Counting 3-Veronese surfaces
Counting 3-Veronese surfaces
An elementary fact in projective geometry states that d+3 general points in d-dimensional projective space determine a unique rational normal curve. Less known is the fact that this enumerative problem is only the 1-dimensional case of a system of problems in every dimension: the problem of counting Veronese images of projective spaces of any dimension.
Arthur Coble, in the 1920's, ingeniously demonstrated that 9 general points in 5-space determine 4 Veronese surfaces. In this paper, about a century later, we prove that 13 general points in 9-space determine 4246 Veronese surfaces. We attack the problem with many techniques, most of them very particular to the given problem. We wonder how much of what we do generalizes to other cases.