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In this paper we present a smallest possible counterexample to the Numerical Terao's Conjecture in the class of line arrangements in the complex projective plane. Our example consists of a pair of two arrangements with $13$ lines. Moreover, we use the newly discovered singular matroid realization spaces to construct new examples of pairs of line arrangements having the same underlying matroid but different free resolutions of the Milnor algebras. Such rare arrangements are called Ziegler pairs in the literature.

We study smoothness of realization spaces of matroids for small rank and ground set. For $\mathbb{C}$-realizable matroids, when the rank is $3$, we prove that the realization spaces are all smooth when the ground set has $11$ or fewer elements, and there are singular realization spaces for $12$ and greater elements. For rank $4$ and $9$ or fewer elements, we prove that these realization spaces are smooth. As an application, we prove that $\text{Gr}^{\circ}(3,n;\mathbb{C})$ -- the locus of the Grassmannian where all Pl\"ucker coordinates are nonzero -- is not sch\"on for $n\geq 12$.

We prove that the open subvariety $\operatorname{Gr}_0(3,8)$ of the Grassmannian $\operatorname{Gr}(3,8)$ determined by the nonvanishing of all Pl\"ucker coordinates is sch\"on, i.e., all of its initial degenerations are smooth. Furthermore, we find an initial degeneration that has two connected components, and show that the remaining initial degenerations, up to symmetry, are irreducible. As an application, we prove that the Chow quotient of $\operatorname{Gr}(3,8)$ by the diagonal torus of $\operatorname{PGL}(8)$ is the log canonical compactification of the moduli space of $8$ lines in $\mathbb{P}^2$, resolving a conjecture of Hacking, Keel, and Tevelev. Along the way we develop various techniques to study finite inverse limits of schemes., Comment: 27 pages, 7 figures

We study valuated matroids, their tropical incidence relations, flag matroids and total positivity. This leads to a characterization of permutahedral subdivisions, namely subdivisions of regular permutahedra into generalized permutahedra. Further, we get a characterization of those subdivisions arising from positive valuated flag matroids., Comment: long version of extended abstract (FPSAC 2022, DMD 2022)

Given a collection of boundary divisors in the moduli space of stable genus-zero n-pointed curves, Giansiracusa proved that their intersection is nonempty if and only if all pairwise intersections are nonempty. We give a complete classification of the pairs (g,n) for which the analogous statement holds in the moduli space of n-pointed curves of genus g., Comment: 10 pages