Inspired by Schubert varieties, given a two-dimensional linear subspace of $n \times n$ matrices $L$, we study the geometry of its compactification $Y_L$ in the Grassmannian $\Gr(n,2n)$. We determine degree and singular locus of the surface $Y_L$, and show that the algebraic invariants of $L$ under $\GL(2)\times \GL(n) \times \GL(n)$ correspond to intersection numbers of special divisors in $Y_L$. This is an ongoing work with F. Gesmundo and H. Keneshlou.

In a recent article Matsubara-Heo and Telen introduced the theory of principle matroid determinants, which parallels the wellstudied concept of principle A-determinants by Gelfand, Kapranov and Zelevinsky. The presented work aims to answer several questions that were left open in this framework, but are known in the classical version of the theory. We identify the conormal varieties of reciprocal linear spaces as a central object in these questions and study their multidegrees. The tropicalisation of the reciprocal conormal variety is shown to be set-theoretically equal to the so called conormal fan of the underlying matroid, a combinatorial object that has been used by Ardila, Denham and Huh to prove several logconcavity conjectures in matroid theory. In this way our work gives a concrete geometric meaning to these combinatorial models in the realisable case.

Speyer's 2005 f-vector conjecture asserted the nonnegativity of the coefficients of a matroid invariant he defined, notably the leading coefficient omega(M). When M is represented by a hyperplane arrangement, Larson identified omega(M) as the Euler characteristic of a certain anti-nef line bundle L^-1 on the wonderful compactification of the arrangement, up to a predictable sign. Eur and Larson used this to show that omega(M) is the leading coefficient in the "h* polynomial" of L, i.e. the numerator of the Hilbert series of the total coordinate ring of the image of the map to projective space defined by L. If the image is sufficiently nice (arithmetically Cohen-Macaulay), then the h* polynomial necessarily has nonnegative coefficients.

The work I'll be talking about completes the argument, giving a proof that omega is positive for all matroids. We show that wonderful varieties degenerate inside the permutahedral toric variety to a Cohen--Macaulay union of torus orbits controlled by a second matroid. This union of torus orbits can be defined when M is not representable, and the needed Euler characteristic can be computed on the degeneration.