Let $G$ be a finite group acting on a scheme $X$. It is shown in SGA1 (Expose 1, Prop 1.8) that a good categorical quotient exists (as a scheme) if and only if $X$ is a union of $G$-invariant open affines. Here we say that a quotient "$X\rightarrow X/G"$ is a good categorical quotient if it is a categorical quotient and satisfies the additional properties that the map $X\rightarrow X/G$ is integral, submersive, the fibers are $G$-orbits, and the point stabilizers surject onto the automorphism groups of the residue field extensions.

There certainly exist examples of finite group actions on schemes whose categorical quotient is not a scheme. My question is:

If $G$ is a finite group acting $S$-linearly on a prestable curve $f : C\rightarrow S$, must a good categorical quotient exist (as a scheme)? I'm happy to assume that $|G|$ is invertible on $S$.

In every situation I've encountered I've been able to avoid having to face this question (often one knows something extra about $C/S$ to ensure the quotient exists), but I have wondered for a long time if it is true in this generality.

Recall that a prestable curve is a proper flat finitely presented morphism whose geometric fibers are reduced schemes of equidimension 1 whose only singularities are ordinary double points.

First, some reductions. Clearly the question is Zariski local on the base, so we may assume $S$ affine. By Noetherian approximation we may also assume $S$ is of finite type over $\mathbb{Z}$. By the criterion above, it would suffice to check that $C$ satisfies property (AF) (any finite set of points is contained in an open affine). By a result of Kollar (Corollary 48, "Quotients by finite equivalence relations"), property (AF) (also called the Chevalley-Kleiman condition) descends along finite surjective morphisms, so we are further reduced to: $S = \text{Spec }A$ where $A$ is a normal domain of finite type over $\mathbb{Z}$.

If one can find a finite set of sections $\sigma_1,\ldots,\sigma_n$ lying in the smooth locus of $C/S$ and meeting every irreducible component of geometric genus $\le 1$ of every fiber, then $\mathcal{\omega}_{C/S}(3\sum_{i=1}^n\sigma_i)$ would be $f$-ample, which would imply property (AF) (EGA II, Corollaire 4.5.4). Such sections clearly exist etale locally, but it's unclear if the (AF) condition is etale local on the target. On the other hand one can try to find such sections finite-surjective-locally by taking the normalization of $S$ inside the residue field of smooth points of the generic fiber (and then induct on dimension), but it's unclear how to make sure the sections constructed this way avoid the singular points of closed fibers.

Another idle question I have is - Do there exist examples of a finite group $G$ acting on a scheme $X$ for which the categorical quotient $X/G$ exists (as a scheme), but a good categorical quotient does not?

  • 2
    $\begingroup$ You can realize Hironaka's example (in the appendix of Hartshorne's Algebraic geometry) as the total space of a proper, flat family of reduced, at-worst-nodal curves. $\endgroup$ Oct 12, 2021 at 23:07
  • $\begingroup$ Sorry if I missed a crucial point here but are not prestable curves quasi-projective (in fact even projective) over the base? In that case, the condition you mention is automatically satisfied. See eg Remark (4.17), p. 56 in page.mi.fu-berlin.de/elenalavanda/BMoonen.pdf . $\endgroup$ Oct 13, 2021 at 13:17
  • $\begingroup$ @DamianRössler I don't think prestable curves are projective (or quasi-projective). If they were, then you'd have a relatively ample sheaf, so Zariski locally on the base, $C$ would satisfy (AF) by EGA II Corollaire 4.5.4, which is enough for the quotient to be a scheme. $\endgroup$
    – Will Chen
    Oct 14, 2021 at 2:51
  • $\begingroup$ I see (my reference says the same thing). So your issue is really that relative prestable curves might not be relatively quasi-projective. $\endgroup$ Oct 14, 2021 at 8:40


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