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Remarks on uniform bundles on projective spaces. (English) Zbl 1401.14188
If \(k\) is an algebraically closed field, a rank \(r\) vector bundle \({\mathcal E}\) on the projective space \({\mathbb P}^d\) is said to be uniform if for all lines \(L\) in \({\mathbb P}^d\), in the decomposition of the restriction \({\mathcal E}|_L\simeq \bigoplus_{1\leq i\leq r}{\mathcal O}_{{\mathbb P}^1}(a_i)\), the set of integers \(a_1,\ldots, a_r\) does not change. A vector bundle on \({\mathbb P}^d\) is said to be homogeneous if it is isomorphic to all its pullbacks by automorphisms of \({\mathbb P}^d\). Every homogeneous bundle is uniform, but there exist uniform bundles that are not homogeneous. When the characteristic of the base field \(k\) is positive, H. Lange [Manuscr. Math. 29, No. 1, 11–28 (1979; Zbl 0439.14001)] and L. Ein [Math. Ann. 254, No. 1, 53–72 (1980; Zbl 0431.14003)] have shown that uniform bundles of rank \(r\leq d\) on \({\mathbb P}^d\) are homogeneous. The main results of the paper under review exhibit uniform non-homogeneous rank \(d+1\) bundles on a projective space \({\mathbb P}^d\), where \(d\geq 2\), over an algebraically closed field of positive characteristic. The examples are obtained by constructing rank \(d+1\) uniform but not homogeneous toric bundles on the projective space \({\mathbb P}^d\), viewed as a toric variety. The bundles in these examples are adequate extensions of the Frobenius pullback of the tangent bundle on \({\mathbb P}^d\). Note that when the base field has characteristic zero, E. Ballico [Tsukuba J. Math. 72, No. 2, 215–226 (1983; Zbl 0532.14006)] has shown that all rank \(d+1\) uniform bundles on \({\mathbb P}^d\) are homogeneous.
MSC:
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14G17 Positive characteristic ground fields in algebraic geometry
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