In a general sense, there are multiple reasons for the horizontal stabilizer to have dihedral/anhedral.
- To clear the tail from the downwash/propwash. A good example is the Cessna 425, which was developed from the Cessna 421. The 421 had a straight horizontal stabilizer. However, with the addition of dihedral, the tail is kept out of the propwash, improving ride quality. From a report citing Cessna engineers:
Cessna engineers say the dihedral improves Dutch roll qualities and moves a major portion of the tail out of the prop wash, thus providing a smoother ride.
The Dassault Falcon 50 had anhedral so as to clear the tail from the wing's downwash. Another good example of this is the F-4 Phantom. The prototypes had a straight tail. However, the anhedral tail was adopted so as to keep it clear of the disturbed air from the wing at high angles of attack while at the same time keeping it out of engine exhaust.
The anhedral in BAE Hawk is explained by the same reason. From The Hawk Story:
This model gave the first indications of high speed problems due to interference between the high intake and the low wing, and the
position of the tail.
This resulted in the adoption of a low intake, ... .The rear fuselage was cambered downwards to bring the tail down, and the tailplane was
given pronounced anhedral for the same purpose.
Stability- The dihedral/anhedral in the stabilizers affects the aircraft's stability. In short, for a conventional tail, an anhedral improves the yaw stability of the aircraft while reducing the pitch stability (slightly), while a dihedral will reduce the tail's contribution to both pitch and yaw stability.
However, the use of dihedral in the horizontal tail influences static roll stability in the same way as that of dihedral in the wing. Dihedral affects the roll response in the same way as that of the wing- it causes the lift on the right and left semispans to respond differently to downwash.
A good example of this is the Vickers Viscount, which had a dihedral horizontal stabilizer, which improved its stability.
Due to this, it is better for the designer to add a dihedral to the horizontal tail rather than wing especially if problems are found in later stages of design as the impact will be lesser.
From Roskam's Aircraft Design Vol III, Chapter 5.1:
The overall lateral stability is usually dominated by the inherent lateral stability designed into the wing. Wing sweep, dihedral angle and (wing) location on the fuselage dominate the magnitude and the sign of the stability derivative. Since the wing is designed by performance and operational considerations, the empennage is often used to fine tune the lateral stability of the aircraft. Examples include the F-4 and the AV-8B.
It should be noted that, as with any other design consideration, the dihedral of the horizontal stabilizer depends on the design choices made and perfomance characteristics required; as a result, the configuration required will change from aircraft to aircraft.
In case of DC-10, the main reason appears to be to keep the stabiliser out of the wing influence, especially when high flap settings are used. The more pronounced dihedral (compared to, say Boeing 737) appears to be used as the stabilizer is mounted lower due to the presence of engine in the tail.