Related papers: On the birational geometry of matroids
A famous construction of Gelfand, Kapranov and Zelevinsky associates to each finite point configuration $A \subset \mathbb{R}^d$ a polyhedral fan, which stratifies the space of weight vectors by the combinatorial types of regular…
It is proved that the group of holomorphic automorphisms of holomorphically homogeneous nondegenerate (finite Bloom-Graham type + holomorphic nondegenaracy) model surface Q is a subgroup of the group of birational automorphisms of the…
The dual graph $\Gamma(h)$ of a regular triangulation $\Sigma(h)$ carries a natural metric structure. The minimum spanning trees of $\Gamma(h)$ recently proved to be conclusive for detecting significant data signal in the context of…
A new class of full fans in an euclidean space - tight fans - is introduced. Such fans are defined using a property of local symmetry in a face of a tiling. Tight fans are related to the theory of parallelotopes in an euclidean space. A…
We study the iterative behavior of the family of 3-step linear fractional recurrences and the family of birational maps they define. We determine all the possible periodicities within this family or, equivalently, the birational maps of…
We construct first examples of Fano varieties with torsion in their third cohomology group. The examples are constructed as double covers of linear sections of rank loci of symmetric matrices, and can be seen as higher-dimensional analogues…
Recently, many examples of smooth fans that admit a transitive homeomorphism have been constructed. For example, a family of uncountably many pairwise non-homeomorphic smooth fans that admit transitive homeomorphisms was constructed. In…
We characterize 2-dimensional complexes associated canonically with basis graphs of matroids as simply connected triangle-square complexes satisfying some local conditions. This proves a version of a (disproved) conjecture by Stephen Maurer…
A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering…
We describe a natural geometric relationship between matroids and underlying flag matroids by relating the geometry of the greedy algorithm to monotone path polytopes. This perspective allows us to generalize the construction of underlying…
In this paper we present some geometrical representations of the Frobenius group of order $21$ (henceforth, $F_{21}$). The main focus is on investigating the group of common automorphisms of two orthogonal Fano planes and the automorphism…
Toric varieties play an important role both in symplectic and complex geometry. In symplectic geometry, the construction of a symplectic toric manifold from a smooth polytope is due to Delzant. In algebraic geometry, there is a more general…
Let $X_\Sigma$ be a smooth, not necessarily compact toric variety. We show that a certain complex, defined in terms of the fan $\Sigma$, computes the integral cohomology of $X_\Sigma$, including the module structure over the homology of the…
The generators of the group of birational automorphisms of any Severi-Brauer surface non-isomorphic over an algebraically non-closed field to the projective plane are explicitly described.
We study automorphism groups of formal matrix algebras. We also consider automorphisms of ordinary matrix algebras (in particular, triangular matrix algebras).
Motivated by the vast literature of quantum automorphism groups of graphs, we define and study quantum automorphism groups of matroids. A key feature of quantum groups is that there are many quantizations of a classical group, and this…
The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we…
Let S be a compact Riemann surfaces of genus g >= 2 and G a conformal automoprhism group of order n acting on S. In this paper we give the definition of an adapted generating set and an adapted basis for the first homology group of such a…
Viewing a fan as a partially ordered set (of cones) we consider a category of sheaves on the fan which corresponds to a category of equivariant sheaves on the corresponding toric variety if the fan is rational. In this category we define an…
Lattices induced by coverings arise naturally in matroid theory and combinatorial optimization, providing a structured framework for analyzing relationships between independent sets and closures. In this paper, we explore the structural…