On the geography of threefolds. (With an appendix by Mei-Chu Chang)
type: Item
instance of: scholarly article
title: On the geography of threefolds. (With an appendix by Mei-Chu Chang)
zbMATH Open document ID: 0941.14014
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1998-04-22T00:00:00Z
review text: The term geography is used to study the distribution of Chern numbers of an algebraic manifold of general type. Such term was introduced by \textit{U. Persson} [Compos. Math. 43, 3-58 (1981; Zbl 0479.14018)], where he was concerned with the surfaces case. For a 3-fold of general type, since the minimal model may not be unique and may not be smooth, to well pose the geography problem one has to fix a unique model in each birational equivalence class. In the paper under review the author considers certain 3-folds of general type with some smooth minimal model and pairs \(({c_1^3\over c_1c_2}, {c_3\over c_1c_2})\) instead of triples \((c_1^3, c_1c_2, c_3)\). This was Sommese's idea in the surface case [\textit{A. J. Sommese}, Math. Ann. 268, 207-221 (1984; Zbl 0521.14016)] where he considered the ratio \({c_1^2}\over{c_2}\) instead of the pair \((c_1^2, c_2)\). Then the question is whether for any such pair, in a suitable region, there exists a 3-fold of general type with those Chern numbers. The author uses double covers of \(\mathbb{P}^1\times F_n\) branched along some special chosen configuration of divisors to construct some smooth minimal 3-folds of general type, where \(F_n\) is the Segre-Hirzebruch rational surface. In the appendix, \textit{Mei-Chu Chang} gives a ``global'' statement of Liu's theorem.
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 14E30
zbMATH DE Number: 1036422
zbMATH Keywords: Chern numbers
zbMATH Keywords: 3-fold of general type
zbMATH Keywords: minimal model
author: Xian-Fang Liu
reviewed by: Maria Lucia Fania
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1036422