an overview of the literature listing Fano fourfolds
Kawamata, Y. (1989). Small contractions of four-dimensional algebraic manifolds. Math. Ann., 284(4), 595–600.
| construction | contraction-theory |
|---|---|
| invariants | contractions |
| links | 10.1007/BF01443353 MR1006374 |
| notes | The exceptional locus of a small contraction is a disjoint union of ℙ²'s with normal bundle 𝒪(−1)⊕𝒪(−1). |
| case | exceptional locus | contraction / result | notes |
|---|---|---|---|
| small (flipping) contraction, dim 4 | The exceptional locus is a disjoint union of irreducible components each isomorphic to $\mathbb{P}^2$, with normal bundle $N_{\mathbb{P}^2/X} \cong \mathcal{O}_{\mathbb{P}^2}(-1) \oplus \mathcal{O}_{\mathbb{P}^2}(-1)$. | The flip $\varphi^+ : X^+ \to Y$ exists and $X^+$ is smooth. Blowing up the locus in $X$ gives an exceptional divisor a disjoint union of components $\cong \mathbb{P}^2 \times \mathbb{P}^1$ (normal bundle $\mathrm{pr}_1^*\mathcal{O}_{\mathbb{P}^2}(-1) \oplus \mathrm{pr}_2^*\mathcal{O}_{\mathbb{P}^1}(-1)$), which can be contracted the other way onto $X^+$. | Setting: $X$ a nonsingular projective $4$-fold, $\varphi : X \to Y$ a small (isomorphism in codimension $1$) birational contraction of a $K_X$-negative extremal ray, i.e. $-K_X$ is $\varphi$-ample. |
Main theorem of Kawamata, 'Small contractions of four-dimensional algebraic manifolds', Math. Ann. 284 (1989) 595-600. The paper proves essentially one structural theorem. Springer full text was not directly accessible; the statement is corroborated verbatim by Andreatta-Wisniewski (arXiv:alg-geom/9605013, intro summary theorem and Lemma 6.1), which restates it as 'the situation studied in [Ka1]'.