  Fix $k,s,n\in \mathbb N$, and consider non-zero integers $c_1,\ldots ,c_s$,
not all of the same sign. Provided that $s\ge k(k+1)$, we establish a Hasse
principle for the existence of lines having integral coordinates lying on the
affine diagonal hypersurface defined by the equation $c_1x_1^k+\ldots
+c_sx_s^k=n$. This conclusion surmounts the conventional convexity barrier
tantamount to the square-root cancellation limit for this problem.
