Asymptotic invariants of ideals with Noetherian symbolic Rees algebra and applications to cover ideals

Abstract

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.

Symbolic defects of edge ideals of unicyclic graphs

“…Here I(C 3 ) = J(K 3 ). We have seen that the quasi-polynomial associated with the symbolic defects of I(C 3 ) obtained by our procedure as in Remark 4.10 coincide with the quasi-polynomial associated with the symbolic defects of J(K 3 ) obtained by [4,Theorem 5.7].…”

“…In [5], F. Galetto et al give an understanding of the symbolic defect of I when I is either the defining ideal of a star configuration or the ideal associated with a finite set of points in P 2 . If I is an ideal whose symbolic Rees algebra is Noetherian, B. Drabkin and L. Guerrieri in [4] prove that sdefect(I, s) is eventually a quasi-polynomial as a function of s. They compute the symbolic defects explicitly for the cover ideals of complete graphs and odd cycles. Also, they find the quasi-polynomial associated with the symbolic defects of the cover ideal of a complete graph.…”

On the resurgence and asymptotic resurgence of homogeneous ideals

“…It may be note that the equality in (3) has already been proved in [9,Corollary 4.4]. Also, in [9, Remark 4.10], it is proved that if the sdefect(J(G), 2) = 1, then the product of all variables is contained in J(G) (2) .…”

On quasi-equigenerated and Freiman cover ideals of graphs

“…The quantity sdef(I, m) is called symbolic defect and it is defined as the minimal number of generators of the module I (m) I m . The paper [1] is devoted to the study of symbolic defect of cover ideals of graphs.…”