  We derive analytic expressions for the critical temperatures of the
superconducting (SC) and pseudogap (PG) transitions of the high-Tc cuprates as
a function of doping. These are in excellent agreement with the experimental
data both for single-layered materials such as LSCO, Bi2201 and Hg1201 and
multi-layered ones, such as Bi2212, Bi2223, Hg1212 and Hg1223. Optimal doping
occurs when the chemical potential vanishes, thus leading to an universal
expression for the optimal SC transition temperatures. This allows for the
obtainment of a quantitative description of the growth of such temperatures
with the number of layers, N, which accurately applies to the $Bi$, $Hg$ and
$Tl$ families of cuprates. We study the pressure dependence of the SC
transition temperatures, obtaining excellent agreement with the experimental
data for different materials and dopings. These results are obtained from an
effective Hamiltonian for the itinerant oxygen holes, which includes both the
electric repulsion between them and their magnetic interactions with the
localized copper ions. We show that the former interaction is responsible for
the SC and the latter, for the PG phases, the phase diagram of cuprates
resulting from the competition of both. The Hamiltonian is defined on a
bipartite oxygen lattice, which results from the fact that only the $p_x$ and
$p_y$ oxygen orbitals alternatively hybridize with the $3d$ copper orbitals.
From this, we can provide an unified explanation for the $d_{x^2-y^2}$ symmetry
of both the SC and PG order parameters and obtain the Fermi pockets observed in
ARPES experiments.
