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Register a new userRegularity of Symbolic Powers of Edge Ideals
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In this article, we prove that for several classes of graphs, the Castelnuovo-Mumford regularity of symbolic powers of their edge ideals coincide with that of their ordinary powers.
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In this article, we study the regularity of integral closure of powers of edge ideals. We obtain a lower bound for the regularity of integral closure of powers of edge ideals in terms of induced matching number of graphs. We prove that the regularity of integral closure of powers of edge ideals of graphs with at most two odd cycles is the same as the regularity of their powers.
Let $A = K[X_1,ldots, X_d]$ and let $I$, $J$ be monomial ideals in $A$. Let $I_n(J) = (I^n colon J^infty)$ be the $n^{th}$ symbolic power of $I$ wrt $J$. It is easy to see that the function $f^I_J(n) = e_0(I_n(J)/I^n)$ is of quasi-polynomial type, say of period $g$ and degree $c$. For $n gg 0$ say [ f^I_J(n) = a_c(n)n^c + a_{c-1}(n)n^{c-1} + text{lower terms}, ] where for $i = 0, ldots, c$, $a_i colon mathbb{N} rt mathbb{Z}$ are periodic functions of period $g$ and $a_c eq 0$. In an earlier paper we (together with Herzog and Verma) proved that $dim I_n(J)/I^n$ is constant for $n gg 0$ and $a_c(-)$ is a constant. In this paper we prove that if $I$ is generated by some elements of the same degree and height $I geq 2$ then $a_{c-1}(-)$ is also a constant.
Let $mathcal{D}$ be a weighted oriented graph and $I(mathcal{D})$ be its edge ideal. In this paper, we show that all the symbolic and ordinary powers of $I(mathcal{D})$ coincide when $mathcal{D}$ is a weighted oriented certain class of tree. Finally, we give necessary and sufficient conditions for the equality of ordinary and symbolic powers of naturally oriented lines.
Let $G$ be a simple graph and $I$ its edge ideal. We prove that $${rm reg}(I^{(s)}) = {rm reg}(I^s)$$ 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*}
We compute the Betti numbers for all the powers of initial and final lexsegment edge ideals. For the powers of the edge ideal of an anti-$d-$path, we prove that they have linear quotients and we characterize the normally torsion-free ideals. We determine a class of non-squarefree ideals, arising from some particular graphs, which are normally torsion-free.
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