Embedded varieties, X-ranks and uniqueness or finiteness of the solutions

IF 0.9 Q2 MATHEMATICS
E. Ballico
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引用次数: 0

Abstract

Let \(X\subset \mathbb {P}^r\) be an integral and non-degenerate variety. For any \(q\in \mathbb {P}^r\) its X-rank \(r_X(q)\) is the minimal cardinality of a finite subset of X whose linear span contains q. The solution set \(\mathcal {S}(X,q)\) of \(q\in \mathbb {P}^r\) is the set of all \(S\subset X\) such that \(\#S=r_X(q)\) and S spans q. We prove that if \(X\ne \mathbb {P}^r\) there is at least one q with \(\#\mathcal {S}(X,q)>1\) and that for almost all pairs (Xq) we have \(\dim \mathcal {S}(X,q)>0\).

嵌入变量、x阶和解的唯一性或有限性
让 \(X\subset \mathbb {P}^r\) 是一个积分且非简并的变量。对于任何 \(q\in \mathbb {P}^r\) 它的x级 \(r_X(q)\) 是X的有限子集的最小基数,它的线性张成空间包含q,解集 \(\mathcal {S}(X,q)\) 的 \(q\in \mathbb {P}^r\) 是所有的集合吗 \(S\subset X\) 这样 \(\#S=r_X(q)\) S张成q,我们证明如果 \(X\ne \mathbb {P}^r\) 至少有一个q \(\#\mathcal {S}(X,q)>1\) 对于几乎所有的(X, q)对 \(\dim \mathcal {S}(X,q)>0\).
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Afrika Matematika
Afrika Matematika MATHEMATICS-
CiteScore
2.00
自引率
9.10%
发文量
96
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