The degree of Fano threefolds with canonical Gorenstein singularities. (English. Russian original) Zbl 1081.14058
Sb. Math. 196, No. 1, 77-114 (2005); translations from Mat. Sb. 196, No. 1, 81-122 (2005).
This paper is devoted to the proof of the following conjecture (due to Fano and Iskovskikh): let \(V\) be a Fano threefold with canonical Gorenstein singularities, then \(-K_X^3 \leq 72\). Moreover, if the equality holds, \(V\) is shown to be isomorphic to a weighted projective space (either \({\mathbb P}(3,1,1,1)\) or \({\mathbb P}(6,4,1,1)\)).
The proof is based on the minimal model program and on the properties of a weak Fano threefold, i.e. a threefold whose anticanonical divisor is a nef and big \({\mathbb Q}\)-Cartier divisor. More precisely, by means of a suitable modification \(\phi: W \to V\), the author reduces the statement to a similar one for a weak Fano threefold \(W\) with terminal factorial singularities. Then the required result follows from a detailed analysis of the structure of the extremal rays of \(W\).
The proof is based on the minimal model program and on the properties of a weak Fano threefold, i.e. a threefold whose anticanonical divisor is a nef and big \({\mathbb Q}\)-Cartier divisor. More precisely, by means of a suitable modification \(\phi: W \to V\), the author reduces the statement to a similar one for a weak Fano threefold \(W\) with terminal factorial singularities. Then the required result follows from a detailed analysis of the structure of the extremal rays of \(W\).
Reviewer: Luciana Picco Botta (Torino)
MSC:
14J45 | Fano varieties |
14J30 | \(3\)-folds |
14E30 | Minimal model program (Mori theory, extremal rays) |