### Degeneracy Loci of Vector Bundle Maps and Ampleness

Jørgen Anders Geertsen
2002 Mathematica Scandinavica
jørgen anders geertsen don't vanish at all, namely those of the form (λ, σ ) where λ ∈ k \ {0}, and the other sections vanish too much, namely on a line, i.e. in codimension 1. So even for very nice globally generated bundles it can happen that no X k (φ) has the 'right' dimension. We define the number t k = t k (X, E, F ) as follows: Note that since we can take for φ the zero-morphism 0 : E → F , which has X k (0) = X for all k, the number t k is well-defined and t k ≤ dim X. Also, the
more » ... . Also, the inequality m k ≤ t k holds for all k (0 ≤ k ≤ min{e, f } as always). For instance, in the above example, m 0 = 0 and t 0 = 1. We are concerned with the following relative situation. Suppose E, F are vector bundles on X as above. Let Y be an irreducible algebraic variety and suppose f : X → Y is a proper surjective map. We say that the bundle E * ⊗ F is ample relative to f , written 'rel f ', if it is ample on all fibers of f , i.e. the restriction E * ⊗ F | f −1 (y) is an ample vector bundle for all (closed) points y ∈ Y . Our purpose is to describe what happens to the images of the degeneracy loci under the map f . The results are the following.