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Let $k$ be an algebraically closed field, $X$ a connected smooth curve over $k$, $G$ a connected reductive group over $k$, and $B \subset G$ a Borel subgroup.

Given a $G$-torsor $E$ on $X$ in the etale topology, I would like to understand why it has a reduction to $B$, in the simple case above (usually one finds some generalizations).

I already understood that on some open dense subset the $G$-torsor trivializes, and hence has a $B$-reduction. The next step is to consider $B\backslash E \to X$, which is a fiber bundle in the etale topology, with fiber $B \backslash G$. If I understand correctly, it is clear what is $B \backslash E$ as an algebraic space - since $E$ is etale-locally trivial and algebraic spaces glue in the etale topology. I also read somewhere that in fact $B \backslash E$ is a scheme, but I don't know the reason, and it says it is not important for the argument.

Now, I would like to take a section of $B\backslash E \to X$ over some dense open subset, and continue it to the missing points, by valuative criterion of properness. The problem is that for algebraic spaces in the criterion there appears some finite field extension, and anyway I don't understand algebraic spaces to well. So the question is - could somebody explain how to technically proceed to show that one can extend the section to the missing points?

Thank you, Sasha

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    $\begingroup$ You ask a pedagogical question not a mathematical one: which way to proceed will be easiest for you to understand? Here is another approach that might be more transparent. Read Iwahori's theorem in "Geometric Invariant Theory". This is the key to the Hilbert-Mumford criterion, so worth knowing anyway. If you lift your rational section $\sigma$ from $E/B$ to a section $\epsilon$ of $E$, Iwahori's theorem constructs a rational section $\tau$ of the maximal torus $T$ in $B$ so that $\tau\cdot\epsilon$ (locally) extends. Since $\tau\cdot \sigma$ equals $\sigma$, $\sigma$ extends. $\endgroup$
    – Jason Starr
    Feb 1, 2016 at 11:00
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    $\begingroup$ Let me take another whack at that. Iwahori's theorem proves that there is a discrete invariant -- an element in the cocharacter group of $B$ -- that measures whether or not $\epsilon$ extends. Because it is discrete, it can be determined after an 'etale base change, so now it is fine to work with algebraic spaces to determine it. Now adjust $\epsilon$ by $\tau$ over the original base, as suggested above, and then check $\tau\cdot \epsilon$ extends after base change. $\endgroup$
    – Jason Starr
    Feb 1, 2016 at 12:12
  • $\begingroup$ Thank you very much. I will try to look up your reference. Can't what you say in your second comment be somehow done for the extension of maps from a curve (to deduce what we want from the extension principle for usual varieties)? $\endgroup$
    – Sasha
    Feb 1, 2016 at 15:21
  • $\begingroup$ @Sasha - is the context of this question the Drinfeld-Simpson theorem? $\endgroup$
    – Elden Elmanto
    Feb 2, 2016 at 14:35
  • $\begingroup$ @EldenElmanto: Well, I just try to understand simple cases. $\endgroup$
    – Sasha
    Feb 2, 2016 at 14:43

2 Answers 2

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It is not too difficult to show that $B\backslash E$ is a scheme -- see e.g. http://arxiv.org/pdf/1308.3078.pdf, Prop. 4.4.

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It seems to me that perhaps one can argue as follows: We know that there is a section on some open dense $U \subset X$. we can assume that $X-U$ is one point. Find etale $V \to X$ which hits that point, and on which $E$ becomes trivial. Thus, on $V$ sections are maps to $B\backslash G$. On $V\times_X U$ we have section, and since $V$ is a smooth curve, we can continue it to a section on $V$. Thus, we get that we have compatible sections on $U$ and on $V$, and thus we get a section on $X$, since maps to $B \backslash E$ are a sheaf on quasi-compact schemes.

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