The exciton and biexciton emissions of a series of single quantum dots of InAs in an AlAs matrix have been studied. These emissions consist of linear cross polarized doublets showing large values of both the biexciton binding energy and the fine-structure splitting. At increasing exciton emission energy, corresponding to decreasing dot size, the biexciton binding energy of 9 meV decreases down to zero, reflecting a possible crossover to an antibinding regime. Simultaneously the fine-structure splitting diminishes from a value of 0.3 meV down to zero, at the same energy, suggesting a common origin for the two effects. © 2006 American Institute of Physics.