Equations for point configurations to lie on a rational normal curve

Abstract

The parameter space of n ordered points in projective d-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in (P d ) n . The resulting variety was used to study the birational geometry of the moduli space M 0,n of n-tuples of points in P 1 . In this paper we turn to a more classical question, first asked independently by both Speyer and Sturmfels: what are the defining equations? For conics, namely d = 2, we find scheme-theoretic equations revealing a determinantal structure and use this to prove some geometric properties; moreover, determining which subsets of these equations suffice set-theoretically is equivalent to a well-studied combinatorial problem. For twisted cubics, d = 3, we use the Gale transform to produce equations defining the union of two irreducible components, the compactified configuration space we want and the locus of degenerate point configurations, and we explain the challenges involved in eliminating this extra component. For d ≥ 4 we conjecture a similar situation and prove partial results in this direction.

Point configurations, phylogenetic trees, and dissimilarity vectors

“…The equations for the Zariski closure of the locus in (P r −1 ) n mentioned in the preceding theorem were studied in ref. 14. While they are not known in full generality, we prove here that a…”

“…Two potential issues arise with this strategy: 1) generators for the ideal of Vr−1,n are not fully known in general and 2) not all of the generators for this ideal are SLr -invariant (remark 4.24 in ref. 14). However, we will establish in this section that the generators that are known from ref.…”

“…We can then write C = C1 ∪ C2 where, by lemma 2.6 in ref. 14, C1 and C2 are connected, possibly reducible, quasi-Veronese curves of positive degrees d1 and d2, respectively, with d1 + d2 = r − 1. The same lemma shows that the projective linear subspace spanned by a degree di quasi-Veronese curve is of dimension di .…”

“…This latter closed subvariety, and the ideal defining it, was the focus of ref. 14, where it is denoted Vr−1,n ⊆ (P r −1 ) n (since it parameterizes configurations on a quasi-Veronese curve).…”

“…However, we will establish in this section that the generators that are known from ref. 14 (all of which are SLr -invariant) suffice to cut out the tropicalization of ϕr (Gr(2, n) • ). We begin by reviewing these equations.…”