摘要翻译:
我们研究了多项式环中齐次理想的连锁类。如果一个理想在每一步只使用齐次正则序列就能链接到一个完全交集,则称它为齐次licci。我们提出一个自然的问题:如果$I$是齐次licci,那么它是否可以通过在每一步使用尽可能小的形式的规则序列来链接到一个完全的交集(我们称这样的理想为最小齐次licci)?在本文中,我们对这个问题作了否定的回答。特别地,对于每$n\geq28$,我们在$\mathbb p^3$中构造了一组$n$点,它们是齐次licci,但不是最小齐次licci。此外,我们证明了不能仅根据Hilbert函数来区分齐次licci理想和非licci理想,也不能仅根据分次Betti数来区分齐次licci理想和极小齐次licci理想。最后,通过取超曲面截面,我们证明了当理想的高度至少为3时,自然问题有一个否定的答案。
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英文标题:
《Minimal Homogenous Liaison and Licci Ideals》
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作者:
Craig Huneke, Juan Migliore, Uwe Nagel and Bernd Ulrich
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最新提交年份:
2007
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分类信息:
一级分类:Mathematics 数学
二级分类:Commutative Algebra 交换代数
分类描述:Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环,模,理想,同调代数,计算方面,不变理论,与代数几何和组合学的联系
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一级分类:Mathematics 数学
二级分类:Algebraic Geometry 代数几何
分类描述:Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇,叠,束,格式,模空间,复几何,量子上同调
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英文摘要:
We study the linkage classes of homogeneous ideals in polynomial rings. An ideal is said to be homogeneously licci if it can be linked to a complete intersection using only homogeneous regular sequences at each step. We ask a natural question: if $I$ is homogeneously licci, then can it be linked to a complete intersection by linking using regular sequences of forms of smallest possible degree at each step (we call such ideals minimally homogeneously licci)? In this paper we answer this question in the negative. In particular, for every $n\geq 28$ we construct a set of $n$ points in $\mathbb P^3$ which are homogeneously licci, but not minimally homogeneously licci. Moreover, we prove that one cannot distinguish between the classes of homogeneously licci and non-licci ideals based only on their Hilbert functions, nor distinguish between homogeneously licci and minimally homogeneously licci ideals based solely on the graded Betti numbers. Finally, by taking hypersurface sections, we show that the natural question has a negative answer whenever the height of the ideal is at least three.
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PDF链接:
https://arxiv.org/pdf/0708.3369