If $f:X\rightarrow Y$ is a morphism of algebraic schemes with reduced and projective source $X$ and nonsingular target $Y$, then first order deformations of the map $f$ with both source and target fixed is in bijection with $H^{0}(X,f^{\*}T\_Y)$. 


How do I understand the condition that $Y$ is nonsingular? Is there a good example to keep in mind where $Y$ is singular and we no longer have the bijection above?


(References: Proposition 3.4.2 in Sernesi's deformation theory book, [Reference Request: Deformations of a map bijective to global sections of the pullback of the tangent sheaf](https://mathoverflow.net/questions/79764/reference-request-deformations-of-a-map-bijective-to-global-sections-of-the-pul))



Deformation of maps to singular target
The first order deformations are in general (say $Y$ has no embedded points in $f(X)$ and the image of every irreducible components of $X$ intersects the smooth locus of $Y$) parametrized by $\mathrm{Hom}\_X(f^\*\Omega\_Y,\mathscr O\_X)$. If $Y$ is smooth, then $\Omega\_Y$ is locally free, so this group is the same as $H^0(X, f^\*T\_Y)$. To find an example that you want, just find one where these two groups are not isomorphic.


