The symbolic generic initial system of almost linear point configurations in $\mathbb P^2$

Abstract

Consider an ideal I ⊆ K [x, y, z] corresponding to a point configuration in P 2 where all but one of the points lies on a single line. In this paper we study the symbolic generic initial system {gin(I (m) )}m obtained by taking the reverse lexicographic generic initial ideals of the uniform fat point ideals I (m) . We describe the limiting shape of {gin(I (m) )}m and, in proving this result, demonstrate that infinitely many of the ideals I (m) are componentwise linear. arXiv:1304.7541v1 [math.AC]

Asymptotic invariants constructed from generic initial ideals

“…We see, for example, from Corollary 7.8 that we can construct a limiting shape for a graded sequence of symbolic powers of ideal with any, but finite number of line segments forming its boundary. Conclusions following from these two theorems coincide with an observation made by S. Mayes (see [29, Observation 5.4]) and contain a partial answer to Question 5.5 in [29].…”

“…As an interesting application, we construct in Corollary 7.8 the first example of limiting shapes for symbolic powers of ideal with arbitrarily high (but finite) number of line segments forming its boundary. To the best of author's knowledge, the biggest number of line segments forming boundary of $$I_{\bullet }=\lbrace I^{(m)}\rbrace _m$$ known previously was two (see [29, Theorem 1.1] and compare [33]).…”

“…The main result of this paper is the construction of limiting shapes with arbitrary high number of line segments which form their boundary. It is done for symbolic powers of ideals defined by carefully chosen 0‐dimensional schemes in $$\mathbb {P}^2$$ (see Corollary 7.8) and provides a partial answer to Question 5.5 in [29]. Understanding all possible limiting shapes $$\Delta (I_{\bullet })$$ investigated in this paper may give a way to approach Questions 5.2 and 5.5 from [29], or Remark 5.4 from [15].…”

“…It is done for symbolic powers of ideals defined by carefully chosen 0‐dimensional schemes in $$\mathbb {P}^2$$ (see Corollary 7.8) and provides a partial answer to Question 5.5 in [29]. Understanding all possible limiting shapes $$\Delta (I_{\bullet })$$ investigated in this paper may give a way to approach Questions 5.2 and 5.5 from [29], or Remark 5.4 from [15]. Theorem For any positive integer $$M>0$$, there exists a set of points Z in $$\mathbb {P}^2$$ such that the limiting shape of the symbolic powers of ideal $$I(Z)$$ has at least M line segments.…”

Bibliometric Analysis: Symbolic Power Publication Trens in Scopus.com

“…The use of protest symbols in the form of slogals requires understanding and thinking power over the guarantee of transparency of protest functions (Barash & Antonovskiy, 2019). In addition, fatpoint is related to the concept of symbolic power regarding the study of generic early systems and lexicography (Mayes, 2016) ).…”