Equations for point configurations to lie on a rational normal curve

Let Vd,n⊂(Pd)n be the Zariski closure ofthe set of n-tuples of points lying on a rational normal curve. The variety Vd,n was introduced because it provides interesting birational models of \barM0,n: namely, the GIT quotients Vd,n // LSLd+1.In this talk our goal is to find the defining equations of Vd,n. In thecase d=2 we have a complete answer. Fortwisted cubics, we use the Gale transform to find equations defining V3,n union the locus of degenerate point configurations. We prove a similar result for d≥4 and n=d+4. This is joint work with Alessio Caminata, Noah Giansiracusa, and Han-Bom Moon.