  Quasi-Monte Carlo methods are designed for integrands of bounded variation,
and this excludes singular integrands. Several methods are known for integrands
that become singular on the boundary of the unit cube $[0,1]^d$ or at isolated
possibly unknown points within $[0,1]^d$. Here we consider functions on the
square $[0,1]^2$ that may become singular as the point approaches the diagonal
line $x_1=x_2$, and we study three quadrature methods. The first method splits
the square into two triangles separated by a region around the line of
singularity, and applies recently developed triangle QMC rules to the two
triangular parts. For functions with a singularity `no worse than
$|x_1-x_2|^{-A}$ for $0<A<1$ that method yields an error of $O(
(\log(n)/n)^{(1-A)/2})$. We also consider methods extending the integrand into
a region containing the singularity and show that method will not improve up on
using two triangles. Finally, we consider transforming the integrand to have a
more QMC-friendly singularity along the boundary of the square. This then leads
to error rates of $O(n^{-1+\epsilon+A})$ when combined with some
corner-avoiding Halton points or with randomized QMC, but it requires some
stronger assumptions on the original singular integrand.
