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Comparision between regularity of small symbolic powers and ordinary powers of an edge ideal (2109.05242v1)
Published 11 Sep 2021 in math.AC
Abstract: Let $G$ be a simple graph and $I$ its edge ideal. We prove that $${\rm reg}(I{(s)}) = {\rm reg}(Is)$$ for $s = 2,3$, where $I{(s)}$ is the $s$-th symbolic power of $I$. As a consequence, we prove the following bounds \begin{align*} {\rm reg} I{s} & \le {\rm reg} I + 2s - 2, \text{ for } s = 2,3, {\rm reg} I{(s)} & \le {\rm reg} I + 2s - 2,\text{ for } s = 2,3,4. \end{align*}