Weddle schemes

Abstract

The classical Weddle surface is the locus of vertices of quadric cones through six points in $\mathbb{P}^3$ in linear general position. Equivalently, it is the closure of the locus of centers of projection from which those six points map to six points on a plane conic. Motivated by this 1850 construction of T. Weddle, we introduce $d$-Weddle schemes for finite point sets $Z\subset \mathbb{P}^n$, defined by an analogous projection-to-degree-$d$ condition. Our main tool is Macaulay duality, which yields a natural multiplication map in an Artinian algebra defined by powers of linear forms. This viewpoint connects $d$-Weddle schemes to unexpected cones and interprets them as non-Lefschetz loci for these multiplication maps. Parallel to this, we give an analysis from the point of view of interpolation matrices, and we explain the connections between these approaches. For a general set $Z\subset \mathbb{P}^n$ of $\binom{d+n}{n}$ points, we show that the $d$-Weddle scheme is a hypersurface and we compute its degree. We also study general sets whose cardinalities are "near" such a binomial coefficient, where the Weddle scheme has higher codimension. Returning to sets of six points (not always in linear general position), we discuss special configurations in which the appropriate Weddle scheme is reducible, or even nonreduced.