Let $X$ be a variety and $\varphi : F_1 \to F_2$ be a morphism of vector bundles over $X$. Then it is easy to check that the locus on $X$ for which $\varphi$ vanishes is a closed subscheme of $X$. Furthermore, the ideal sheaf cutting out this locus can be described locally by trivializing both vector bundles.

Let us try to do this in families now.

Take a family of varieties $\pi:\mathcal{X} \to B$ over some base scheme $B$. We will not assume $B$ is reduced but we can assume the usual nice things about $\pi$. Lets say $\pi$ is flat, proper and of finite presentation with nice fibers. Take a morphism of vector bundles $\varphi: F_1 \to F_2$ on $\mathcal{X}$.

We want to solve the following:

Vaguely: What is the locus in $B$ like over which $\varphi$ is precisely zero.

Observe that this is the same situation as in the very first sentence of this post, if we take $\pi = \operatorname{id}_{\mathcal{X}}$. But in our current generality we need to make things a little more precise for the question to make sense.

Define the functor $G \subset \hom(\_,B)$ so that $T\to B$ is in $G$ iff $\varphi|_{\mathcal{X}_T}$ is the zero morphism.

Precisely: Is $G$ representable? If it is a closed subscheme, what is the ideal defining it?

  • $\begingroup$ $\varphi$ is just a section of $F_1^\vee\otimes F_2$, so you are asking about whether the zero locus of a section of a vector bundle is closed, right? $\endgroup$ – Denis Nardin Dec 23 '16 at 0:08
  • $\begingroup$ This should follow from Corollaire 7.7.8 of EGA III. $\endgroup$ – Jason Starr Dec 23 '16 at 0:08
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    $\begingroup$ @DenisNardin. It is a bit more complicated than that. If you look at the zero locus of the morphism as a closed subset $Z$ of $\mathcal{X}$, the OP is, essentially, asking about the subset of $B$ of points $b$ such that the fiber $\mathcal{X}_b$ is completely contained in $Z$. $\endgroup$ – Jason Starr Dec 23 '16 at 0:10
  • $\begingroup$ @JasonStarr Ah right I misunderstood the meaning of zero locus. $\endgroup$ – Denis Nardin Dec 23 '16 at 0:43

I am just writing my comment above as an answer. This follows from Corollaire 7.7.8, EGA III.

Grothendieck, Alexander
Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Seconde partie.
Publications Mathématiques de l'IHÉS, 17 (1963), p. 5-91

In EGA, the result is only established if $\pi$ is projective (stronger than proper). However, the result has been significantly generalized both in the work of Martin Olsson and the work of Max Lieblich. By now, the projective hypothesis is unnecessary.

The point is, there exists a coherent sheaf $\mathcal{N}$ on $B$ and an equivalence between the functor $\mathcal{M}\mapsto \textit{Hom}_{\mathcal{O}_B}(\mathcal{N},\mathcal{M})$ and the functor $\mathcal{M}\mapsto\pi_*\textit{Hom}_{\mathcal{O}_{\mathcal{X}}}(F_1,F_2\otimes \pi^*\mathcal{M})$. In particular, for $\mathcal{M}$ equal to $\mathcal{O}_B$, the section $\phi$ of $\textit{Hom}_{\mathcal{O}_{\mathcal{X}}}(F_1,F_2)$ is equivalent to a morphism $\psi:\mathcal{N}\to \mathcal{O}_B$. The image of $\psi$ is the ideal sheaf of the closed subscheme of $B$ that represents your functor $G$.

Edit. I should add; I first learned about this particular question when I happened on it myself as a student reading through Hartshorne's book in the following variant: for a closed subscheme $C$ of $\mathcal{X}$, is the subfunctor $H$ of the Yoneda functor $h_B$ is representable by a closed subscheme, where $T\to B$ factors through $H$ if and only if $C\times_B T \to \mathcal{X}\times_B T$ is an isomorphism. If $\phi:F_1\to \mathcal{O}_X$ is a resolution of the ideal sheaf of $C$ by a locally free sheaf, then this is a special case of your question. I believe at the time that Brian Conrad referred me to EGA III.

The references for the generalized result of Lieblich and Olsson are as follows. The deduction from Olsson's paper requires first translating your question into an equivalent question about the Quot functor.

MR2233719 (2008c:14022) Reviewed
Lieblich, Max(1-PRIN)
Remarks on the stack of coherent algebras.
Int. Math. Res. Not. 2006, Art. ID 75273, 12 pp.

MR2183251 (2006h:14003) Reviewed
Olsson, Martin C.(1-IASP-SM)
On proper coverings of Artin stacks. (English summary)
Adv. Math. 198 (2005), no. 1, 93–106.

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    $\begingroup$ Thank you for all the references and taking the time to explain it. I dared not hope for a complete solution, you made my day! $\endgroup$ – Emre Dec 23 '16 at 13:10

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