Comparison of the x-regularities of symbolic and ordinary Rees algebras

Determine whether the x-regularity of the symbolic Rees algebra of a monomial ideal is always bounded above by the x-regularity of its ordinary Rees algebra, namely whether reg_Rs(I) ≤ reg_R(I), and characterize the conditions under which this inequality holds.

Background

The paper introduces the x-regularity of the symbolic Rees algebra as a quantity controlling upper bounds for the regularity of symbolic powers. The ordinary Rees algebra has x-regularity zero for various important ideals, including edge ideals with linear resolution.

The authors ask whether the symbolic Rees algebra’s x-regularity is generally no larger than that of the ordinary Rees algebra, or, if not universally, which hypotheses guarantee the inequality. This issue is not resolved in the paper.

References

Therefore, in view of Theorem 1.2(d), we are led to speculate whether in general we have reg_R$(I) ≤ reg_R(I), or to determine under which conditions this inequality holds.

— Symbolic powers of polymatroidal ideals  (2502.19998 - Ficarra et al., 27 Feb 2025) in Introduction, paragraph following Question D