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Spinor bundles on quadrics. (English) Zbl 0657.14006
By means of the spinor varieties (parametrising the families of k-planes in a smooth quadric $$Q_{2k}$$ or $$Q_{2k+1}$$ in $${\mathbb{C}}P^{n+1})$$, we can embed the quadric in a Grassmannian. A spinor bundle on Q is what the author calls the pull back of the universal bundle of the Grassmannian under this embedding. This gives a stable vectorbundle S of degree $$2^{k-1}$$ $$(n=2k$$ or 2k-1). - Actually, in the case of n even, there are two spinor bundles corresponding to the two different families of linear spaces of maximal dimension in the quadric. These generalise the universal bundle and the dual of the quotient bundle on $$Q_ 4\simeq Gr(1,3).$$
Work concerning these two bundles on Q by R. Hernandez and I. Sols [J. Reine Angew. Math. 360, 124-135 (1985)] is extended to $$Q_ 5$$ and $$Q_ 6$$ using the new spinor bundles. In particular, every rank 3 stable vectorbundle on $$Q_ 5$$, whose Chern classes are all 2, arises as the quotient E of the dual of the $$Q_ 5$$-spinor bundle S in a sequence $$0\to {\mathcal O}\to S^*\to E\to 0.$$
Finally $$CP^ 7\setminus Q_ 6$$ is the fine moduli space for all such stable bundles on $$Q_ 5$$ (as well as for stable bundles on $$Q_ 6$$ with $$c_ 1=2=c_ 2$$, $$c_ 3=(2,0)$$ or (0,2)).
Reviewer: P.Bryant

##### MSC:
 14D20 Algebraic moduli problems, moduli of vector bundles 14M15 Grassmannians, Schubert varieties, flag manifolds 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14D22 Fine and coarse moduli spaces
##### Keywords:
Grassmannian; spinor bundle; fine moduli space
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