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Let $f: X \rightarrow Y$ be a morphism between smooth projective varieties over an algebraic closed field. The graph of $f$ namely $\Gamma_f$ is a cycle inside $X \times Y$. When is $\Gamma_f$ nef? When is it a complete intersection of ample divisors? Are there some useful criterions or restrictions?

If $X=Y$ and $f=id$ then this is a problem about the diagonal cycle, for example in the curve case the diagonal is ample iff genus of $X$ is zero. How about the surface or threefold case?

In the general morphism case with $Y$ being a curve (or just $\mathbb P^1$), the graph is a divisor, maybe the property can be described in a simple way.

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    $\begingroup$ @abx Is this obvious? Do you have a reference? It's clear for the case of curves, but what about the higher dimensional case? $\endgroup$
    – Yosemite Stan
    Commented May 5, 2019 at 7:25
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    $\begingroup$ This paper (arxiv.org/abs/1707.08659) of Lehmann and Ottem certainly seems relevant. $\endgroup$
    – Yosemite Stan
    Commented May 5, 2019 at 7:34
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    $\begingroup$ @Yosemite Stan: you are right, I was too hasty. I delete my comment. $\endgroup$
    – abx
    Commented May 5, 2019 at 8:09

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