sometimes you can fix this problem by putting some sandpaper (a medium fine grain) between the string and the nut. Put the gritty side down in the groove, put the string on top, and cut the ends of the paper off. This lifts the string a tad and gets it up off the frets.
It's not a good fix, but it will hold you for a while.
A good guitar tech can fix this problem thru straightening the neck, raising the action, sanding the frets, whatever the guy thinks is the right solution for your problem.
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