论文标题
投影空间中的积分及其预测的配置
Configurations of points in projective space and their projections
论文作者
论文摘要
我们称一组点$ z \ subset {\ mathbb p}^{3} _ {\ mathbb c} $ an $（a，b）$ - geproci set（对于一般投影而言，如果将其从一般点$ p $转换为飞机上的一般交叉点）是$ $ $ $ a $ a $ a $ $ a $ a $ $ b y和$ b和$ b y和$ b和$ b和$ b。自2011年以来，我们称之为网格的示例。在这里，对于任何$ 4 \ leq a \ leq b $，我们以系统的方式构建了非构建非排效公司$（a，b）$ - geproci set。我们还表明，$ a = 3 $的唯一示例是来自$ d_4 $ root System的$（3,4）$ - geproci set，我们详细描述了$ d_4 $ configuration。我们还考虑了Geproci集的等价（各种意义）的问题，以及在真实物质上发生哪些集合，哪些集合发生的。我们确定了具有有趣属性的Geproci集的其他示例。我们还探讨了意外的锥与Geproci集之间的关系，并介绍了$ D $ - 伍德尔计划的概念，这是由有限积分的特殊投影引起的。这项工作启动了对古典几何领域的新观点的探索。我们在最后一章中提出并讨论了一系列开放问题。
We call a set of points $Z\subset{\mathbb P}^{3}_{\mathbb C}$ an $(a,b)$-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point $P$ to a plane is a complete intersection of curves of degrees $a$ and $b$. Examples which we call grids have been known since 2011. The only nongrid nondegenerate examples previously known had $ab=12, 16, 20, 24, 30, 36, 42, 48, 54$ or $60$. Here, for any $4 \leq a \leq b$, we construct nongrid nondegenerate $(a,b)$-geproci sets in a systematic way. We also show that the only such example with $a=3$ is a $(3,4)$-geproci set coming from the $D_4$ root system, and we describe the $D_4$ configuration in detail. We also consider the question of the equivalence (in various senses) of geproci sets, as well as which sets occur over the reals, and which cannot. We identify several additional examples of geproci sets with interesting properties. We also explore the relation between unexpected cones and geproci sets and introduce the notion of $d$-Weddle schemes arising from special projections of finite sets of points. This work initiates the exploration of new perspectives on classical areas of geometry. We formulate and discuss a range of open problems in the final chapter.
