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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)

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  • $\begingroup$ What you write does not seem completely correct, $H^{0}(X,f^{*}T_Y\otimes I)$, where $I$ is your square zero ideal, makes more sense to me. Moreover you do only have a pseudo-torsor under this group. $\endgroup$
    – Niels
    Commented Nov 24, 2016 at 7:55
  • $\begingroup$ For non smooth schemes the natural answer is that you need the whole cotangent complex, that was built especially for this purpose. $\endgroup$
    – Niels
    Commented Nov 24, 2016 at 7:58

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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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    $\begingroup$ A small addition: a good reference is Kollár's Rational curves on algebraic varieties, ch. I, Theorem 2.16. $\endgroup$
    – abx
    Commented Nov 24, 2016 at 10:48

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