The Line of Sovereignty is the demarcation point between the Home Territory and the Central area of the board.
The primary importance of the Sovereignty Line is that a pawn's possible behavior is wholly influenced by its location. Issues which determine a pawn's movement are whether it is in a Home Territory, in the Central area of the board (has crossed its own Sovereignty Line, but not an opponent's Sovereignty Line), or has crossed an opponent's Sovereignty Line. In the picture below, the blue arrows indicate the directions in which a Blue pawn may move. The green arrows indicate the directions in which a Blue pawn may capture. The red arrows indicate directions of movement or capture which are prohibited due to the Blue pawn's location at the moment.
Note that a Blue pawn's possible forward movement when in its own Home Territory is exactly the opposite of the movement possible when the same pawn is in an opponent Home Territory (e.g. Red), and that "sideways" movement (remaining on the same rank while moving) is not permitted in any Home Territory. Note also that once a pawn crosses a Sovereignty Line, "sideways" as a direction technically does not exist until a pawn crosses another Sovereignty Line. Finally, also note that once a pawn crosses any Sovereignty Line, it may not cross that Sovereignty Line again.
The pawn is the most complex piece in modern chess, and this concept becomes even more true in ImmortalStarMasters. The rewards (and dangers) for this complexity are many, and they begin to multiply exponentially as the pawn assumes an even greater value and power in the central area of the board. This will be considered a boon to some and a crushing bane to others (and sometimes both to the same player).
See the "Capturing en passant" link below for a unique feature of pawn movement which is frequently very valuable but also very much unappreciated in modern chess.