Generalized Weddle Loci

Chiantini and others in Arxiv:2026.25060v1 introduced the Weddle locus W(Z,d) of a finite set in the framework of Veronese embeddings of Pn. It is the Zariski closure in Pn of an even more interesting algebraic object, W0(Z,d). In this paper, we give several generalizations of this definition using zero-dimensional schemes: an abstract set-up and the extensions of the classical case to multiprojective spaces and projections from more than one point. One of our results includes “the sets are as small as possible for the given numerical data”. We raise a few open questions. Some of these are for general finite sets with prescribed cardinality. For the classical Weddle locus, we show that W(Z,d)=∅ if Z is a general subset of Pn with at least n+n+d−1n−1 points. For arbitrary zero-dimensional schemes Z, we describe it if deg(Z)≤2d+1. Over a smaller range, we describe it for multiprojective spaces.