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Monomial principalization in the singular setting

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 `c.i. monomial'. We prove that c.i. monomial schemes in arbitrarily singular varieties can be principalized by a sequence of blow-ups at codimension 2 c.i. monomial centers.