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Symbolic Defect of Monomial Ideals

Benjamin R. Oltsik

Abstract

Given a monomial ideal $I$, we study two functions that quantify ways to measure the difference between symbolic powers and usual powers of $I$. In many cases we determine the asymptotic growth rate of these two functions. We also perform explicit computations by using the symbolic polyhedron.

Symbolic Defect of Monomial Ideals

Abstract

Given a monomial ideal , we study two functions that quantify ways to measure the difference between symbolic powers and usual powers of . In many cases we determine the asymptotic growth rate of these two functions. We also perform explicit computations by using the symbolic polyhedron.
Paper Structure (8 sections, 20 theorems, 41 equations, 5 figures)

This paper contains 8 sections, 20 theorems, 41 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Growth of Symbolic Defect of Monomial Ideals
  4. Integral Closure and Symbolic Defect
  5. Calculating Symbolic Defect with the Symbolic Polyhedron
  6. Calculating Symbolic Defect from the Symbolic Polyhedron, pt. II
  7. Open Questions
  8. Acknowledgements

Key Result

Theorem 1.1

Let $I$ be a monomial ideal in $k[x_1, \ldots, x_r]$ without embedded primes. Then $\operatorname{sdef}_I(n) = O(n^{r-2})$.

Figures (5)

  • Figure 1: ${\rm NP}(xy, yz, xz)$
  • Figure 2: $\operatorname {SP}(xy, yz, xz)$
  • Figure 3: ${\rm NP}(I)$
  • Figure 4: ${\rm SP}(I)$
  • Figure 5: ${\rm SP}(I)\setminus{\rm NP}(I)$

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Example 1.3
  • Example 1.4
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 2.4
  • Definition 2.5: Galetto-Geramita-Shin-Van Tuyl GGSV
  • Lemma 2.6
  • ...and 50 more