Realizable triples and sharp multiplicity bounds for minimal lc centers

Characterize the realizable triples (d,c,b) associated with smooth germs and positive-dimensional minimal log canonical centers, and determine the maximal multiplicity (M_{d,c}(b)) for each such triple; in particular, determine precisely when the multiplicity bound in Corollary \ref{cor:discrete-lc-center} is attained.

Background

The paper proves a multiplicity bound for a positive-dimensional minimal log canonical center $W=W_x(X,\Delta)$ through a point $x$, expressed in terms of $d=\dim W$, the embedding codimension $c=\operatorname{edim}\mathcal O_{W,x}-d$, and the local discrepancy $b=b_x(X,\Delta)$. The appendix derives a discrete refinement of this estimate and gives an example in which the refined bound is sharp.

The authors define a triple $(d,c,b)$ to be realizable if there exist a smooth germ $x\in X$, a pair $(X,\Delta)$ that is log canonical but not klt at $x$, and a positive-dimensional minimal lc center $W=W_x(X,\Delta)$ with $\dim W=d$, embedding codimension $c$, and local discrepancy $b_x(X,\Delta)=b$. For each realizable triple, $M_{d,c}(b)$ is defined as the maximal possible value of $\operatorname{mult}_xW$. The unresolved problem is to classify all such triples and determine these maxima, including exactly when the discrete multiplicity bound is sharp. A solution could improve the numerical estimate used to obtain the constant $C_0$ in the paper's linear bound for Fujita's freeness conjecture.