Expected resurgences and symbolic powers of ideals

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 Harbourne's 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.Comment: Final version to appear in the Journal of the London Mathematical Societ

Edge ideals: algebraic and combinatorial properties

Let C be a clutter and let I(C) be its edge ideal. This is a survey paper on the algebraic and combinatorial properties of R/I(C) and C, respectively. We give a criterion to estimate the regularity of R/I(C) and apply this criterion to give new proofs of some formulas for the regularity. If C is a clutter and R/I(C) is sequentially Cohen-Macaulay, we present a formula for the regularity of the ideal of vertex covers of C and give a formula for the projective dimension of R/I(C). We also examine the associated primes of powers of edge ideals, and show that for a graph with a leaf, these sets form an ascending chain

Limit Behavior of the Rational Powers of Monomial Ideals

We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo-Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this we show the before-now unknown convergence of Stanley depths of integral closure powers. In addition, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals

Integral closures of powers of sums of ideals

Let $k$ be a field, let $A$ and $B$ be polynomial rings over $k$, and let $S= A \otimes_k B$. Let $I \subseteq A$ and $J \subseteq B$ be monomial ideals. We establish a binomial expansion for rational powers of $I+J \subseteq S$ in terms of those of $I$ and $J$. Particularly, for a positive rational number $u$, we prove that $(I+J)_u = \sum_{0 \le \omega \le u, \ \omega \in \mathbb{Q}} I_\omega J_{u-\omega},$ and that the sum on the right hand side is a finite sum. This finite sum can be made more precise using jumping numbers of rational powers of $I$ and $J$. We further give sufficient conditions for this formula to hold for the integral closures of powers of $I+J$ in terms of those of $I$ and $J$. Under these conditions, we provide explicit formulas for the depth and regularity of $\overline{(I+J)^k}$ in terms of those of powers of $I$ and $J$.Comment: revision added connections to jumping numbers; 14 page