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We study certain moduli spaces of stable vector bundles of rank two on cubicand quartic threefolds. In many cases under consideration, it turns out thatthe moduli space is complete and irreducible and a general member has vanishingintermediate cohomology. In one case, all except one component of the modulispace has such vector bundles.

On some Moduli spaces of stable vector bundles on cubic and quartic threefolds