Exceptional Sets and Fiber Products

Abstract

Exceptional sets where fibers have dimensions higher than the generic fiber dimension are of interest in mathematics and in application areas, such as the theory of overconstrained mechanisms.We show that fiber products promote such sets to become irreducible components, whereupon they can be found using techniques from numerical algebraic geometry for computing the irreducible decomposition. However, such a decomposition may contain components other than the exceptional loci we seek. We show that each irreducible component of the exceptional loci gives rise to a main component in a fiber product of sufficiently high order, and we give procedures for identifying these components. The methods are illustrated by finding the rulings of a general quadricin C³.