A primitive multiple curve is a Cohen-Macaulay irreducible projective curve Y that can be
locally embedded in a smooth surface, and such that C = Y _{red} is smooth. In this case,
L = _{C}∕_{C}^{2} is a line bundle on C. If Y is of multiplicity 2, i.e. if _{
C}^{2} = 0, Y is called a
ribbon. If Y is a ribbon and h^{0}(L^{-2}) ⁄= 0, then Y can be deformed to smooth curves, but in
general a coherent sheaf on Y cannot be deformed in coherent sheaves on the smooth
curves.

A ribbon with associated line bundle L such that deg(L) = -d < 0 can be deformed to reduced curves having 2 irreducible components if L can be written as