TITLE: Deformation of maps to singular target
QUESTION [2 upvotes]: 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)

REPLY [3 votes]: 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.
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