an overview of the literature listing Fano fourfolds
Fujita, K. (2014). On a generalization of the Mukai conjecture for Fano fourfolds. Tokyo J. Math., 37(2), 319–333.
| construction | contraction-theory |
|---|---|
| invariants | contractions |
| links | 10.3836/tjm/1422452796 MR3304684 |
| notes | Contains useful information on extremal contractions of Fano fourfolds. |
| case | exceptional locus | contraction / result | notes |
|---|---|---|---|
| The invariant $s(X)$ and its bound | all extremal rays $R \subset \overline{NE}(X)$ | For an $n$-dimensional Fano manifold $X$, set $s(X) := \sum_{R} (l(R)-1)$ over all extremal rays $R$, where $l(R) = \min\{(-K_X \cdot C)\}$ over rational curves $C$ with $[C] \in R$. Theorem 1.4 (Main Theorem): $s(X) \le n$ whenever $n \le 4$. | Refines the Mukai and Tsukioka conjectures since $\rho_X(r_X-1) \le \rho_X(l_X-1) \le s(X)$, where $l_X = \min_R l(R)$. For $n \ge 5$ there exist Fano manifolds with $s(X) > n$ (Remark 1.5). |
| Equality $s(X)=n$ in dimension $\le 3$ | contraction of each extremal ray | Theorem 1.4(i): for $n \le 3$, equality $s(X)=n$ holds iff $X \cong \prod_{R} \mathbb{P}^{\,l(R)-1}$, a product of projective spaces indexed by the extremal rays. | |
| Equality $s(X)=4$ for Fano fourfolds | extremal contractions of the fourfold | Theorem 1.4(ii): for $n=4$, equality $s(X)=4$ holds iff either $X \cong \prod_{R} \mathbb{P}^{\,l(R)-1}$ (a product of projective spaces), or $X \cong \mathrm{Bl}_{p,q}(Q^4)$, the blow-up of a smooth quadric fourfold $Q^4 \subset \mathbb{P}^5$ at two distinct points $p,q$ whose joining line $pq$ is not contained in $Q^4$. | The blow-up $\mathrm{Bl}_{p,q}(Q^4)$ arises as the $d=2$, $n=4$ case of the blow-up analysis (Theorem 3.4 / Remark 3.5(iii)), where $s = 2n-4 = 4$. |
| Generalized Mukai conjecture in dimension $\le 4$ | minimal-length extremal ray | Corollary 1.6: Tsukioka Conjecture 1.2 holds for $n \le 4$, i.e. $\rho_X(l_X-1) \le n$ with equality iff $X \cong (\mathbb{P}^{\,l_X-1})^{\rho_X}$. | Tsukioka had proved the inequality for $n=4$; Fujita settles the equality case. |
Statements from the Main Theorem 1.4 and Corollary 1.6, with the extremal-contraction case analysis of section 3; read from the full text (arXiv:1206.1990v2).