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Let $X$ be a projective algebraic variety over a (perfect) field $k$.

Let $Aut(X):k \text{-}Alg \to Grp$ be the functor of points defined by

$$Aut(X) : A \mapsto Aut_{Spec (A)}(X \times_{k} Spec (A))$$

In this situation $Aut(X)$ is representable by an open subscheme of the Hom scheme $\mathsf{Hom}(X,X)$. It is therefore a group scheme locally of finite type over $k$.

Question 1: Can $Aut(X)$ be non-reduced? ($k$ has positive characteristic here of course) if so, what's an example of this (preferably one where $X$ is itself reduced)?

Let $Aut_0(X)$ be the connected component of the identity. As far as I understand the functor of points of the quotient $Aut(X)/Aut_0(X)$ is also representable by a scheme at this level of generality, lets denote it by $MCG(X)$.

Question 2: Is $MCG(X)(k)$ finitely presented as an abstract group?

Finally, are there any non-trivial examples for which $MCG(X)$ has been calculated?

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    $\begingroup$ For a projective manifold $X$, $Aut(X)$ is reduced due to Cartier as far as I know (in characteristic =0) but for positive Characteristic, see sciencedirect.com/science/article/pii/S0022404904000763 $\endgroup$
    – user21574
    Commented Nov 24, 2017 at 12:41
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    $\begingroup$ The etale component group of ${\rm{Aut}}_{X/k}$ can fail to be finitely generated even for smooth projective $X$ over $k = \mathbf{C}$ (and such examples also exist over $\mathbf{Q}$). The first examples were found by John Lesieutre: arxiv.org/abs/1609.06391 $\endgroup$
    – nfdc23
    Commented Nov 24, 2017 at 12:51
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    $\begingroup$ There are definitely smooth projective surfaces with non-smooth Aut-scheme over any algebraically closed field of positive characteristic. The topic of criteria for smoothness of such Aut-schemes is the focus of arxiv.org/pdf/1506.08843.pdf, which aids in where to look for non-smooth examples, and section 11 of that paper (and references therein!) discusses how to make some examples (including some interesting singular surfaces in low positive characteristic). $\endgroup$
    – nfdc23
    Commented Nov 24, 2017 at 13:19
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    $\begingroup$ @nfdc23 Sounds exactly like what I was looking for! These might answer both of my questions... $\endgroup$
    – Saal Hardali
    Commented Nov 24, 2017 at 14:15

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