Curves in cages: an algebro-geometric zoo
| dc.creator | Katz, Gabriel | |
| dc.date | 2005-08-03 | |
| dc.date.accessioned | 2026-07-07T05:22:12Z | |
| dc.date.available | 2026-07-07T05:22:12Z | |
| dc.description | 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. | |
| dc.description | 16 pages, 9 figures in the EPS format, and boxedeps.sty macro | |
| dc.identifier | https://arxiv.org/abs/math/0508076 | |
| dc.identifier | http://arxiv.org/abs/math/0508076 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/75971 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Combinatorics | |
| dc.title | Curves in cages: an algebro-geometric zoo | |
| dc.type | text |