  Uncertainties in our knowledge of the properties of dense matter near and
above nuclear saturation density are among the main sources of variations in
multi-messenger signatures predicted for core-collapse supernovae (CCSNe) and
the properties of neutron stars (NSs). We construct 97 new finite-temperature
equations of state (EOSs) of dense matter that obey current experimental,
observational, and theoretical constraints and discuss how systematic
variations in the EOS parameters affect the properties of cold nonrotating NSs
and the core collapse of a $20\,M_\odot$ progenitor star. The core collapse of
the $20\,M_\odot$ progenitor star is simulated in spherical symmetry using the
general-relativistic radiation-hydrodynamics code GR1D where neutrino
interactions are computed for each EOS using the NuLib library. We conclude
that the effective mass of nucleons at densities above nuclear saturation
density is the largest source of uncertainty in the CCSN neutrino signal and
dynamics even though it plays a subdominant role in most properties of cold NS
matter. Meanwhile, changes in other observables affect the properties of cold
NSs, while having little effect in CCSNe. To strengthen our conclusions, we
perform six octant three-dimensional CCSN simulations varying the effective
mass of nucleons at nuclear saturation density. We conclude that neutrino
heating and, thus, the likelihood of explosion is significantly increased for
EOSs where the effective mass of nucleons at nuclear saturation density is
large.
