If you consider that only the atoms "count" for existing, then the line does not exist. (Neither do the stones, by the way - since they are constantly exchanging minor quantities of atoms and electrons with the rest of the environment, the set that defines what each stone is is constantly changing. But let's ignore that bit). The line is a "mathematical abstraction", layered on top of the "real" world.
Now consider that you measure and define exactly where that line is, according to the system of coordinates that you prefer.
Now imagine that you remove the three stones. Now there's no atoms "sustaining" that the line any more. But in a sense the line is "still there", it's defined by the coordinates you wrote. It can be argued that it has always "existed" - before you put the stones there, before Earth existed, that particular line in space already "was", in the same sense that numbers "are" - that the line has existed in some way or another since Space itself started existing.
You now pick 3 bottle caps and put them on top of the line again. Now the line is again defined by 3 clumps of matter. Does it "exist" any more than it existed in the previous two cases?
It all comes down to how we use the verb "exist" - it means several things depending on the context. "Exist" in the atoms world, or "exist" as information.
I believe that mind and consciousness is a bit of the same thing, just many orders of magnitude bigger. Instead of 3 stones what defines a mind is trillions of neurons arranged in a certain way. I am materialistic, but I believe in the be encoding of information. To me this means that each mind, at some point in time, can absolutely be "encoded" into a big enough number (huge, billions of trillions of digits). So in that sense minds are "defined by atoms" but also "exist independently of atoms, and are eternal".