Twistor spaces of algebraic dimension two associated to a connected sum of projective planes

For any integer m greater than or equal to 4 we construct twister spaces of algebraic dimension a = 2 associated to some self-dual structures on mP(2), the connected sum of m, complex projective planes P-2. Together with previously known results this implies that for any m greater than or equal to 5 all the possible values for algebraic dimension 0 less than or equal to a less than or equal to 3 are attained for some such twistor spaces. Our method is to construct such twistor spaces by small deformations of Joyce twistor spaces.