Tulane University（杜兰大学）；University of California at Riverside（加州大学河滨分校）
Sankhaneel Bisui, Eloísa Grifo, Huy Tài Hà, Thái Thành Nguyên
备注：
v3 changes: improved the containments shown for ideals of minors of generic matrices, minors of generic symmetric matrices, and pfaffians of generic skew symmetric matrices
We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective spaces. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly's bound, and prove that a general version of that containment holds for generic determinantal ideals and defining ideals of star configurations.
Given a homogeneous ideal $I \subseteq k[x_0,\dots,x_n]$, the Containment problem studies the relation between symbolic and regular powers of $I$, that is, it asks for which pair $m, r \in \mathbb{N}$, $I^{(m)} \subseteq I^r$ holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne -- Huneke Conjecture and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in $\mathb…
The symbolic powers $I^{(n)}$ of a radical ideal $I$ in a polynomial ring consist of the functions that vanish up to order $n$ in the variety defined by $I$. These do not necessarily coincide with the ordinary algebraic powers $I^n$, but it is natural to compare the two notions. The containment problem consists of determining the values of $n$ and $m$ for which $I^{(n)} \subseteq I^m$ holds. When $I$ is an ideal of height $2$ in a regular ring, $I^{(3)} \subseteq I^2$ may fail, but we show that this containment does hold for the defining ideal of the space monomial curve $(t^a, t^b, t^c)$. Mor…
The containment problem for symbolic and ordinary powers of ideals asks for what values of $a$ and $b$ we have $I^{(a)} \subseteq I^b$. Over a regular ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede partially answers this question, but the containments it provides are not always best possible. In particular, a tighter containment conjectured by Harbourne has been shown to hold for interesting classes of ideals - although it does not hold in general. In this paper, we develop a Fedder (respectively, Glassbrenner) type criterion for $F$-purity (respectively, strong $F$-re…
