We describe a compactification by stable pairs (also known as KSBA compactification) of the 4-dimensional family of Enriques surfaces which arise as the Z22-covers of the blow up of P2 at three general points branched along a configuration of three pairs of lines. Up to a finite group action, we show that this compactification is isomorphic to the toric variety associated to the secondary polytope of the unit cube. We relate the KSBA compactification considered to the Baily–Borel compactification of the same family of Enriques surfaces. Part of the KSBA boundary has a toroidal behavior, another part is isomorphic to the Baily–Borel compactification, and what remains is a mixture of these two. We relate the stable pair compactification studied here with Looijenga’s semitoric compactifications.

Pascal's theorem gives a synthetic geometric condition for six points a, horizontal ellipsis ,f in P2 to lie on a conic. Namely, that the intersection points ab over bar boolean AND de over bar , af over bar boolean AND dc over bar , ef over bar boolean AND bc over bar are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for d+4 points in Pd to lie on a degree d rational normal curve? In this paper we find many of these conditions by writing in the Grassmann-Cayley algebra the defining equations of the parameter space of d+4-ordered points in Pd that lie on a rational normal curve. These equations were introduced and studied in a previous joint work of the authors with Giansiracusa and Moon. We conclude with an application in the case of seven points on a twisted cubic.

Projective duality identifies the moduli spaces B-n and X(3, n) parametrizing linearly general configurations of n points in P-2 and n lines in the dual P-2, respectively. The space X(3, n) admits Kapranov's Chow quotient comp actification (X) over bar (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 P-2 with n "broken lines". Gerritzen and Piwek proposed a dual perspective, a compact moduli space parametrizing certain reducible degenerations of P-2 with n smooth points. We investigate the relation between these approaches, answering a question of Kapranov from 2003.

It is known that some GIT compactifications associated to moduli spaces of either points in the projective line or cubic surfaces are isomorphic to Baily-Borel compactifications of appropriate ball quotients. In this paper, we show that their respective toroidal compactifications are isomorphic to moduli spaces of stable pairs as defined in the context of the MMP. Moreover, we give a precise mixed-Hodge-theoretic interpretation of this isomorphism for the case of eight labeled points in the projective line. (c) 2021 Elsevier Inc. All rights reserved.

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and U(2) circle plus D-4(circle plus 2) lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of P-1 x P-1 branched along a specific (4, 4) curve. We show that, up to a finite group action, this stable pairs compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient (P-1)(8)//SL2 with the symmetric linearization.

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.

We explore the maximum likelihood degree of a homogeneous polynomial F on a projective variety X, MLD_F(X), which generalizes the concept of Gaussian maximum likelihood degree. We show that MLD_F(X) is equal to the count of critical points of a rational function on X, and give different geometric characterizations of it via topological Euler characteristic, dual varieties, and Chern classes.