I have a manifold X covered by a family of elliptic curves, some of which have
non-reduced structure (like multiple fibers on elliptic surfaces; such non-reduced curves C are members of my family, but their reductions C_{red}
are not). For some reason I know that the normal bundle of each curve in my
family is trivial (meaning, in the non-reduced case, that I_C/(I_C)^2 is a 
direct sum of a few copies of the structure sheaf). Now for a general member
of my family I know by differential geometry ("tubular neighbourhood lemma") that its small deformations do not intersect (locally around this general member my X is fibered in those elliptic curves). I wonder whether this is
also true around a multiple fiber: what would replace the tubular neighbourhood lemma in the algebraic case?