The matroid stratification of the Hilbert scheme of points on P 1
The matroid stratification of the Hilbert scheme of points on P 1
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P 1 上点的希尔伯特格式的拟阵分层
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通讯作者：
Nicolas Trotignon
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作者：
Frédéric Maffray;Nicolas Trotignon
。给出了f中系数非零的单项式的集合，对于每一次d和对每一多项式ffii d，都是f中具有非零coef∈fi性质的单项式集。这些数据共同构成了i的热带化。热带理想在任何(多分次)fi格式上都会产生一个“拟阵Strati Hilbert阳离子”。人们对这些Stratifi阳离子的结构知之甚少。在这篇文章中，我们探索了许多拟阵地层的例子，包括一些具有有趣的组合结构的例子，并给出了一种方便的可视化方法。证明了(P1)[k]点的希尔伯特格式中的拟阵fi正离子是由k个变量的Schur多项式生成的。最后，我们给出了(A2)[n]的T-图问题的一个应用；对这个图的分类是一个长期存在的公开问题，并且我们证明了infinite边类的存在性。
. Givenahomogeneousideal I inapolynomialringoverafield,onemayrecord,for each degree d and for each polynomial f ∈ I d , the set of monomials in f with nonzero coefficients. These data collectively form the tropicalization of I . Tropicalizing ideals induces a “matroid stratification” on any (multigraded) Hilbert scheme. Very little is known about the structure of these stratifications. In this paper, we explore many examples of matroid strata, including some with interesting combinatorial structure, and give a convenient way of visualizing them. We show that the matroid stratification in the Hilbert scheme of points ( P 1 ) [ k ] is generated by all Schur polynomials in k variables. We end with an application to the T -graph problem of ( A 2 ) [ n ] ; classifying this graph is a longstanding open problem, and we establish the existence of an infinite class of edges.