Uniform Steiner bundles
The well known result of Grothendieck says us every vector bundle on \(\mathbb{P}^1\) should be direct sum of line bundles. Therefore, if you restrict a rank \(r\) vector bundle \(\mathcal{E}\) on a variety to a line \(L\), then it should split as \(\mathcal{E}|_L = \bigoplus_{i}^{r} \mathcal{O}_L (a_i)\) where \(a_1 \leq \ldots \leq a_r\). If \(a_i\) is fixed for all lines on the variety then \(\mathcal{E}\) is called uniform. Among the uniform bundles, there are special classes which are called \(k\) uniform. A vector bundle is \(k\) uniform if \((a_1, \ldots, a_r)=(-k^{l_k}, -k+1^{l_{k-1}}, \ldots, 0^{l_0})\) where \(-i^{l_i}\) means we have \(l_i\) direct summand \(\mathcal{O}_{\mathbb{P}^1}(-i)\) in the splitting. In this paper, authors study \(k\)-type uniform Steiner bundles on \(\mathbb{P}^n\). A vector bundle \(\mathcal{E}\) on \(\mathbb{P}^n\) is called Steiner if it is defined as \[ 0 \rightarrow \mathcal{E} \rightarrow \mathcal{O}_{\mathbb{P}^n}^x \rightarrow \mathcal{O}_{\mathbb{P}^n}^c (1) \rightarrow 0. \] First, they give a sharp interval for the possible rank of a 1-type uniform Steiner bundle on \(\mathbb{P}^2\) with a fixed first Chern class (which is less than $-1$) and then prove the existence of them for any rank in this interval. Then, they prove a similar result for \(\mathbb{P}^n\). Their final result on 1-type uniform Steiner bundle on \(\mathbb{P}^n\) is that they can be obtained as an extension of a 1-type uniform Steiner bundle of possible minimum rank with the direct sum of trivial bundles. It is known that the lower rank uniform bundles on \(\mathbb{P}^n\) are homogenous. So, someone can try to find the largest integer \(r(n)\) such that any uniform vector bundle on \(\mathbb{P}^n\) of rank \(r(n)\) is homogeneous. In the paper, authors show that the constructed 1-type uniform Steiner bundle on \(\mathbb{P}^n\) with the possible minimum rank is not homogenous if the first Chern class is less than \(-3\). Therefore, they find an upper bound for \(r(n)\). They finalize their paper by obtaining results on \(k\)-type uniform Steiner bundles where \(k>1\). They give bounds (which are not sharp) on the rank of \(k\)-type uniform Steiner bundle on \(\mathbb{P}^2\) in terms of their first Chern classes and the number \(k\). Also, they show the existence of uniform Steiner bundles of different \(k\)-type on \(\mathbb{P}^n\).
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