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The category theoretical definition of stacks (as given for instance in Giraud: Cohomologie non-abélienne) allow for arbitrary categories as targets (the stack condition only involves isomorphisms how …

I am currently teaching a course in algebraic geometry where one of the aims is to give an overview of the Enriques-Kodaira classification of surfaces. I am trying to throw in some modern aspects so I …

Certainly the fact that the ring of differential operators is non-Noetherian is an inconvenience but it is not clear if it is more than that. For instance one can define the notion of holonomic module …

Recall that if $G\rightarrow S$ is a flat group scheme, then a $G$-torsor is an $S$-scheme $X\rightarrow S$ with a $G$-action $G\times_SX\rightarrow X$ such that $G\times_SX\rightarrow X\times_SX$ giv …

I don't understand why the usual proof over a field base doesn't work over a (local) artinian base $R$ for a flat finite type group scheme over $R$: Let $A$ be the affine algebra of $G$. Take any basi …

Also for the étale fundamental group there is in fact always some universal cover. However, in the abstract way that Grothendieck formulated the theory of coverings a universal cover would only exist …

The same thing is true in positive characteristic, the degree of $c_n$ is equal to the Euler characteristic (except if you consider de Rham cohomology where it only is the Euler characteristic mod $p$ …

Yes, it is true, though an algebraic proof seems (there may be a simpler proof however) somewhat tricky. Such a line bundle lies in $\mathrm{Pic}^\tau(X)$. This is a general fact as a line bundle l …

You have to be careful with what you mean here. As your equations involve complex conjugation they do not define a complex variety. They do define a real algebraic variety. However, then you have to b …

I would be a little bit nervous about things when the group scheme is not smooth (there may not be any problems though) but you are interested in a smooth case anyway. To me it seems that the most nat …

Let me try an argument different from Christian's: $\sigma$ does not act freely as $\chi(\mathcal O_X)=1$ and hence not divisible by $2$. At a fixed point $x$, $\sigma$ acts by $\pm1$ on the fibre of …

If we have a central extension of group schemes $1\rightarrow B \rightarrow C\rightarrow A\rightarrow1$ with $A$ abelian, then we get a commutator mapping $\Lambda^2A\rightarrow B$ (of sheaves as $\La …

This is not an answer to the questions but some general comments. One should be aware that the relation between vector fields and Hasse derivations in characteristic $p$ is not at all analogous to the …

This is an attempt to realise Sándor's program of getting an example based on Kodaira vanishing or non-vanishing varying in a family. It will be done by keeping the surface fixed but varying the line …

We have that $\mathcal F^\ast$ is, by the pairing induced by the exterior algebra, canonically isomorphic to $\Lambda^{d-1}\mathcal F\bigotimes(\Lambda^d\mathcal F)^{-1}$. Now, in general if $\mathcal …