Let $X$ be a smooth variety and $D \subset X$ be an effective, reduced, irreducible divisor. My question is the following.


1.What is the first order deformation and obstrution for the pair $(X, D)$? 


2.In particular, if $D$ is a singular divisor and has a global smoothing such that the line bundle $\mathcal{O}(E)\vert\_E$ also extends to a general smoothing, then under what conditions can we deform the pair so that the induced deformation of the pair gives a smoothing of $D$.



deformation and obstruction of a pair (X, D)
From another point of view, the isomorphism classes of first order deformations of the pair (X,D) are isomorphic to the 1st hypercohomology group of the 2 step complex from the tangent sheaf of X to the normal sheaf of D in X.


The 1st order deformations of just D are given by the 1st hypercohomology group of the 2 step complex from the restriction of the tangent sheaf of X to D, to the same normal sheaf of D in X.


By the long exact sequence induced by the natural map of these complexes, the forgetful map from T^1(X,D) to T^1(D) is an isomorphism whenever the first 2 hypercohomology groups vanish for the 2 step complex from the ideal sheaf of D in X times the tangent sheaf of X, to zero.


This happens for instance if X is a principally polarized abelian variety of dimension 3 or more, and D is the theta divisor, by Kodaira (or abstractly Mumford) vanishing.


ref: Compositio Math 76(1990) pp. 367-398.


