AI 中文总结
研究拟阵的斯坦利 - 赖斯纳理想的渐近复苏，通过与面理想渐近复苏的等价性，利用拟阵收缩的面理想的瓦尔施密特常数证明公式，得出渐近复苏与弱序的关系及相关上界，还证明了许多类拟阵面理想渐近复苏的下界等式并计算部分结果。
AI 中文摘要
拟阵配置是由杰拉米塔、哈伯恩、米廖雷和纳格尔引入的射影簇，它推广了所谓的星配置，其定义理想通过适当特化拟阵的斯坦利 - 赖斯纳理想得到。受此联系的启发，我们研究拟阵的斯坦利 - 赖斯纳理想的渐近复苏。比利亚雷亚尔的一个结果表明，研究面理想的渐近复苏与之等价。我们根据拟阵收缩的面理想的瓦尔施密特常数证明了拟阵面理想渐近复苏的一个公式。结果表明，渐近复苏尊重同秩拟阵上的弱序。因此，给定拟阵的面理想的渐近复苏以同秩所谓几乎均匀拟阵的面理想的渐近复苏为上界，我们明确计算了该上界。瓜尔多、哈伯恩和范图伊尔表明，理想的渐近复苏以理想的初始次数与其瓦尔施密特常数的比值为下界。我们证明，对于许多类拟阵的面理想，这个下界是等式，包括在大小为\(n\geq2k\)的基集上秩为\(k\)且其对偶是铺砌的拟阵、完美拟阵设计以及由施泰纳系统产生的稀疏铺砌拟阵。对于后两类，我们明确计算了渐近复苏。
英文摘要
Matroid configurations -- introduced by Geramita, Harbourne, Migliore, and Nagel -- are projective varieties which generalize so-called \textit{star configurations} and whose defining ideals are obtained by appropriately specializing the Stanley-Reisner ideal of a matroid. Motivated by this connection, we study the asymptotic resurgence of the Stanley-Reisner ideals of matroids. A result of Villareal shows that it is equivalent to study the asymptotic resurgence of facet ideals. We prove a formula for the asymptotic resurgence of the facet ideal of a matroid in terms of the Waldschmidt constant of facet ideals of the contractions of the matroid. As a consequence, we show that asymptotic resurgence respects the weak order on matroids of the same rank. Therefore, the asymptotic resurgence of the facet ideal of a given matroid is bounded above by the asymptotic resurgence of the facet ideal of a so-called \textit{almost-uniform} matroid of the same rank, which we compute explicitly. Guardo, Harbourne, and Van Tuyl showed that the asymptotic resurgence of an ideal is bounded below by the ratio of the initial degree of the ideal by its Waldschmidt constant. We prove that this lower bound is an equality for facet ideals of many classes of matroids, including matroids of rank $k$ on a ground set of size $n\ge 2k$ whose dual is paving, perfect matroid designs, and sparse paving matroids arising from Steiner systems. For the latter two classes, we explicitly compute the asymptotic resurgence.
Comments39 pages (including figures), comments welcome
