Row Ideals and Fibers of Morphisms

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. Since we are interested in the rational map, we may remove common divisors of W , and thus assume that I has codimension at least 2. A point q in the target P = P(W ) is a codimension 1 subspace Wq of W . We write Iq for the ideal generated by Wq. If F → G = S ⊗ W is a free presentation of I, then the composition F → G → S ⊗ (W/Wq) is called the generalized row corresponding to q, and its image is called the generalized row ideal corresponding to q. It is the ideal generated by a row in the presentation matrix after a change of basis. From this we see that the row ideal is simply Iq : I. The rational map φ is a morphism away from the algebraic set V (I), and as such we may form the fiber over a point q ∈ P. The saturated ideal of the schemetheoretic closure of this fiber is Iq : I ∞, which we call the morphism fiber ideal associated to q. In Section 3 we use these ideas to give bounds on the analytic spread of I by interpreting the analytic spread as 1 plus the dimension of the image of φ. The rational map φ also gives a correspondence Γ ⊂ P × P, which is the closure of the graph of the morphism induced by φ. There are projections