an overview of the literature listing Fano fourfolds
Ou, W. (2018). Fano varieties with Nef(X)=Psef(X) and ρ(X)=\dim X-1. Manuscripta Math., 157(3–4), 551–587.
| construction | contraction-theory |
|---|---|
| invariants | contractions |
| links | 10.1007/s00229-018-1015-2 MR3858418 |
| notes | Section 7 treats dimension four; discusses when the nef cone is simplicial. |
| case | exceptional locus | contraction / result | notes |
|---|---|---|---|
| Setting in dimension four | the Mori cone $\overline{NE}(X)$, of dimension $\rho(X)=3$ | Section 7 studies Fano fourfolds $X$ with locally factorial canonical singularities, smooth in codimension 2, with $\rho(X)=3$ and $\mathrm{Nef}(X)=\mathrm{Psef}(X)$. Each $2$-dimensional face $V_2$ of $\overline{NE}(X)$ meets exactly two other $2$-faces $V_1,V_3$ along extremal rays. | This is the $n=4$, $\rho = \dim X - 1$ case of the paper's general hypotheses. |
| Existence of a fibration to $\mathbb{P}^1$ | contraction of an extremal ray | Proposition 7.1 (main result of the section): every such fourfold $X$ admits a fibration $X \to \mathbb{P}^1$. This anchors the induction that proves the general classification Theorem 1.3. | Proved by contradiction: a hypothetical $X$ with no $\mathbb{P}^1$-fibration is ruled out via Lemmas 7.2-7.6. |
| Structure when no $\mathbb{P}^1$-fibration is assumed | three fibrations $f_i: X \to B_i$ from three $2$-faces $V_1,V_2,V_3$ | Lemma 7.2: if $X$ had no fibration to $\mathbb{P}^1$, then each base $B_i \cong \mathbb{P}^2$ and $f_1 \times f_2 \times f_3 : X \to \mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^2$ is finite onto its image $Z$, where $Z = D \cap E$ is the intersection of hypersurfaces of degree $(r,s,0)$ and $(0,a,b)$ with $r,s,a,b \in \{1,2\}$. Lemma 7.5 forces $r=s=a=b=2$. | Hypothetical configuration only; shown impossible in Proposition 7.1. |
| Nef/Mori cone not simplicial in the excluded case | the cone $\overline{NE}(X)$ | Lemma 7.3: a fourfold satisfying the hypotheses with no $\mathbb{P}^1$-fibration cannot have simplicial $\overline{NE}(X)$; a simplicial cone (three $2$-faces, three extremal rays) contradicts the degree constraints on $Z$. Combined with $-K_Z \equiv H_1 - H_2 + H_3$ and $-K_Z(H_1+H_3)^3 = 0$, this contradicts $-K_Z$ being big, completing the proof. | The non-simplicial conclusion is what makes the finiteness of $f_1 \times f_3$ incompatible with $-K_Z$ big. |
| Resulting dimension-four classification | Mori fibration $h \times g : X \to Y \times \mathbb{P}^1$ | Via Proposition 7.1 and the global Theorem 1.3, such a fourfold decomposes as $X \cong X_1 \times X_2$ where $X_1$ is a point or a Theorem-1.1 factor, and $X_2$ is a finite cover of degree $1,2,4$ over $(\mathbb{P}^1)^s \times \mathbb{P}^2$ or $(\mathbb{P}^1)^s \times W$ with $W$ a normal hypersurface in $\mathbb{P}^2 \times \mathbb{P}^2$; explicitly $X_2$ is $\mathbb{P}^2$, a Theorem-1.2 threefold, or a variety from Construction 4.1 or 4.6. | Theorem 1.3 requires $X$ smooth in codimension 2; the statement can fail without this hypothesis. |
Section 7 (dimension four, rho=3): Proposition 7.1 and Lemmas 7.2-7.6, plus the dimension-4 specialization of the global Theorem 1.3; read from the full text (arXiv:1508.00182).