an overview of the literature listing Fano fourfolds
Bonavero, L., Campana, F., & Wiśniewski, J. A. (2002). Variétés complexes dont l’éclatée en un point est de Fano. C. R. Math. Acad. Sci. Paris, 334(6), 463–468.
| construction | contraction-theory |
|---|---|
| invariants | contractions |
| links | 10.1016/S1631-073X(02)02284-7 MR1890634 |
| notes | If the blow-up of a point in X is Fano, then X is ℙ⁴, Q⁴, or the blow-up of ℙ⁴ along a smooth surface of degree ≤ 4 contained in a hyperplane. |
| case | exceptional locus | contraction / result | notes |
|---|---|---|---|
| Projective space | blow-up at any point $a \in X$ | $X \cong \mathbb{P}^4$ | The point $a$ is arbitrary. |
| Quadric | blow-up at any point $a \in X$ | $X \cong \mathbb{Q}^4$ | The point $a$ is arbitrary. Corollaire 1: $\mathbb{Q}^n$ is the only $X$ whose blow-up at two distinct points (not on a common line of the quadric) is again Fano. |
| Blow-up of $\mathbb{P}^4$ along a surface in a hyperplane | blow-up at a point $a \notin H$ | $X \cong V_d$, the blow-up of $\mathbb{P}^4$ along a smooth surface $Y$ (dimension $n-2 = 2$) of degree $d$ | The surface $Y$ lies in a hyperplane $H \subset \mathbb{P}^4$, the blown-up point $a$ is off $H$, and $1 \le d \le n = 4$. |
Bonavero-Campana-Wiśniewski, Théorème 1 (arXiv:math/0106047, CRAS 334 (2002) 463-468). Classification of smooth connected complex projective $X$ ($\dim X = n \ge 3$) such that the blow-up $\tilde X \to X$ at a point $a$ is Fano; items specialized to $n=4$. Full arXiv text used.