  We investigate the chiral phase structure of three flavor QCD in a background
$U(1)$ magnetic field using the standard staggered action and the Wilson
plaquette gauge action. We perform simulations on lattices with a temporal
extent of $N_\tau=4$ and four spatial extents of $N_\sigma = 8,16, 20$ and 24.
We choose a smaller-than-physical quark mass in lattice spacing as $am = 0.030$
such that there exists a crossover transition at vanishing magnetic fields, and
adopt two values of magnetic field strength in lattice spacing $a \sqrt{
e{B}}\simeq 1.5$ and 2. We find that the transition becomes stronger in the
presence of a background magnetic field, and turns into a first order as seen
from the volume scaling of the order parameter susceptibility as well as the
metastable states in the time history of the chiral condensate. On the other
hand, the chiral condensate and transition temperature always increase with $B$
even within the regime of a first order phase transition. This suggests that
the discrepancy in the behavior of chiral condensates and transition
temperature as a function of $B$ between earlier lattice studies using
larger-than-physical pion masses with standard staggered fermions and those
using physical pions with improved staggered fermions is mainly due to lattice
cutoff effects.
