  We study the vertical structure of polytropic, $P\propto \rho^\Gamma$,
centrifugally-supported gaseous discs embedded in cold dark matter (CDM) halos.
At fixed radius $R$, the shape of the vertical density profile depends only
weakly on whether the disc is self-gravitating (SG) or not (NSG). The disc
thickness, set by the midplane sound speed and circular velocity, $(c_s/V_c)R$,
in the NSG case, and by the sound speed and surface density, $c_s^2/G\Sigma$,
in SG discs, is smaller than either of these scales. SG discs are typically
Toomre unstable, NSG discs are stable. Exponential discs in CDM halos with
roughly flat circular velocity curves generally "flare" outwards. For the
polytropic equation of state of the EAGLE simulations, discs whose mass and
size match observational constraints are stable (NSG) for $M_d< 3\times 10^9\,
M_\odot$ and unstable (SG) at higher masses, if fully gaseous. We test these
analytic results using a set of idealized SPH simulations and find excellent
agreement. Our results clarify the role of the gravitational softening on the
thickness of simulated discs, and on the onset of radial instabilities. EAGLE
low-mass discs are non-self-gravitating so the softening plays no role in their
vertical structure. High-mass discs, on the other hand, are expected to be
self-gravitating and unstable, and may be artificially thickened and stabilized
unless gravity is well resolved. Simulations with spatial resolution high
enough to not compromise the vertical structure of a disc also resolve the
onset of their instabilities, but the converse is not true: resolving
instabilities does not guarantee that the vertical structure is resolved.
