Let I be an ideal whose symbolic Rees algebra is Noetherian. For m ≥ 1, the m-th symbolic defect, sdefect(I, m), of I is defined to be the minimal number of generators of the module I (m) I m . We prove that sdefect(I, m) is eventually quasi-polynomial as a function in m. We compute the symbolic defect explicitly for certain monomial ideals arising from graphs, termed cover ideals. We go on to give a formula for the Waldschmidt constant, an asymptotic invariant measuring the growth of the degrees of generators of symbolic powers, for ideals whose symbolic Rees algebra is Noetherian. MSC: 13F20; 05C25.
Many facts about group theory can be generalized to the context of the theory of association schemes. In particular, association schemes with fewer than 6 elements are all commutative. While there is a nonabelian group with 6 elements which is unique up to isomorphism, there are infinitely many isomorphism classes of noncommutative association schemes with 6 elements. All examples previously known to us are imprimitive, and fall into three classes which are reasonably well understood. In this paper, we construct a fourth class of noncommutative, imprimitive association schemes of rank 6.
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