Appears in collection : 2026 - T1 - WS3 - Integrating Research and Illustration in Number Theory

A cubic surface can be realized as the projective plane blown up at six points, and the twenty-seven lines on the surface can be deduced from the coordinates of those six points. This relationship is practically begging for a dynamic visualization, allowing a user to drag around the six points and watch the associated cubic surface (and lines) deform. Building this visualization requires finding the implicit equation of a cubic surface in real time (ideally 60 frames per second). This talk will discuss how searching for ways to quickly solve this implicitization problem led us to some interesting facts about Gröbner bases which have applications more generally to the implicitization of rational projective hypersurfaces with base points.