Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion of his 65th birthday. Abstract We study the fibers of projective morphisms and rational maps. We characterize the analytic spread of a homogeneous ideal through properties of its syzygy matrix. Powers of linearly presented ideals need not be linearly presented, but we identify a weaker li...

Table of contents 1. Primary ideals 1 2. Primary modules 3 3. Primary decompositions 6 4. More ways to get associated primes 11 5. Witnesses 14 6. Canonical primary decomposition 17 7. Associated primes of powers of an ideal 18 8. Primary decompositions of powers of an ideal 20 9. An algorithm for computing primary decompositions 24 10. Complexity of associated primes 28 11. Exercises 31 Biblio...