Search for: All records

We describe a geometric, stable pair compactification of the moduli space of Enriques surfaces with a numerical polarization of degree 2, and identify it with a semitoroidal compactification of the period space. 

Abstract Projective duality identifies the moduli spaces $$\textbf{B}_n$$ and $$\textbf{X}(3,n)$$ parametrizing linearly general configurations of $$n$$ points in $$\mathbb{P}^2$$ and $$n$$ lines in the dual $$\mathbb{P}^2$$, respectively. The space $$\textbf{X}(3,n)$$ admits Kapranov’s Chow quotient compactification $$\overline{\textbf{X}}(3,n)$$, studied also by Lafforgue, Hacking, Keel, Tevelev, and Alexeev, which gives an example of a KSBA moduli space of stable surfaces: it carries a family of certain reducible degenerations of $$\mathbb{P}^2$$ with $$n$$ “broken lines”. Gerritzen and Piwek proposed a dual perspective, a compact moduli space parametrizing certain reducible degenerations of $$\mathbb{P}^2$$ with $$n$$ smooth points. We investigate the relation between these approaches, answering a question of Kapranov from 2003. 

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.