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Let $X$ be the Deligne-Mumford compactification of $\mathcal{M}_{0,n}$. Suppose I have two (big) line bundles $L$ and $L'$ on $X$ and that I want to show that they are the same element of $Pic(X)$. Of course one can check if they have same intersections with the F-curves, or just check that they coincide over an open subset of $X$ whose complement has codimension at least 2. Do you see any other reasonable way? Perhaps considering the maps induced by their global sections?

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  • $\begingroup$ How are the line bundles presented/given to you? $\endgroup$
    – Kevin H. Lin
    Commented Oct 8, 2011 at 1:15
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    $\begingroup$ Your first suggestion (compare their restrictions to F-curves) seems a natural approach. Alternatively, you could use one of the explicit blow-up constructions of $\overline{M}_{0, n}$, which also produce a natural basis for $\mathrm{Pic}$. Or if they are given as linear combinations of boundary divisors, it might be most convenient to use the known explicit relations between them (which are all pull-backs of the relation "All boundary divisors on $M_{0, 4}$ have the same divisor class."). $\endgroup$
    – Arend Bayer
    Commented Oct 8, 2011 at 18:50
  • $\begingroup$ Actually the two line bundles that I am considering are GIT-line bundels. That is: the map to the projective space that they induce via global sections yields a map to a GIT-compactification of $\overline{M}_{0,n}$ (non functorial compactification, due to the existence of strictily semi-stable points). Thank you for your comments! $\endgroup$
    – IMeasy
    Commented Oct 10, 2011 at 8:53

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