Nagoya Mathematical Journal

On classification of $\mathbb{Q}-Fano 3-folds of Gorenstein index 2. I

Hiromichi Takagi

Full-text: Open access

Abstract

We formulate a generalization of K. Takeuchi's method to classify smooth Fano $3$-folds and use it to give a list of numerical possibilities of $\mathbb{Q}$-Fano $3$-folds $X$ with Pic $X = \mathbb{Z} (-2K_{X})$ and $h^{0} (-K_{X}) \geq 4$ containing index $2$ points $P$ such that $(X, P) \simeq (\{xy +z^{2}+u^{a}=0\} / \mathbb{Z}_{2}(1, 1, 1, 0), o)$ for some $a \in \mathbb{N}$. In particular we prove that then $(-K_{X})^{3} \leq 15$ and $h^{0} (-K_{X}) \leq 10$. Moreover we show that such an $X$ is birational to a simpler Mori fiber space.

Article information

Source
Nagoya Math. J., Volume 167 (2002), 117-155.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114649295

Mathematical Reviews number (MathSciNet)
MR1924722

Zentralblatt MATH identifier
1048.14022

Subjects
Primary: 14J45: Fano varieties
Secondary: 14E05: Rational and birational maps 14E30: Minimal model program (Mori theory, extremal rays) 14J30: $3$-folds [See also 32Q25]

Citation

Takagi, Hiromichi. On classification of $\mathbb{Q}-Fano 3-folds of Gorenstein index 2. I. Nagoya Math. J. 167 (2002), 117--155. https://projecteuclid.org/euclid.nmj/1114649295


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