A quation id-i-eq1"> quation-source format-t-e-x" xmlns:search="http://marklogic.com/appservices/search">\(\mathbb {Q}\) -factorial complete toric variety is a quotient of a poly weighted space

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• 作者：Michele Rossi ; Lea Terracini
• 关键词：\(\mathbb {Q}\) ; factorial complete toric varieties ; Connectedness in codimension 1 ; Gale duality ; Weighted projective spaces ; Hermite normal form ; Smith normal form
• 刊名：Annali di Matematica Pura ed Applicata (1923 -)
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：196
• 期：1
• 页码：325-347
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：1618-1891
• 卷排序：196

文摘

We prove that every \(\mathbb {Q}\)-factorial complete toric variety is a finite abelian quotient of a poly weighted space (PWS), as defined in our previous work (Rossi and Terracini in Linear Algebra Appl 495:256–288, 2016. doi:10.1016/j.laa.2016.01.039). This generalizes the Batyrev–Cox and Conrads description of a \(\mathbb {Q}\)-factorial complete toric variety of Picard number 1, as a finite quotient of a weighted projective space (WPS) (Duke Math J 75:293–338, 1994, Lemma 2.11) and (Manuscr Math 107:215–227, 2002, Prop. 4.7), to every possible Picard number, by replacing the covering WPS with a PWS. By Buczyńska’s results (2008), we get a universal picture of coverings in codimension 1 for every \(\mathbb {Q}\)-factorial complete toric variety, as topological counterpart of the \(\mathbb {Z}\)-linear universal property of the double Gale dual of a fan matrix. As a consequence, we describe the bases of the subgroup of Cartier divisors inside the free group of Weil divisors and the bases of the Picard subgroup inside the class group, respectively, generalizing to every \(\mathbb {Q}\)-factorial complete toric variety the description given in Rossi and Terracini (2016, Thm. 2.9) for a PWS.