Sets of points which project to complete intersections

2019/04/03 by Luca Chiantini, Chiantini, Luca, Juan Migliore +1 · 3 citations
Engineering · Mathematics · Computer Science · #Advanced Numerical Analysis Techniques #Algebraic Geometry and Number Theory #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1904.02047

Abstract

The motivating problem addressed by this paper is to describe those non-degenerate sets of points Z in \mathbb P³ whose general projection to a general plane is a complete intersection of curves in that plane. One large class of such Z is what we call (m,n)-grids. We relate this problem to the \em unexpected cone property \mathcal C(d), a special case of the unexpected hypersurfaces which have been the focus of much recent research. After an analysis of \mathcal C(d) for small d, we show that a non-degenerate set of 9 points has a general projection that is the complete intersection of two cubics if and only if the points form a (3,3)-grid. However, in an appendix we describe a set of 24 points that are not a grid but nevertheless have the projection property. These points arise from the F₄ root system. Furthermore, from this example we find subsets of 20, 16 and 12 points with the same feature.

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