Let $f:X \to Y$ be a finite morphism, where $X$ is an affine, non-singular curve and $Y$ is an affine, non-singular surface over $\mathbb{C}$. Denote by $\Gamma\_f \subset X \times Y$ the graph of the morphism $f$. When is $\Gamma\_f$ a local complete intersection subscheme of $X \times Y$? Any hint/reference will be most welcome.



When is a graph morphism a regular embedding?
All we need here is smoothness of $Y$. In general if $X$ and $Y$ are varieties over a field $k$, with $Y$ smooth, and if $f: X \to Y$ is a morphism, then $ \Gamma\_f \subset X \times Y$ is a local complete intersection, with co-normal bundle isomorphic to $f^\* \Omega\_Y$. 


To see this, note that $\pi\_X: X \times Y \to X$ is a smooth morphism (since $Y$ is smooth) and $\mathrm{id} \times f: X \to X \times Y$ gives a section, with image $\Gamma\_f$. Since $\Gamma\_f$ is certainly smooth over $X$ (the projection $\Gamma\_f \to X$ is an isomorphism) a relative version of Hartshorne II.8.17 implies the conormal sequence 
$$
 0 \to \mathcal{I}\_{\Gamma\_f}/\mathcal{I}\_{\Gamma\_f}^2 \to \Omega\_{X \times Y | X} |\_{\Gamma\_f} \to \Omega\_{\Gamma\_f | X} =0 \to 0 
$$ 
is exact. Hence 
$$
\mathcal{I}\_{\Gamma\_f}/\mathcal{I}\_{\Gamma\_f}^2 \simeq \Omega\_{X \times Y | X} |\_{\Gamma\_f} \simeq (\mathrm{id} \times f)^\* \pi\_Y^\* \Omega\_Y \simeq f^\* \Omega\_Y
$$


Appendix B.7 of Fulton's *Intersection Theory* contains a more detailed discussion, plus some generalizations (l.c.i. morphisms).


