Castelnuovo-Mumford regularity of simplicial toric rings. (Q1867303)

Eisenbud and Goto conjectured that the Castelnuovo-Mumford regularity \(\text{reg}(R)\) of a standard graded domain over an algebraically closed field is bounded by the difference \(\text{deg}(R) -\text{codim}(R)\). So far the conjecture has been proved only in a few low-dimensional cases. The authors provide a bound for \(\text{reg}(R)\) in the case where \(R\) is the toric ring defined by a simplicial affine semigroup, namely \[ \text{reg}(R)\leq\min ( \text{codim}(R)(\deg(R)-1),\dim(R)(\text{deg}(R)-\text{codim}(R)-2)+2 ) \] provided \(\deg(R)\geq \text{codim}(R)+2\). Though the bound is certainly weaker than the conjectured one, it can be considered a very good bound since it is linear in \(\text{deg}(R)\). In its proof the authors use various approaches to control the reduction number and the local cohomology of \(R\).