Tested on Linux:

#!/usr/bin/perl
use strict;
use warnings;
use Cwd 'abs_path';

sub in_dir {
    $_ = shift;
    m|(.*)/.*$|;  ## we take advantage of greediness
    $1 || "/";
}

my $self = abs_path $0;
my $stripped = in_dir in_dir $self;
print "$self -> $stripped\n";

__OUTPUT__

$ # there's a copy in my home and other in /
$ ./a.pl 
/home/hue/a.pl -> /home
$ /a.pl 
/a.pl -> /
$ cd /
$ ./a.pl 
/a.pl -> /
[download]

UPDATE: I knew I was reinventing the wheel, I just didn't find that File::Basename thingy.