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ACM bundles, quintic threefolds and counting problems. (English) Zbl 1274.14019

On a general quintic threefold \(X\) in the projective space \(\mathbb P^4\), one can find indecomposable bundles of rank \(2\) which are Arithmetically Cohen Macaulay (i.e., the intermediate cohomology vanishes). These bundles are linked to a description of \(X\) as the vanishing locus of the Pfaffian of a skew-symmetric matrix. It is known that ACM bundles on a general quintic are rigid, and their Chern classes fit in a finite list (up to twist).
The authors improve our knowledge on ACM bundles on quintic threefolds, by finding the possible numerical shapes for the associated matrices. Moreover, they prove that all the possibilities are realized on a general quintic. This last result is achieved by a computer aided procedure. The authors also attach the problem of finding the (finite) number of ACM bundles on a general quintic threefold.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)

Software:

Macaulay2
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Full Text: DOI

References:

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