Hilbert Series of Generic Ideals in Products of Projective Spaces

Pub Date : 2022-10-02 DOI:10.1080/10586458.2021.1925999
R. Fröberg
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Abstract

Abstract If , k a field, is a standard graded algebra, then the Hilbert series of R is the formal power series . It is known already since Macaulay which power series are Hilbert series of graded algebras. A much harder question is which series are Hilbert series if we fix the number of generators of I and their degrees, say for ideals , . In some sense “most” ideals with fixed degrees of their generators have the same Hilbert series. There is a conjecture for the Hilbert series of those “generic” ideals, see below. In this article we make a conjecture, and prove it in some cases, in the case of generic ideals of fixed degrees in the coordinate ring of , which might be easier to prove.
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射影空间积中一般理想的Hilbert级数
如果域k是一个标准的分级代数,则R的Hilbert级数就是形式幂级数。自麦考利以来,人们已经知道哪些幂级数是分级代数的希尔伯特级数。一个更难的问题是哪个级数是希尔伯特级数如果我们确定I的生成函数的个数和它们的度数,比如说理想值。在某种意义上,“大多数”理想具有固定度的发生器具有相同的希尔伯特级数。对于那些“一般”理想的希尔伯特级数有一个猜想,见下文。本文提出了一个猜想,并在某些情况下证明了它,在定度的一般理想的情况下,这可能更容易证明。
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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