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Register a new userSymbolic powers of certain cover ideals of graphs
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In this paper, we compute the regularity and Hilbert series of symbolic powers of the cover ideal of a graph $G$ when $G$ is either a crown graph or a complete multipartite graph. We also compute the multiplicity of symbolic powers of cover ideals in terms of the number of edges.
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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.
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.
We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbournes conjecture holds for such ideals. This criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning when the third symbolic power is in the square of an ideal, and a strong resurgence bound for some classes of space monomial curves.
An equigenerated monomial ideal $I$ in the polynomial ring $R=k[z_1,ldots, z_n]$ is a Freiman ideal if $mu(I^2)=ell(I)mu(I)-{ell(I)choose 2}$ where $ell(I)$ is the analytic spread of $I$ and $mu(I)$ is the number of minimal generators of $I$. In this paper we classify all simple connected unmixed bipartite graphs whose cover ideals are Freiman ideals.
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