CURVES AND VECTOR BUNDLES ON QUARTIC THREEFOLDS

Abstract

In this paper we study arithmetically Cohen-Macaulay (ACM for short) vector bundles <TEX>$\varepsilon$</TEX> of rank k <TEX>$\geq$</TEX> 3 on hypersurfaces <TEX>$X_r\;{\subset}\;{\mathbb{P}}^4$</TEX> of degree r <TEX>$\geq$</TEX> 1. We consider here mainly the case of degree r = 4, which is the first unknown case in literature. Under some natural conditions for the bundle <TEX>$\varepsilon$</TEX> we derive a list of possible Chern classes (<TEX>$c_1$</TEX>, <TEX>$c_2$</TEX>, <TEX>$c_3$</TEX>) which may arise in the cases of rank k = 3 and k = 4, when r = 4 and we give several examples.