  We numerically analyze superconductivity (SC) in the cuprate superconductors
by using ab initio effective Hamiltonians consisting of the antibonding
combination of Cu $3d_{x^2-y^2}$ and O $2p_{\sigma}$ orbitals. We perform
variational Monte Carlo calculations for the four carrier doped cuprates with
diverse experimental optimal SC critical temperature $T_{c}^{\rm opt}$:
CaCuO$_2$ ($T_{c}^{\rm opt} \sim 110$ K), Bi$_2$Sr$_2$CuO$_6$ ($T_{c}^{\rm opt}
\sim 10$-$40$ K), Bi$_2$Sr$_2$CaCu$_2$O$_8$ ($T_{c}^{\rm opt} \sim 85$-$100$
K), and HgBa$_2$CuO$_4$ ($T_{c}^{\rm opt} \sim 90$ K). Materials and hole
doping concentration ($\delta$) dependencies of the SC order parameter $F_{\rm
SC}$ and the competition with spin/charge order show essential and quantitative
agreements with the available experiments in the following points: (1) The
ground state is commonly the SC state, which is severely competing with the
charge/spin stripe and antiferromagnetic states. (2) $F_{\rm SC}$ shows
amplitude consistent with the superfluid density measured in the muon spin
resonance and its dome structure found in $\delta$ dependence shows consistency
with that of the SC gap in the tunneling and photoemission measurements. We
further find insights into the universal SC mechanism: (I) $F_{\rm SC}$
increases with the ratio $U/|t_1|$, indicating that $U/|t_1|$ is the principal
component controlling the SC. Here, $U$ and $t_1$ are the onsite Coulomb
repulsion and the nearest neighbor hopping, respectively, in the Hamiltonians.
(II) A universal scaling $T_{c}^{\rm opt}\sim 0.16 \lvert t_1 \rvert F_{\rm
SC}$ holds. (III) SC is enhanced and optimized if $U$ is increased beyond the
real available materials. It is further enhanced by decreasing the offsite
interaction. The present findings provide useful clues for the design of new SC
materials with even higher $T_{c}^{\rm opt}$.
