Assume that $X$ is a quasiprojective scheme with a fixed imbedding into a smooth scheme M given by an ideal sheaf $\mathcal{I}$. Assume further that there is a locally free sheaf $E_X$ on $X$ that is a restriction of a sheaf $E_M$ on $M$. The last piece of data is a surjective morphism $f:E_M\to \mathcal{I}/\mathcal{I}^2$.

Locally it is always possible to lift $f$ to a morphism 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 $\mathcal{I}/\mathcal{I}^2$ will in general only be a cone, not a bundle.

Assume that $X$ is a quasiprojective scheme with a fixed imbedding into a smooth scheme M given by an ideal sheaf $\mathcal{I}$. Assume further that there is a locally free sheaf $E_X$ on $X$ that is a restriction of a sheaf $E_M$ on $M$. The last piece of data is a surjective morphism $f:E_M\to \mathcal{I}/\mathcal{I}^2$.

Locally it is always possible to lift $f$ to a morphism from $E_M$ 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 $\mathcal{I}/\mathcal{I}^2$ will in general only be a cone, not a bundle.

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_M on X that is a restriction of a sheaf on M. The last piece of data is a surjective morphism f from E_M to I/I².

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/I² will in general only be a cone, not a bundle.

Assume that $X$ is a quasiprojective scheme with a fixed imbedding into a smooth scheme M given by an ideal sheaf $\mathcal{I}$. Assume further that there is a locally free sheaf $E_X$ on $X$ that is a restriction of a sheaf $E_M$ on $M$. The last piece of data is a surjective morphism $f:E_M\to \mathcal{I}/\mathcal{I}^2$.

Locally it is always possible to lift $f$ to a morphism 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 $\mathcal{I}/\mathcal{I}^2$ will in general only be a cone, not a bundle.