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ag.algebraic geometry – Finite decision by semi-stable bundles
Devlin Mallory’s strategy is actually rectify, and in reality the status is plane a miniature higher than he urged.
Let $V$ breathe an capricious vector bundle of rank $n$. Let $W$ breathe a steady bundle of rank $n+1$ (if one exists) or a semistable bundle. Fix an copious bundle $mathcal O(1)$.
I pretense that for $m$ sufficiently sizable, there’s a map $W(-m) to V$ that’s surjective and whose kernel is a line bundle (and thus is mechanically steady).
It suffices to take $m$ sizable sufficient that $Votimes W^vee otimes mathcal O(m)$ is globally generated. We can perceive sections of this line bundle as maps $W(-m) to V$, and the fiber of the part at a degree corresponds to the fiber of the map at a degree.
Maps from an $m+1$-dimensional vector area to an $m$-dimensional vector area have complete rank outdoors of a codimension $2$ locus. Thus amongst world sections of $V otimes W^vee otimes mathcal O(m)^vee$, these with out complete rank at a given level have codimension $2$. So these which fail to have complete rank at a number of factors of $C$ have codimension $1$. Thus we are able to discover a part which has complete rank at each level, which suggests it’s surjective and its kernel is a line bundle.
Over finite fields, this codimension dispute would not labor, however a similar counting dispute does.
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