an overview of the literature listing Fano fourfolds
Tsukioka, T. (2010). Fano manifolds obtained by blowing up along curves with maximal Picard number. Manuscripta Math., 132(1–2), 247–255.
| construction | contraction-theory |
|---|---|
| Picard rank $\rho_X$ | 5 |
| invariants | contractions, picard_rank |
| links | 10.1007/s00229-010-0346-4 MR2609296 |
| notes | Extremal contractions blowing up a smooth curve; classifies the cases of Picard number 5, which is maximal. |
| case | exceptional locus | contraction / result | notes |
|---|---|---|---|
| Maximal Picard number, linear-space center | blow-up of $Y$ along $C$ = strict transform of the line $\overline{pq}$ | $Y$ is the blow-up of $\mathbb{P}^n$ whose center is the union of two points $p, q$ and a linear subspace $\mathbb{P}^{n-2}$ disjoint from $\overline{pq}$ | $X$ Fano with $\rho(X) = 5$ (the maximum), $\dim Y = n \ge 4$. Case (1) of Theorem 1 (= example (8) of the finer Theorem 2). |
| Maximal Picard number, quadric center | blow-up of $Y$ along $C$ = strict transform of the line $\overline{pq}$ | $Y$ is the blow-up of $\mathbb{P}^n$ whose center is the union of two points $p, q$ and a smooth quadric $\mathbb{Q}^{n-2}$ disjoint from $\overline{pq}$ | $X$ Fano with $\rho(X) = 5$ (the maximum), $\dim Y = n \ge 4$. Case (2) of Theorem 1 (= example (9) of the finer Theorem 2). |
Tsukioka, Theorem 1 (arXiv:0904.2268, Manuscripta Math. 132 (2010) 247-255). A Fano manifold $X$ ($\dim \ge 4$) that is the blow-up of a smooth projective $Y$ along a smooth curve $C$ has Picard number $\le 5$ (Casagrande); Theorem 1 classifies the maximal case $\rho(X)=5$ into exactly two pairs $(Y,C)$, both with $Y$ a blow-up of $\mathbb{P}^n$ and $C$ the strict transform of the line $pq$. Full arXiv text used.