Coherent sheaves on P-1: Their families and their deformations related to a behavioral approach to singular systems

Recently, Lomadze, Ravi, Rosenthal and Schumacher gave a natural and geomet rically meaningful one-to-one correspondence between abstract linar behavio rs and coherent sheaves on P-1. Motivated by their result, here we give a c omplete picture of the deformation theory of coherent sheaves on P-1. We us e our results to study deformations over the field R obtaining a connectedn ess result for the real locus of the algebraic parameter spaces.