笼中曲线：代数几何动物园
Curves in cages: an algebro-geometric zoo
摘要：
本文研究了在射影平面上含有给定的特殊有限集X的平面代数曲线族。我们专注于这样的情况下，其中的集合X是由横向相交的线对从两个给定的有限的基数d家庭。两个族的所有直线的并集称为笼形，交点X由第一个族的直线与第二个族的直线相交的d^2个点组成。X的点称为笼的节点。我们研究节点集X的子集A，使得任何包含A的d次平面代数曲线C也必须包含X。作为推论，我们得到了著名的帕斯卡定理的一些推广，这些推广使用了内切于二次曲线的多边形（而不是帕斯卡定理中的六边形）。我们的结果与经典的Chasles和Bacharach定理密切相关。虽然节点集X所产生的笼是相当特殊的比较，更一般的完整的交叉研究Bacharach，笼为我们提供了一个更好的把握组合的子集A的X与上述属性。
英文摘要：
The paper is concerned with families of plane algebraic curves that contain a given and quite special finite set X of points in the projective plane. We focus on the case in which the set X is formed by transversally intersecting pairs of lines selected from two given finite families of cardinality d. The union of all lines from both families is called a cage, and the intersection X consists of d^2 points at which a line from the first family intersects a line from the second. The points of X are called the nodes of the cage. We study the subsets A of the nodal set X such that any plane algebraic curve C of degree d that contains A must contain X as well. As a corollary, we get a few generalizations of the famous Pascal theorem, generalizations that employ polygons (instead of hexagons as in Pascal's theorem) inscribed in a quadratic curve. Our results are closely related to the classical theorems of Chasles and Bacharach. Although the nodal sets X produced by our cages are quite special in comparison to more general complete intersections studied by Bacharach, the cages provide us with a much better grip on the combinatorics of subsets A of X with the property that is described above.
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