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Simplicity and exceptionality of syzygy bundles over \(\mathbb{P}^n\). (English) Zbl 1349.14062
A syzygy bundle \(F\) on \(\mathbb P^n\) is a vector bundle which arises by splitting of a resolution on \(\mathbb P^n\) of the type: \[ 0\rightarrow \mathcal O^{b_{n+1}}(-d_{n+1})\rightarrow\dots\rightarrow\mathcal O^{b_1}(-d_1)\rightarrow\mathcal O^{b_0}(-d_0)\rightarrow 0 \] for integers \(d_0<d_1<\dots<d_{n+1}\).
The authors find criteria of simplicity and exceptionality of syzygy bundles. A vector bundle \(F\) is simple if \(\mathrm{Hom}(F,F)=\mathbb C\), and \(F\) is exceptional if it is simple and moreover \(\mathrm{Ext}^i(F,F)=0\) for all \(i>0\).
The property of exceptionality is interesting also for its relation with the stability of a vector bundle: a long-standing conjecture says that every exceptional bundle on \(\mathbb P^n\) is stable, see [J. M. Drezet and J. Le Potier, Ann. Sci. Éc. Norm. Supér. (4) 18, 193–243 (1985; Zbl 0586.14007)] and [M. C. Brambilla, Math. Nachr. 281, No. 4, 499–516 (2008; Zbl 1156.14013)] for more details.

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
13D02 Syzygies, resolutions, complexes and commutative rings
16E05 Syzygies, resolutions, complexes in associative algebras
Software:
BoijSoederberg
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References:
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