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Nef cone of flag bundles over a curve. (English) Zbl 1302.14025

Summary: Let \(X\) be a smooth projective curve defined over an algebraically closed field \(k\), and let \(E\) be a vector bundle on \(X\). Let \(\mathcal O_{\mathrm{Gr}_r(E)}(1)\) be the tautological line bundle over the Grassmann bundle \(\mathrm{Gr}_r(E)\) parameterizing all the \(r\)-dimensional quotients of the fibers of \(E\). We give necessary and sufficient conditions for \(\mathcal O_{\mathrm{Gr}_r(E)}(1)\) to be ample and nef, respectively. As an application, we compute the nef cone of \(\mathrm{Gr}_r(E)\). This yields a description of the nef cone of any flag bundle over \(X\) associated to \(E\).

MSC:

14H60 Vector bundles on curves and their moduli
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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References:

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