an overview of the literature listing Fano fourfolds
Szurek, M., & Wiśniewski, J. A. (1990). Fano bundles over P3 and Q₃. Pacific J. Math., 141(1), 197–208.
| count | 5 cases |
|---|---|
| construction | projective-bundle |
| Picard rank $\rho_X$ | 2 |
| entry numbering | case in Theorem 2.1 |
| invariants | construction |
| links | journal MR1028270 |
| notes | Rank-2 Fano bundles over P³ (5 cases: 4 split + the null-correlation bundle); their ℙ¹-bundles are Fano fourfolds. Companion classification over Q³. |
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 |
|---|---|---|
| SW90a:1 | 2 | $\mathcal{E}=\mathcal{O}\oplus\mathcal{O}$ over $\mathbb{P}^3$ ($c_1=0$); $\mathbb{P}(\mathcal{E})=\mathbb{P}^3\times\mathbb{P}^1$ |
| SW90a:2 | 2 | $\mathcal{E}=\mathcal{O}\oplus\mathcal{O}(-1)$ over $\mathbb{P}^3$ ($c_1=-1$) |
| SW90a:3 | 2 | $\mathcal{E}=\mathcal{O}(-1)\oplus\mathcal{O}(1)$ over $\mathbb{P}^3$ ($c_1=0$) |
| SW90a:4 | 2 | $\mathcal{E}=\mathcal{O}(-2)\oplus\mathcal{O}(1)$ over $\mathbb{P}^3$ ($c_1=-1$) |
| SW90a:5 | 2 | $\mathcal{E}=$ null-correlation bundle over $\mathbb{P}^3$ (stable, $c_1=0$, $c_2=1$) |
Bundle classification (Theorem 2.1) from Pacific J. Math. 141 (1990); P(E) is the Fano fourfold. No per-case numerical invariants are tabulated in the paper.