an overview of the literature listing Fano fourfolds
Andreatta, M., & Wiśniewski, J. A. (1998). On contractions of smooth varieties. J. Algebraic Geom., 7(2), 253–312.
| construction | contraction-theory |
|---|---|
| invariants | contractions |
| links | MR1620110 |
| notes | General theory of contractions of smooth varieties. Part II (computations and applications) is MR1638131. |
| case | exceptional locus | contraction / result | notes |
|---|---|---|---|
| small (non-divisorial) birational, dim 4 | Exceptional locus $E = F \cong \mathbb{P}^2$ with normal bundle $N_{F/X} = \mathcal{O}(-1)\oplus\mathcal{O}(-1)$ (isolated $2$-dimensional fibre). | The contraction is small and its flip exists (recovering the case of Kawamata [Ka1]). | Blow-up of $X$ at $F$ equals blow-up of $Z$ at $z$; the induced map is an isomorphism exactly in this small case. |
| divisorial birational, dim 4, fibre $\mathbb{P}^2$ (type I) | Isolated $2$-dimensional fibre $F \cong \mathbb{P}^2$ with conormal bundle $N^*_{F/X} \cong T_{\mathbb{P}^2}(-1)\oplus\mathcal{O}(1)$ (mod $\mathcal{O}$); trivial scheme fibre structure $I_F = \varphi^{-1}(m_z)\mathcal{O}_X$. | Outside $F$, $\varphi$ is a simple blow-down of the divisor $E$ onto the surface $S=\varphi(E)\subset Z$. $Z$ has a cone-over-$\mathbb{Q}^3$ singularity at $z$; $S$ is smooth at $z$. | One of the three divisorial subcases from the intro summary table / Thm 5.7.6. |
| divisorial birational, dim 4, fibre $\mathbb{P}^2$ (type II) | Isolated $2$-dimensional fibre $F \cong \mathbb{P}^2$ with conormal bundle $N^*_{F/X} \cong \mathcal{O}^{\oplus 4}/\mathcal{O}(-1)^{\oplus 2}$. | Outside $F$, $\varphi$ is a simple blow-down of $E$ onto $S=\varphi(E)$. $Z$ is smooth at $z$; $S$ is a cone over the rational twisted cubic at $z$. | Second divisorial subcase (Thm 5.7.6). |
| divisorial birational, dim 4, fibre quadric | Isolated $2$-dimensional fibre $F$ a quadric (possibly singular or reducible, $\mathbb{P}^1\times\mathbb{P}^1$, quadric cone $S_2$, or $\mathbb{P}^2\cup\mathbb{P}^2$) with conormal $N^*_{F/X}$ the spinor bundle $\mathcal{S}(1)$ from $\mathbb{Q}^4$. | Outside $F$, $\varphi$ is a simple blow-down of $E$ onto $S=\varphi(E)$. $Z$ is smooth at $z$; $S$ is non-normal at $z$. | Third divisorial subcase (Thm 5.7.6). Refined conormal descriptions per quadric type given in the paper. |
| conic fibration (fibre type), dim 4, isolated 2-dim fibre | Irreducible isolated $2$-dimensional fibre $F$. If $F\cong\mathbb{P}^2$ then $N^*_{F/X}\cong\mathcal{O}^{\oplus 3}/\mathcal{O}(-2)$; if $F$ is an irreducible quadric then $N^*_{F/X}$ is the pullback of $T_{\mathbb{P}^2}(-1)$ via a double cover of $\mathbb{P}^2$. | In both cases the scheme fibre structure coincides with the geometric one and $Z$ is smooth at $z$ (Thm 5.9.6). | Remaining (reducible / non-lci) conic-fibration cases deferred to a sequel. Fibre-type isolated fibres are also classified by Kachi. |
| birational adjoint contraction, dim $n \ge 5$ | Isolated fibre $F$ of dimension $n-2$: then $F \cong \mathbb{P}^{n-2}$ with $N_{F/X}\cong\mathcal{O}(-1)\oplus\mathcal{O}(-1)$. | The contraction is necessarily small and a flip exists (Cor 5.8.1). | For $\varphi$ supported by $K_X+(n-3)H$ with $H$ $\varphi$-ample; obtained by slicing down to the 4-dimensional case. |
| classification of isolated 2-dim fibres (Prop 4.11) | Complete list of pairs $(F, L_F)$ for an isolated $2$-dim fibre of a good contraction of a smooth $4$-fold. | Birational: $(\mathbb{P}^2,\mathcal{O}(1))$, $(\mathbb{F}_0, C_0+f)$, $(S_2,\mathcal{O}(1))$, $(\mathbb{P}^2\cup\mathbb{P}^2,\mathcal{O}(1))$. Fibre type, $F$ irreducible: $(\mathbb{P}^2,\mathcal{O}(1))$, $(\mathbb{P}^2,\mathcal{O}(2))$, $(S_3,\mathcal{O}(1))$, $(S_2,\mathcal{O}(1))$, $(\mathbb{F}_1, C_0+2f)$, $(\mathbb{F}_0, C_0+f)$. Fibre type, $F$ reducible ($\le 3$ components): $\mathbb{P}^2\cup\mathbb{P}^2$, $\mathbb{P}^2\cup\mathbb{F}_0$, $\mathbb{P}^2\cup_{C_0}\mathbb{F}_1$, $\mathbb{P}^2\cup S_2$, $\mathbb{P}^2\cup\mathbb{P}^2\cup\mathbb{P}^2$, $\mathbb{P}^2\cup_{\ldots}\mathbb{P}^2$, plus the exceptional $\mathbb{P}^2\bullet\mathbb{P}^2$ meeting at a point. | $S_r$ = cone over the $r$-uple Veronese of $\mathbb{P}^1$; $\mathbb{F}_r$ = Hirzebruch surface. Components meet along lines unless indicated otherwise. |
Andreatta-Wisniewski, 'On contractions of smooth varieties' (J. Algebraic Geom. 7 (1998); arXiv:alg-geom/9605013). General theory of good contractions (proper surjective with connected fibres and $-K_X$ relatively ample) whose fibres have dimension <= 2. Items below capture the low-dimensional structure theorems: the classification of isolated 2-dimensional fibres (Prop 4.11), the birational 4-fold description (intro summary, Thm 5.7.6, Lemma 6.1), the 4-fold conic-fibration description (Thm 5.9.6), and the higher-dimensional adjoint generalization (Cor 5.8.1).