We prove a generalization of both Pascal's Theorem and its converse, the Braikenridge-Maclaurin Theorem: if two sets of k lines meet in k 2 distinct points and if dk of those points lie on an irreducible curve C of degree d, then the remaining k(k − d) points lie on a unique curve S of degree k − d. If S is a curve of degree k − d produced in this manner using a curve C of degree d we say that S is d-constructible. For fixed degree d, we show that almost every curve of high degree is not d-constructible. In contrast, almost all curves of degree 3 or less are d-constructible. The proof of this last result uses the group structure on an elliptic curve and is inspired by a construction due to Möbius. The exposition is embellished with several exercises designed to amuse the reader.
Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebraic properties of the edge ideal from the deck of G, that is, from the collection of subgraphs obtained by removing a vertex from G. These properties include the Krull dimension, the Hilbert function, and all the graded Betti numbers i,j where j < n. We also state many further questions that arise from our study.
Traves and Wehlau [9] recently gave a straightedge construction that checks whether 10 points lie on a plane cubic curve. They also highlighted several open problems in the synthetic geometry of cubics. Hermann Grassmann investigated incidence relations among points on cubic curves in three papers [4,5,6] appearing in Crelle's Journal from 1846 to 1856. Grassmann's methods give an alternative way to check whether 10 points lie on a cubic. Using Grassmann's techniques, we solve the synthetic geometry problems introduced by Traves and Wehlau. In particular, we give straightedge constructions that find the intersection of a line with a cubic, find the tangent line to a cubic at a given point, and find the third point of intersection of this tangent line with the cubic. As well, given 5 points on a conic and a cubic and 4 additional points on the cubic, a straightedge construction is given that finds the sixth intersection point of the conic and the cubic. The paper ends with two open problems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.