d+4 points on a rational normal curve

Castelnuovo's lemma states that d+3 points in general linear position in d-dimensional projective space uniquely determine a rational normal curve. However, the problem becomes more intricate for d+4 points, as it involves configurations of points in special position. The first notable examples of this phenomenon are Pascal’s and Brianchon’s theorems, which describe specific arrangements of six points on a conic in the projective plane. Pascal’s theorem asserts that six points A,...,F lie on a conic if the intersection points of the lines AB and DE, AF and CD, and EF and BC are collinear. Brianchon's theorem asserts that if six tangent lines to a conic are divided into two groups of three, the six vertices of the triangles formed by these groups also lie on a conic.

In this talk, we will present algebraic and combinatorial proofs of these classic results and discuss how these approaches can be extended to generalize Pascal’s and Brianchon’s theorems. Specifically, we will explore analogous statements for d+4 points on a rational normal curve of degree d. The results discussed are based on joint works with E. Carlini, N. Giansiracusa, H. Moon, and L. Schaffler.