Are there any general results on when a closed subscheme X of a quasi-projective smooth scheme M can be written as the zero-set of a section of a vector bundle E on M?
To put it in a diagram: When is X the fiber product of M -> E <- M , where one arrow is the zero section and the other arrow is the section I'm looking for. 


If this is not possible, can X be written as a degeneracy locus?



When is a scheme a zero-set of a section of a vector bundle?
As for the first question, the class of X has to be the product of the Chern roots of the bundle, so in the Chow ring, it is the class of a complete intersection.


As for the second question, you would have to find classes that will solve the class of X in the Thom-Porteus formula, see Fulton's intersection theory 14.4


