an overview of the literature listing Fano fourfolds
Takagi, H. (1999). Classification of extremal contractions from smooth fourfolds of (3,1)-type. Proc. Amer. Math. Soc., 127(2), 315–321.
| construction | contraction-theory |
|---|---|
| invariants | contractions |
| links | 10.1090/S0002-9939-99-05114-X MR1637436 |
| notes | For (3,1)-type contractions the exceptional divisor is a ℙ²-bundle or quadric bundle over a smooth curve, and the contraction is the blow-up along that curve. |
| case | exceptional locus | contraction / result | notes |
|---|---|---|---|
| (3,1)-type, $\mathbb{P}^2$-bundle, general fibre $(\mathbb{P}^2,\mathcal{O}(1))$ | Exceptional divisor $E$ a $\mathbb{P}^2$-bundle over the smooth curve $C = f(E)$; general fibre $(F, -K_X|_F) \cong (\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(1))$, with $\mathcal{O}_X(-E)|_F \cong \mathcal{O}_{\mathbb{P}^2}(1)$. | $f$ is the blow-up of $Y$ along the smooth curve $C$; $Y$ is smooth along $C$. | The classical smooth blow-up of a curve. Here $f|_E$ is a $\mathbb{P}^2$-bundle. |
| (3,1)-type, $\mathbb{P}^2$-bundle, general fibre $(\mathbb{P}^2,\mathcal{O}(2))$ | Exceptional divisor $E$ a $\mathbb{P}^2$-bundle over the smooth curve $C$; general fibre $(F, -K_X|_F) \cong (\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(2))$, with $\mathcal{O}_X(-E)|_F \cong \mathcal{O}_{\mathbb{P}^2}(1)$. | $f$ is the blow-up of $Y$ along the smooth curve $C$; $Y$ is a one-parameter family of $\tfrac{1}{2}(1,1,1)$ quotient singularities along $C$. | The $\mathbb{P}^2$-bundle structure is detected here via $L = \mathcal{O}_E(-E)$ (the case $-K_X|_F = \mathcal{O}(2)$). |
| (3,1)-type, quadric bundle, general fibre $(\mathbb{P}^1\times\mathbb{P}^1,\mathcal{O}(1,1))$ | Exceptional divisor $E$ a quadric bundle (Definition 1.4) over the smooth curve $C$; general fibre $(F, -K_X|_F) \cong (\mathbb{P}^1\times\mathbb{P}^1, \mathcal{O}(1,1))$. Special fibres may be a quadric cone $F_{2,0}$ or a union of two planes in $\mathbb{P}^3$. | $f$ is the blow-up of $Y$ along the smooth curve $C$; $Y$ is locally a hypersurface in $\mathbb{C}^5$ and $C$ a local complete intersection. | Quadric bundle associated to $L = \mathcal{O}_E(-K_X)$; e.g. $Y : x^2+y^2+z^2+w^2=0$. |
| (3,1)-type, quadric bundle, general fibre $(F_{2,0},\mathcal{O}_{\mathbb{P}^3}(1)|_{F_{2,0}})$ | Exceptional divisor $E$ a quadric bundle over the smooth curve $C$; general fibre $(F, -K_X|_F) \cong (F_{2,0}, \mathcal{O}_{\mathbb{P}^3}(1)|_{F_{2,0}})$, the projective quadric cone. | $f$ is the blow-up of $Y$ along the smooth curve $C$; $Y$ is locally a hypersurface in $\mathbb{C}^5$ (e.g. $x^2+y^2+z^2+w^3=0$). | Fourth of the four possibilities for the general fibre in Theorem 1.1; a quadric-bundle case. |
Takagi, 'Classification of extremal contractions from smooth fourfolds of (3,1)-type', Proc. AMS 127 (1999) 315-321; text taken from the arXiv preprint 'On Classification of the extremal contraction from a smooth fourfold' (arXiv:alg-geom/9604013). '(3,1)-type' = divisorial contraction of a smooth 4-fold contracting a divisor (dim 3) onto a curve (dim 1). Main Theorem: C is smooth, f|_E : E -> C is a P^2-bundle or quadric bundle, and f is the blow-up of Y along C. Cases below split by the general fibre F (Thm 1.1).