Minimal resolutions of dominant and semidominant ideals

We introduce new classes of monomial ideals: dominant, p-semidominant, and GNP ideals. The families of dominant and 1-semidominant ideals extend those of complete and almost complete intersections, while the family of GNP ideals extends that of generic ideals. We show that dominant ideals give a complete characterization of when the Taylor resolution is minimal, 1-semidominant ideals are Scarf, and the minimal resolutions of 2-semidominant ideals can be obtained from their Taylor resolutions by eliminating faces and facets of equal multidegree, in arbitrary order. We also generalize a theorem of Bayer, Peeva and Sturmfels by proving that GNP ideals are Scarf.