Let V be a linear space of dimension over a field K. By an arrangement we shall mean a finite collection of affine subspaces of V . If all of the subspaces in an arrangement A have codimension k then we say that A is an ( , k)arrangement. If k = 1 and so A is a hyperplane arrangement then we shall say that A is an -arrangement. Let A be an arrangement and S the coordinate ring for V . For each ...

We show that standard arguments for deformations based on dimension counts can also be applied over a (not necessarily Noetherian) valuation ring A of rank 1. Key intermediate results are a principal ideal theorem for schemes of finite type over A, and a theorem on subadditivity of intersection codimension for schemes smooth over A.

In this note we study the fibers of a rational map from an algebraic point of view. We begin by describing two interpretations of the word ‘fiber’. Let S = k[x0, . . . , xn] be a polynomial ring over an infinite field k, I ⊂ S an ideal generated by an r+1-dimensional vector space W of forms of the same degree, and φ the associated rational map P → P = P(W ). We will use this notation throughout...