A formula for symbolic powers (2112.12588v2)
Abstract: Let $S$ be a Cohen-Macaulay ring which is local or standard graded over a field, and let $I$ be an unmixed ideal that is also generically a complete intersection. Our goal in this paper is multi-fold. First, we give a multiplicity-based characterization of when an unmixed subideal $J \subseteq I{(m)}$ equals the $m$-th symbolic power $I{(m)}$ of $I$. Second, we provide a saturation-type formula to compute $I{(m)}$ and employ it to deduce a theoretical criterion for when $I{(m)}=Im$. Third, we establish an explicit linear bound on the exponent that makes the saturation formula effective, and use it to obtain lower bounds for the initial degree of $I{(m)}$. Along the way, we prove a conjecture (in fact, a generalized version of it) due to Eisenbud and Mazur about ${\rm ann}_S(I{(m)}/Im)$, and we propose a conjecture connecting the symbolic defect of an ideal to Jacobian ideals.