Let $f \colon X \to Y$ be a morphism from an lci scheme to a smooth scheme; then we can construct the conormal complex $N_f^\vee$ as $f^* \Omega_X \to \Omega_Y$, and the normal complex $N_f = RHom(N_f^\vee, \mathcal{O}_Z)$.

First order deformations of $f$ leaving the target $Y$ fixed, and obstructions to lifting them, should then be parameterized by the hypercohomology groups $\mathbb{H}^0(N_f)$ and $\mathbb{H}^1(N_f)$.

When $f$ is unramified, the hypercohomology reduces to ordinary cohomology of the normal sheaf, and the result is in Sernesi's book...

Moreover, when $X$ is a curve, and $Z \subset X$ a finite set of points, then first-order deformations of $f$ and obstructions, leaving $(f(Z) \subset Y)$ fixed, but with $Z \subset X$ variable, should be parameterized by $\mathbb{H}^0(N_f(-Z))$ and $\mathbb{H}^1(N_f(-Z))$. (Probably this is a special case of a more general statement about maps between pairs, but I'm not exactly sure what that would be --- the case of $Y$ a point is answered here: deformation and obstruction of a pair (X, D) --- but I'm not sure what the more general statement would look like.)

Does someone know references for these facts?


  • $\begingroup$ There is a useful Proposition 21.2 math.berkeley.edu/~robin/math274root.pdf $\endgroup$ – user21574 May 10 '17 at 21:24
  • $\begingroup$ For local complete intersections, as in your question, an excellent and readable reference is Angelo Vistoli's, "The deformation theory of local complete intersections" : arxiv.org/pdf/alg-geom/9703008.pdf Your statement follows from Theorem 4.4, although you need to do some work (e.g., you can use the blowing up of the graph of $f$ in $X\times Y$ to related your question to Theorem 4.4). The analogous statement for more general schemes $X$ and $Y$ follows from Prop., Luc Illusie, "Complexe cotangent et deformations I". $\endgroup$ – Jason Starr May 11 '17 at 18:51

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