We study the geometry of standard graded Hilbert schemes of polynomial rings and exterior algebras. Our investigation is motivated by a famous theorem of Reeves–Stillman for the Grothendieck Hilbert scheme, which states that the lexicographic point is smooth. By contrast, we show that, in standard graded Hilbert schemes of polynomial rings and exterior algebras, the lexicographic point can be singular, and it can lie in multiple irreducible components. We answer questions of Peeva–Stillman and of Maclagan–Smith.

中文翻译：

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关于 Hilbert 方案字典序点的平滑性

我们研究多项式环和外代数的标准分级希尔伯特方案的几何。我们的研究受到格洛腾迪克希尔伯特方案的一个著名的 Reeves-Stillman 定理的启发，该定理指出字典点是平滑的。相比之下，我们表明，在多项式环和外代数的标准分级希尔伯特方案中，词典点可以是奇异的，并且可以位于多个不可约分量中。我们回答 Peeva-Stillman 和 Maclagan-Smith 的问题。