Related papers: Towards a combinatorial Intersection Cohomology fo…
We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. With this…
N.C.Leung and V.Reiner showed that certain convexity conditions on a complete rational simplicial fan determine the sign of the signature of the Poincar\'e pairing on the cohomology of the associated toric variety. The purpose of the…
For a finite Coxeter group W and a Coxeter element c of W, the c-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the c-sortable elements of W. The main result of…
The $\mathbf{g}$-vector fan of a finite-dimensional algebra is a fan whose rays are the $\mathbf{g}$-vectors of its $2$-term presilting objects. We prove that the $\mathbf{g}$-vector fan of a tame algebra is dense. We then apply this result…
We characterise and investigate co-Higgs sheaves and associated algebraic and combinatorial invariants on toric varieties. In particular, we compute explicit examples.
We are interested in two classes of varieties with group action, namely toric varieties and spherical embeddings. They are classified by combinatorial objects, called fans in the toric setting, and colored fans in the spherical setting. We…
We continue the study of engineered complete intersections (ECI) -- an umbrella generality for a number of important objects in combinatoiral and applied algebraic geometry (such as nondegenerate toric complete intersections, critical loci…
In this paper, we investigate birational toric morphisms between quantum toric stacks -- namely, toric (analytic) stacks associated with fans whose cones may be irrational -- focusing on two primary classes of examples: weighted blow-ups…
Using the homogeneous coordinate ring construction of a toric variety IP defined by a complete simplicial fan and the methods of local cohomology theory we develop a framework for the calculation of cohomology groups H^{*}(IP, O(p)) of…
In a finite-dimensional real vector space furnished with a rational structure with respect to a subfield of the field of real numbers, every (simplicial) rational semifan is contained in a complete (simplicial) rational semifan. In this…
For an arbitrary finite Coxeter group W we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we…
We generalise the notion of Gr\"obner fan to ideals in R[[t]][x_1,...,x_n] for certain classes of coefficient rings R and give a constructive proof that the Gr\"obner fan is a rational polyhedral fan. For this we introduce the notion of…
Let $sigma$ be a complete simplicial fan in finite dimensional real Euclidean space $V$, and let $G$ be a cyclic subgroup of $GL(V)$ which acts properly on $\sigma$. We show that the representation of $G$ carried by the cohomology of…
A fan $F$ is \emph{endpoint-homogeneous} if for any two endpoints $e,e'$ of $F$, there is a homeomorphism $h: F \rightarrow F$ such that $h(e) = e'$. We prove there are uncountably many distinct homeomorphism types of endpoint-homogeneous…
By analogy with algebraic geometry, we define a category of non-linear sheaves (quasi-coherent homotopy-sheaves of topological spaces) on projective toric varieties and prove a splitting result for its algebraic K-theory, generalising…
Torsion sensitive intersection homology was introduced to unify several versions of Poincare duality for stratified spaces into a single theorem. This unified duality theorem holds with ground coefficients in an arbitrary PID and with no…
Let $\Sigma$ be a fan inside the lattice $\mathbb{Z}^n$, and $\mathcal{E}:\mathbb{Z}^n \rightarrow \operatorname{Pic}{S}$ be a map of abelian groups. We introduce the notion of a principal toric fibration $\mathcal{X}_{\Sigma, \mathcal{E}}$…
We consider subtorus actions on divisorial toric varieties. Here divisoriality means that the variety has many Cartier divisors like quasiprojective and smooth ones. We characterize when a subtorus action on such a toric variety admits a…
We conjecture that any connected component $Q$ of the moduli space of triples $(X,E=E_1+\dots+E_n,\Theta)$ where $X$ is a smooth projective variety, $E$ is a normal crossing anti-canonical divisor with a 0-stratum, every $E_i$ is smooth,…
A new construction of the associahedron was recently given by Arkani-Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct…