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Let $X$ be a scheme locally of finite type over a sufficiently "nice" base scheme $S$ (nice in sense of reasonable "finiteness conditions", for sake of simplicity let's start as simple as possible and assume $S$ affine with Noetherian ring), and $G$ a affine (#edited later, to keep the story simpler) $S$-group scheme acting on $X$, ie there exist $S$ morphism $m:G \times X \to X$ satisfying usual compatibility stuff [main reference: D. Mumford's GIT]

Let $f: T \to X$ be any $T$-valued point of $X$. Then we define the orbit of $f$ with respect to this action to be the image of the map $\psi_f: G \times_S T \to X \times_S T$, where the map is defined as composition $\psi_f:=(m \cdot (1_G \times f), pr_2)$.

Question: If one talks in context of GIT about "the orbit", does one assumes implicitly that it carries certain "natural" scheme structure? Which one? And what one considers then literally, the set-theoretical orbit as underlying set (with this hypothetical mysterious scheme structure) or the scheme theoretic closure (see below on clarification about the latter object)?

Note, that one can always associate canonically to an image a scheme theoretic closure, an object with scheme structure induced by smallest quasi-coherent ideal sheaf contained in $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to f_*\mathcal{O}_ X)$.

In nice enough situations (eg map quasi-compact and quasi-separated) the underlying topological space of this ideal sheaf coincides with the topological closure of set theoretic image.

But well, what is by convention in context of GIT "the orbit" as scheme? Does (maybe in nice situations) the set theoretic image carry a "canonical" scheme structure which one tacitly assumes in literature, or does one by "orbit" mean in this context always the schematic theoretic closure described above, which in turn has intrinsically given scheme structure?

My motivation is based one my observation that in the literature on GIT one uses often the orbit as "existing object", but I nowhere found a profound discussion which scheme structure it should carry, if one seriously want to study it with algebro-geometric methods.

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    $\begingroup$ The orbit can be nonreduced, e.g., if $G$ itself is nonreduced. Is that what you are asking? $\endgroup$
    – Jason Starr
    Commented Oct 19, 2023 at 12:28
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    $\begingroup$ Note that for algebraic groups over a field, the orbit is always locally closed, so it makes sense to view it as an open subscheme of the scheme-theoretic image. I'm not sure if local closedness is true over a general base. I do expect that the situation for affine group schemes is much better (which is often the case you're interested in when doing geometric invariant theory). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Oct 19, 2023 at 12:29
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    $\begingroup$ The orbit can also be thought of as $G/H$ where $H$ is the stabiliser, making it a question about representability of quotients. There is quite a bit to be said about this; a starting point is SGA 3$_\text{I}$, Exp. VI$_\text{B}$, §9. I am not really an expert in this, so I don't know when it can and when it cannot be done. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Oct 19, 2023 at 12:31
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    $\begingroup$ @JasonStarr: not exactly, I posed a much simpler question, namely just if the orbit in context of GIT carries a "natural" scheme structure. I saw very often in literature (just to pick one example, eg this one , see in Def 1.4, page 107. There one tacitly regards an orbit as a scheme (... since one talks about it's smoothness), and my question is simply what is the $\endgroup$
    – user267839
    Commented Oct 19, 2023 at 19:52
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    $\begingroup$ scheme structure the orbit is endowed with? You remarked in your comment, that a orbit can be non reduced. Sure, but that's the point! When you say that it " is nonreduced", then you already know it's scheme structure, right? And the question is from "where" does one know the scheme structure of the orbit carries? Is there a canonical way to give the orbit a scheme structure from knowledge of the action and the involved spaces? $\endgroup$
    – user267839
    Commented Oct 19, 2023 at 19:59

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