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Traditional models of core collapse suggest the issue of successful versusfailed supernova explosions and neutron star versus black hole formationdepends monotonically on the mass (and metallicity) of the progenitor star.Here we argue that the issue of success or failure of the explosion or otherpossible outcomes may depend non--monotonically on the rotation of theprogenitor star even at fixed progenitor mass and composition. We have computed"shellular" models of core collapse for a star of 15 M_solar with initialcentral angular velocity, Omega_0, in the range 0.1 -- 8 rad/s until a fewhundred ms after bounce to explore qualitative trends. The non--monotonicbehavior will be manifested in the rotation of the proto--neutron star andhence in the strength of the associated magnetic field that will be generatedby shear in that rotating environment. We estimate that our maximally rotatingand shearing models generate toroidal fields approaching or exceeding 10^17G,strengths nearing dynamical significance.Comment: 20 pages, 6 figures, accepted for publication in The Astrophysical  Journal. Few typos are correcte

The Nonmonotonic Dependence of Supernova and Compact Remnant Formation on Progenitor Rotation