Assume that X is a quasiprojective scheme with a fixed imbedding into a smooth scheme M given by an ideal sheaf I. Assume further that there is a locally free sheaf E<sub>M</sub> on X that is a restriction of a sheaf on M. The last piece of data is a surjective morphism f from E<sub>M</sub> to I/I<sup>2</sup>.

Locally it is always possible to lift f to a morphsim from E to I, which both are sheaves on M, not X anymore. Does anybody happen to know whether this also possible globally?

Feel free to modify M as much as you want. Actually I would be happy if i could find one specific M such that the locally free sheaf on X is a restriction and the morphism to the conormal sheaf lifts. The only thing that is fixed is X. What is also important to me is that I can not assume X to be a locally complete intersection, it can be arbitrarily bad. So I/I<sup>2</sup> will in general only be a cone, not a bundle.