Let $A$ be a finite type $k$-algebra and $I\subset A$ a nonzero ideal. Here $k$ is an algebraically closed field.
Question. When is there an injective $A$-linear map $I\to A/I$? When is there none?
Some context. The vector space $\textrm{Hom}_A(I,A/I)$ is the set of first order deformations of $X=\textrm{Spec }A/I$ inside $Y=\textrm{Spec }A$. I was wondering if the injective maps $I\to A/I$ would have a special meaning in terms of deformation theory, but first I would need to understand when there is any at all. Thinking about the identification $$\textrm{Hom}_A(I,A/I)=\textrm{Hom}_{A/I}(I/I^2,A/I)$$ is not helping me in finding examples.
Added. I am particularly interested in the case when $I$ is the ideal of a smooth curve $X$ in a variety $Y=\textrm{Spec }A$ of dimension 3.
Thanks for your kind help.