering | case number |
| invariants | construction |
| links | 10.1017/S0027763000003275 MR1086572 |
| notes | In dimension four there are 12 cases (ℙ²-bundles over ℙ²). The other extremal contraction is described. |
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 | $\rho_X$ | description |
|---|---|---|
| SW90b:1 | 2 | $(c_1,c_2)=(0,0)$: $\mathcal{E}=\mathcal{O}^{\oplus 3}$; $X=\mathbb{P}^2\times\mathbb{P}^2$ |
| SW90b:2 | 2 | $(c_1,c_2)=(1,0)$: $\mathcal{E}=\mathcal{O}(1)\oplus\mathcal{O}^{\oplus 2}$; $X=\mathrm{Bl}_{\mathbb{P}^2}\mathbb{P}^4$ |
| SW90b:3 | 2 | $(c_1,c_2)=(1,1)$: $\mathcal{E}=T(-1)\oplus\mathcal{O}$; $X$ a divisor of degree $(1,1)$ in $\mathbb{P}^2\times\mathbb{P}^3$ |
| SW90b:4 | 2 | $(c_1,c_2)=(2,0)$: $\mathcal{E}=\mathcal{O}(2)\oplus\mathcal{O}^{\oplus 2}$; $X=$ blow-up of the cone over the Veronese $\mathbb{P}^2$ in $\mathbb{P}^7$ along its vertex |
| SW90b:5 | 2 | $(c_1,c_2)=(2,1)$: $\mathcal{E}=\mathcal{O}(1)^{\oplus 2}\oplus\mathcal{O}$; $X=$ blow-up of the cone over $\mathbb{P}^1\times\mathbb{P}^2$ in $\mathbb{P}^6$ along its vertex |
| SW90b:6 | 2 | $(c_1,c_2)=(2,2)$ special: $\mathcal{E}=\mathcal{O}^{\oplus 2}\oplus E_2$; $X=$ blow-up of a cone over $Q_3$ along a $\mathbb{P}^2$ through the vertex |
| SW90b:7 | 2 | $(c_1,c_2)=(2,2)$ general: $\mathcal{E}=T(-1)\oplus\mathcal{O}(1)$; $X=$ blow-up of a cone over $Q_3$ along a $\mathbb{P}^2$ through the vertex |
| SW90b:8 | 2 | $(c_1,c_2)=(2,3)$ special: $\mathcal{E}=\mathcal{O}\oplus E_2$; $X=$ blow-up of the cone over a twisted cubic in $\mathbb{P}^4$ (scroll $S(0,3)$) |
| SW90b:9 | 2 | $(c_1,c_2)=(2,3)$ general: $\mathcal{E}=E_3^s$ (non-split); $X=\mathrm{Bl}_{S(1,2)}\mathbb{P}^4$ |
| SW90b:10 | 2 | $(c_1,c_2)=(2,4)$: $\mathcal{E}$ in $0\to\mathcal{O}\to\mathcal{O}(-2)\to\mathcal{O}^{\oplus 4}\to\mathcal{E}\to0$; $X$ a divisor of degree $(2,1)$ in $\mathbb{P}^2\times\mathbb{P}^3$ |
Bundle classification (main Theorem, rank 3) from Nagoya Math. J. 120 (1990). 10 printed entries; the paper's count of 12 resolves the (2,4) sub-family further. No per-case numerical invariants are tabulated.