It is not exactly curvature in fringes which affects the phase of the OAM beam. Relative fringe shift, width of fringe, number of fringe are the prime factors for the desired phase structuring of output beams from diffraction gratings.

As you might be knowing, the desired output from grating is 1st order diffraction mode. First image shows the conventional gratings, it wont phase structure 1st order mode. output will have constant phase.

Now consider second image.

For the sake of simplicity i will first discuss about Hermite Gaussian(HG) beam generation(first row). HG(0,1) mode is a phase structured beam with two lobes which are out of phase by 180 degree. As evident from HG(0,1) grating 180 degree phase shift can be imparted by shifting upper half of the fringes in grating by distance of half the fringe width.

Second row shows OAM beam generation with topological charge(TC) 1. Addition of one fringe at the top half region of grating can impart 0 - 2pi variation of azimuthal phase.

Third row shows fractional OAM beam generation with TC 1.5 . In this case there is fringe shift as well as fringe addition. Fringe shift causes a phase jump of pi and fringe addition causes 2pi phase variation. Hence total azimuthal phase variation as seen in phase distribution plot is pi/2+2pi+pi/2

you can simulate the effect of grating patterns by taking its 2d fast fourier transform.

I hope this allowed you to develop some intuitive understanding on Fork grating patterns