PUB - Publikationen an der Universität Bielefeld

We describe all the schematic limits of families of divisors associated to a given family of rank-r linear series on a one-dimensional family of projective varieties degenerating to a connected reduced projective scheme X defined over any field, under the assumption that the total space of the family is regular along X. More precisely, the degenerating family gives rise to a special quiver Q, called a Zn-quiver, a special representation L of Q in the category of line bundles over X, called a maximal exact linked net, and a special subrepresentation V of the representation H0(X, L) induced from L by taking global sections, called a pure exact finitely generated linked net of dimension r + 1. Given g = (Q, L, V) satisfying these properties, we prove that the quiver Grassmanian LP(V) of subrepresentations of V of pure dimension 1, called a linked projective space, is local complete intersection, reduced and of pure dimension r. Furthermore, we prove that there is a morphism LP(V) → HilbX , and that its image parameterizes all the schematic limits of divisors along the degenerating family of linear series if g arises from one.