Do coarse moduli spaces respect Galois actions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T01:20:31Z http://mathoverflow.net/feeds/question/69064 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69064/do-coarse-moduli-spaces-respect-galois-actions Do coarse moduli spaces respect Galois actions? Makhalan Duff 2011-06-28T22:52:17Z 2011-06-30T22:30:54Z <p>To explain, I will use the following concrete example: Let $\mathcal{M}_g$ be the functor for the moduli problem of classifying genus $g$ smooth projective curves (taking a scheme $S$ to the set of ways that $S$ parametrizes genus $g$ curves). This, as is well known, has a coarse moduli space: $M_g$.</p> <p>Let $\sigma \in Aut(\mathbb{C})$, and say that we begin with a specific $\mathbb{C}$-curve of genus $g$. This $\sigma$ may act in two ways on this curve (by act I mean turn it into a different curve, not act as meaning an automorphism): 1. By taking a fiber product over the automorphism $Spec(\mathbb{C}) \rightarrow Spec(\mathbb{C})$ (this is the usual action people use, and has nothing to do with the moduli space) 2. By letting $\sigma$ act on $M_g(\mathbb{C})$, the point corresponding to our curve, will be taken to another $\mathbb{C}$-point, which in turn corresponds to another curve.</p> <p>The question is: do these two actions agree in general? Are there conditions that need to be fulfilled for this to be true?</p> <p>As an aside, this is true if the moduli space is fine. For example, if the moduli of genus $g$ curves were fine (which it isn't), then the curve corresponding to $Spec(\mathbb{C})\rightarrow M_g$ will be given simply by taking the fiber product with the universal family $E_g\rightarrow M_g$. Since fiber products commute, we get that the two actions above agree. This reasoning breaks down, however, if the moduli space is not fine.</p> http://mathoverflow.net/questions/69064/do-coarse-moduli-spaces-respect-galois-actions/69071#69071 Answer by Ramsey for Do coarse moduli spaces respect Galois actions? Ramsey 2011-06-29T00:01:54Z 2011-06-29T00:01:54Z <p>If $X$ is the coarse moduli scheme associated to a functor $F$ on schemes, then in particular, there is a natural transformation $F\to h_X$, where $h_X$ is the functor of points of $X$. </p> <p>Unless I am missing something, if you apply the naturality of this transformation to the map $\mathrm{Spec}(\mathbb{C}) \to \mathrm{Spec}(\mathbb{C})$ associated to your automorphism $\sigma$, this implies that the two actions coincide in your case, since the associated map $F(\mathrm{Spec}(\mathbb{C})) \to F(\mathrm{Spec}(\mathbb{C}))$ is given by the fiber product of the curve, and the associated map $X(\mathrm{Spec}(\mathbb{C})) \to X(\mathrm{Spec}(\mathbb{C}))$ is just the action on the points. I wish I could make the obvious 2-by-2 commutative diagram...</p> http://mathoverflow.net/questions/69064/do-coarse-moduli-spaces-respect-galois-actions/69220#69220 Answer by Felipe Voloch for Do coarse moduli spaces respect Galois actions? Felipe Voloch 2011-06-30T22:30:54Z 2011-06-30T22:30:54Z <p>There are curves (with automorphisms) for which the field of definition is not the field of moduli. Concretely, there are curves $C$ with corresponding point $P \in M_g$, such that $C$ cannot be defined over $K =\mathbb{Q}(P)$, only over larger fields. That means that there are automorphisms of $\mathbb{C}$ (fixing $K$) whose action don't change $P$ but change $C$.</p>