Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In the present paper we investigate the more general problem of determining the structure of symbolic powers of a wide generalization of star configurations of points (introduced by Geramita, Harbourne, Migliore and Nagel) called star configurations of hypersurfaces in $\mathbb P_k^n$. Here (1) we provide explicit minimal generating sets of the symbolic powers $I^{(m)}$ of these ideals $I$, (2) we introduce a notion of $\delta$-c.i. quotients, which generalize ideals with linear quotients, and show that $I^{(m)}$ have $\delta$-c.i. quotients, (3) we show that the shape of the Betti tables of these symbolic powers is determined by certain "Koszul" strands and we prove that a little bit more than the bottom half of the Betti table has a regular, almost hypnotic, pattern, and (4) we provide a closed formula for all the graded Betti numbers in these strands. As a special case of (2) we deduce that symbolic powers of ideals of star configurations of points have linear quotients. We also improve and extend results by Galetto, Geramita, Shin and Van Tuyl, and provide explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations. Finally, inspired by Young tableaux, we introduce a technical tool which may be of independent interest: it is a "canonical" way of writing any monomial in any given set of polynomials. Our methods are characteristic--free.

中文翻译：

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超曲面星构型符号幂的结构和自由分辨率

点的星形配置是在 $\mathbb P_k^n$ 中所有点配置中具有已知（和推测）极值行为的配置；额外的兴趣来自它们丰富的结构，这允许使用代数几何、组合学、交换代数和表示理论的工具来研究它们。在本文中，我们研究了更一般的问题，即确定点的星形配置（由 Geramita、Harbourne、Migliore 和 Nagel 引入）的广泛推广的符号权力结构，称为 $\mathbb P_k^n$ 超曲面的星形配置. 这里 (1) 我们提供了这些理想 $I$ 的符号幂 $I^{(m)}$ 的显式最小生成集，(2) 我们引入了 $\delta$-ci 商的概念，它概括了理想线性商，并证明 $I^{(m)}$ 具有 $\delta$-ci 商，(3) 我们证明了这些符号权力的 Betti 表的形状是由某些“Koszul”链决定的，我们证明了比 Betti 表的下半部分多一点点有一个规则的，几乎是催眠的模式，并且 (4) 我们为这些链中的所有分级 Betti 数字提供了一个封闭的公式。作为（2）的一个特例，我们推导出点的星形配置理想的符号幂具有线性商。我们还改进和扩展了 Galetto、Geramita、Shin 和 Van Tuyl 的结果，并为最小数量的生成器和星形配置的符号缺陷提供了明确的新通用公式。最后，受 Young tableaux 的启发，我们介绍了一种可能具有独立兴趣的技术工具：它是一个“规范的” 在任何给定多项式集合中编写任何单项式的方法。我们的方法是无特征的。