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In 2004, Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map with a weighted variant, we show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter and Speyer envisioned. Moreover, we provide a geometric interpretation in terms of configurations of points on rational normal curves and construct a finite tropical basis that yields an explicit characterization of weighted dissimilarity vectors. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Here we study (in positive characteristic) integral curves X ⊂ ℙr with secant degree one, i.e., for which a general P ∈ Seck-1 (X) is in a unique k-secant (k - 1)-dimensional linear subspace. [ABSTRACT FROM AUTHOR]

By using a computer we are able to pose a conjecture for the expected number of generators of the ideal of a non-special general irreducible curve in Pr with degree d, genus g, for d≥r+g. We prove the conjecture for C of degree d≤60. [Copyright &y& Elsevier]