I think it has less to do with “the object isn’t flat”, and more to do with how you’re mapping the marble onto the object. I’ve also seen times where blender wants a paricular level of face/edge count, to properly get a clean UV line.
Sharing the file will help diag it. We don’t need the entire scene, just that piece and the texture.
I am going to guess that you don’t know UV unwrapping? If you don’t, that would be your next topic to learn. It allows controlling how a texture is applied to an object.