Weil divisors on rational normal scrolls

Abstract

The aim of this paper is to study Weil divisors on a singular rational normal scroll X. In particular the author describes explicitly the group of divisorial sheaves associated to Weil divisors on X, via the direct image of the Picard group of the canonical resolution of X, which is very well known. The analysis naturally splits in two cases, depending on the codimension of the vertex of X (>2 or =2). When the codimension of the vertex is >2, then the two groups are isomorphic and there is a natural intersection form in X inherited from the one in the canonical resolution. When the codimension of the vertex is 2 this is no longer true and to overcome this problem the author introduces the concept of integral total transform of a divisor in the resolution of X. In this case the author defines a (non-linear) intersection number of two divisors, that represents the degree of their scheme-theoretic intersection if they are effective with no common components. Lastly there are some examples and applications.