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I am aware that this is not a esearch question, but I don't know where else to ask. I have come across the fact that the stack of bundles of rank r and degree d over a curve of genus g with a rational point over a field k has a generic point of trdeg $(g-1)r^2+1$. First of all, I guess that we are cosidering by defintion the representative of the generic pint of smallest trdeg. I also know that the dimension of the stack is $(g-1)r^2$. In general, give a group G acting on a variety X, how can I compute the trdeg of the generic point of the quotient stack [X/G]?

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    $\begingroup$ $\dim(X)-\dim(G)+\dim(\text{generic stabilizer})$. This can be understood without stacks (or without generic points, for this matter): what is the smallest value of $\dim(Y)$ for a subvariety $Y\subset X$ such that $G\cdot Y\subset X$ is dense? $\endgroup$
    – t3suji
    Commented Jun 5, 2017 at 17:56

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