On the irreducibility of the family of algebraic curves in complex projective space Pr

Let I ' (d,g,r) be the union of the irreducible components of the Hilbert scheme H-d,H-g,H-r whose general points correspond to smooth nondegenerate curves of degree d and genus g in P-r. Suppose the Brill-Noether number rho (d, g, r) = g - (r + 1) (g - d + r) > 0. Set A(0)(g,r) = 2r-4/r+1 g + r+13/r+1, A(1) (g, r) = 2r-6/r+1 g + 2r+26/r+1 and 2g - 2 - [A(1) (g, r)] equivalent to a(mod 4), 1 < a < 4. We prove; if r greater than or equal to 8 and d > A(0)(g, r) then I ' (d,g,r) * irreducible, if r greater than or equal to 6 and 2g+4r+10/3 less than or equal to d less than or equal to A(1) (g,r) - 4 + a then I ' (d,g,r) is reducible, and if r greater than or equal to 15 and d > A(1) (g,r), then I ' (d,g,r) is reducible if and only if A(1) (g,r) < d less than or equal to A(0) (g,r) with 2g - 2 - d equivalent to 0(mod 3).