Monomial principalization in the singular setting

Abstract

We generalize an algorithm by Goward for principalization of monomial ideals in nonsingular varieties to work on any scheme of finite type over a field. The normal crossings condition considered by Goward is weakened to the condition that components of the generating divisors meet as complete intersections. This leads to a substantial generalization of the notion of monomial scheme; we call the resulting schemes `\textit{regular crossings} (r.c.) \textit{monomial}.' We prove that r.c.~monomial subschemes in arbitrarily singular varieties can be principalized by a sequence of blow-ups at codimension~2 r.c.~monomial centers.