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?

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?