I don't understand what you mean by "curl up." You're supposed to sew down the arcs of the circles to the front anyway so I'm not sure why curl would affect that.
Is this a raggy circle quilt or the method where you sew 2 circles, right sides together,cut a slit in the "front" one and turn right side out? A picture would help.
I just finished a baby size, denim & corduroy faux CW and I had previously done a large throw size all denim raggy faux CW. When I sewed the circles together, they do tend to curl inward but it just makes it easier to sew the arcs down for me.