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Sets of points which project to complete intersections, and unexpected cones
Transactions of the American Mathematical Society ( IF 1.3 ) Pub Date : 2021-01-20 , DOI: 10.1090/tran/8290
Luca Chiantini , Juan Migliore

The motivating problem addressed by this paper is to describe those non-degenerate sets of points $Z$ in $\mathbb P^3$ 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 $F_4$ root system. Furthermore, from this example we find subsets of $20$, $16$ and $12$ points with the same feature.

中文翻译:

投影以完成交叉点的点集,以及意外的锥体

本文解决的激励问题是描述 $\mathbb P^3$ 中的那些非退化点 $Z$ 集,它们对一般平面的一般投影是该平面中曲线的完全交集。一大类这样的 $Z$ 就是我们所说的 $(m,n)$-grids。我们将这个问题与 {\em 意外锥属性} ${\mathcal C}(d)$ 联系起来,这是意外超曲面的一个特例,它一直是最近研究的重点。在对小 $d$ 的 ${\mathcal C}(d)$ 进行分析后,我们表明 $9$ 点的非退化集合具有一般投影,即两个三次方的完全交集当且仅当这些点形成一个 $(3,3)$-网格。然而,在附录中,我们描述了一组 $24$ 的点,它们不是网格但仍然具有投影属性。这些点来自 $F_4$ 根系统。此外,
更新日期:2021-01-20
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