an overview of the literature listing Fano fourfolds
Tsukioka, T. (2010). A remark on Fano 4-folds having (3,1)-type extremal contractions. Math. Ann., 348(3), 737–747.
| construction | contraction-theory |
|---|---|
| invariants | contractions |
| links | 10.1007/s00208-010-0497-3 MR2677902 |
| notes | Characterizes the blow-up of an elliptic quartic curve in ℙ⁴. |
| case | exceptional locus | contraction / result | notes |
|---|---|---|---|
| Fano 4-fold with a $(3,1)$-type extremal contraction (main theorem) | exceptional divisor $F$ of $\varphi$ (contracts a divisor $F$ to a curve $B$), assumed smooth; $E$ is $\varphi$-ample | the blow-up data must be $Y \cong \mathbb{P}^4$ and $C$ a smooth complete intersection of a hyperplane and two hyperquadrics, i.e. an elliptic curve of degree $4$ in $\mathbb{P}^4$ | Setting: $\pi : X \to Y$ is the blow-up of a smooth projective 4-fold $Y$ along a smooth curve $C$, $X$ Fano with an elementary $(3,1)$-type extremal contraction $\varphi : X \to Z$ ($\dim \mathrm{Exc}(\varphi)=3$, image a curve) such that the $\pi$-exceptional divisor $E$ is $\varphi$-ample. Smoothness of $F$ is the key hypothesis. |
| The example realizing the characterization | blow-up of $\mathbb{P}^4$ along the elliptic quartic $C$ | $X = \mathrm{Bl}_C(\mathbb{P}^4)$ has a $(3,1)$-type extremal contraction onto a complete intersection of two hyperquadrics in $\mathbb{P}^6$ (singular along a line) | $C \subset \mathbb{P}^4$ is a smooth complete intersection of one hyperplane and two hyperquadrics. The paper shows this is the unique example when $\mathrm{Exc}(\varphi)$ is smooth. |
Tsukioka, Theorem 1 and the motivating example of §2 (arXiv:0710.1719, Math. Ann. 348 (2010) 737-747). Characterizes the blow-up of an elliptic quartic curve in $\mathbb{P}^4$ among Fano 4-folds carrying a $(3,1)$-type extremal contraction. Full arXiv text used.