  We investigated how the magnetic field in solar active regions (ARs) controls
flare activity, i.e., whether a confined or eruptive flare occurs. We analyzed
44 flares of GOES class M5.0 and larger that occurred during 2011--2015. We
used 3D potential magnetic field models to study their location (using the
flare distance from the flux-weighted AR center $d_{\mathrm{FC}}$) and the
strength of the magnetic field in the corona above (via decay index $n$ and
flux ratio). We also present a first systematic study of the orientation of the
coronal magnetic field, using the orientation $\varphi$ of the flare-relevant
polarity inversion line as a measure. We analyzed all quantities with respect
to the size of the underlying dipole field, characterized by the distance
between the opposite-polarity centers, $d_{\mathrm{PC}}$. Flares originating
from underneath the AR dipole $(d_{\mathrm{FC}}/d_{\mathrm{PC}}<0.5$) tend to
be eruptive if launched from compact ARs ($d_{\mathrm{PC}}\leq60$ Mm) and
confined if launched from extended ARs. Flares ejected from the periphery of
ARs ($d_{\mathrm{FC}}/d_{\mathrm{PC}}>0.5$) are predominantly eruptive. In
confined events the flare-relevant field adjusts its orientation quickly to
that of the underlying dipole with height ($\Delta\varphi\gtrsim40^\circ$ until
the apex of the dipole field), in contrast to eruptive events where it changes
more slowly with height. The critical height for torus instability,
$h_{\mathrm{crit}}=h(n=1.5)$, discriminates best between confined
($h_{\mathrm{crit}}\gtrsim40$ Mm) and eruptive flares
($h_{\mathrm{crit}}\lesssim40$ Mm). It discriminates better than
$\Delta\varphi$, implying that the decay of the confining field plays a
stronger role than its orientation at different heights.
