Colength, multiplicity, and ideal closure operations II

Let $(R,\mathfrak{m})$ be a Noetherian local ring. This paper concerns several extremal invariants arising from the study of the relation between colength and (Hilbert–Samuel or Hilbert–Kunz) multiplicity of an $\mathfrak{m}$-primary ideal. We introduce versions of these invariants by restricting to various closures and ``cross-pollinate’’ the two multiplicity theories by asking for analogues invariants already established in one of the theories. On the Hilbert–Samuel side, we prove that the analog of the Stückrad–Vogel invariant (that is, the infimum of the ratio between the multiplicity and colength) for integrally closed $\mathfrak{m}$-primary ideals is often $1$ under mild assumptions. We also compute the supremum and infimum of the relative drops of multiplicity for (integrally closed) $\mathfrak{m}$-primary ideals. On the Hilbert–Kunz side, we study several analogs of the Lech–Mumford and Stückrad–Vogel invariants.