Toric variety
View article: Complete toric varieties with semisimple automorphism group
Complete toric varieties with semisimple automorphism group Open
Let $X$ be a complete toric variety. We give a criterion to decide whether $X$ decomposes as a product of complete toric varieties by analyzing the $1$-skeleton of its fan. More precisely, we prove that any direct-sum decomposition of the …
View article: Complex slices on a real variety
Complex slices on a real variety Open
Let $X$ be a real algebraic variety with set of complex points $X_{\mathbb C}$ and set of real points $X_{\mathbb R}$. A complex slice of $X$ is a transverse intersection of $X_{\mathbb R}$ with a complex subvariety $V$ of $X_{\mathbb C}$.…
View article: Complex slices on a real variety
Complex slices on a real variety Open
Let $X$ be a real algebraic variety with set of complex points $X_{\mathbb C}$ and set of real points $X_{\mathbb R}$. A complex slice of $X$ is a transverse intersection of $X_{\mathbb R}$ with a complex subvariety $V$ of $X_{\mathbb C}$.…
View article: The Tropical Geometry of Graph Spanning Trees
The Tropical Geometry of Graph Spanning Trees Open
This paper explores the deep connections between the combinatorial structure of spanning trees in a graph and the geometric framework of tropical geometry. We demonstrate that the set of all spanning trees of a given graph can be represent…
View article: The Tropical Geometry of Graph Spanning Trees
The Tropical Geometry of Graph Spanning Trees Open
This paper explores the deep connections between the combinatorial structure of spanning trees in a graph and the geometric framework of tropical geometry. We demonstrate that the set of all spanning trees of a given graph can be represent…
View article: On a Computational Approach to the Nash Blowup Problem
On a Computational Approach to the Nash Blowup Problem Open
In this paper we describe the implementation that led to the counterexamples to the Nash blowup conjectures recently discovered by the authors. We also provide new examples of toric varieties with prescribed singularities that are not reso…
View article: On a Computational Approach to the Nash Blowup Problem
On a Computational Approach to the Nash Blowup Problem Open
In this paper we describe the implementation that led to the counterexamples to the Nash blowup conjectures recently discovered by the authors. We also provide new examples of toric varieties with prescribed singularities that are not reso…
View article: Symmetry notions for toric Fanos
Symmetry notions for toric Fanos Open
We survey various notions of symmetry for toric varieties. These notions range from algebraic geometric, complex geometric, representation theoretic, combinatorial, convex geometric, to geometric stability. The main theorem gives the relat…
View article: $SU(n)$-structures through quotient by torus actions
$SU(n)$-structures through quotient by torus actions Open
We show that if $(X, g, J, ω)$ is a Kähler manifold with an $SU (n+s)$-structure and a Hamiltonian holomorphic action of a compact torus $T^s$ , then the usual symplectic quotient $Y$ inherits an $SU (n)$-structure provided the existence o…
View article: Symmetry notions for toric Fanos
Symmetry notions for toric Fanos Open
We survey various notions of symmetry for toric varieties. These notions range from algebraic geometric, complex geometric, representation theoretic, combinatorial, convex geometric, to geometric stability. The main theorem gives the relat…
View article: Classifying Fibers and Bases in Toric Hypersurface Calabi-Yau Threefolds
Classifying Fibers and Bases in Toric Hypersurface Calabi-Yau Threefolds Open
We carry out a complete analysis of the toric elliptic and genus-one fibrations of all 474 million reflexive polytopes in the Kreuzer-Skarke database. Earlier work with Huang showed that all but 29,223 of these polytopes have such a fibrat…
View article: Classifying Fibers and Bases in Toric Hypersurface Calabi-Yau Threefolds
Classifying Fibers and Bases in Toric Hypersurface Calabi-Yau Threefolds Open
We carry out a complete analysis of the toric elliptic and genus-one fibrations of all 474 million reflexive polytopes in the Kreuzer-Skarke database. Earlier work with Huang showed that all but 29,223 of these polytopes have such a fibrat…
View article: Effective Resistance in Simplicial Complexes as Bilinear Forms: Generalizations and Properties
Effective Resistance in Simplicial Complexes as Bilinear Forms: Generalizations and Properties Open
The concept of effective resistance, originally introduced in electrical circuit theory, has been extended to the setting of graphs by interpreting each edge as a resistor. In this context, the effective resistance between two vertices qua…
View article: Effective Resistance in Simplicial Complexes as Bilinear Forms: Generalizations and Properties
Effective Resistance in Simplicial Complexes as Bilinear Forms: Generalizations and Properties Open
The concept of effective resistance, originally introduced in electrical circuit theory, has been extended to the setting of graphs by interpreting each edge as a resistor. In this context, the effective resistance between two vertices qua…
View article: Bott manifolds of Bott--Samelson type and assemblies of ordered partitions
Bott manifolds of Bott--Samelson type and assemblies of ordered partitions Open
A Bott manifold is a smooth projective toric variety having an iterated $\mathbb{C} P^1$-bundle structure. A certain family of Bott manifolds is used to understand the structure of Bott--Samelson varieties (or Bott--Samelson--Demazure--Han…
View article: Bott manifolds of Bott--Samelson type and assemblies of ordered partitions
Bott manifolds of Bott--Samelson type and assemblies of ordered partitions Open
A Bott manifold is a smooth projective toric variety having an iterated $\mathbb{C} P^1$-bundle structure. A certain family of Bott manifolds is used to understand the structure of Bott--Samelson varieties (or Bott--Samelson--Demazure--Han…
View article: Closed-string mirror symmetry for dimer models
Closed-string mirror symmetry for dimer models Open
For all punctured Riemann surfaces arising as mirror curves of toric Calabi--Yau threefolds, we show that their symplectic cohomology is isomorphic to the compactly supported Hochschild cohomology of the noncommutative Landau--Ginzburg mod…
View article: Toric Geometry
Toric Geometry Open
Toric varieties provide a rich class of examples in algebraic geometry that benefit from deep and fruitful interactions with combinatorics. This workshop highlighted recent interactions between toric geometry and mirror symmetry, matroids,…
View article: Closed-string mirror symmetry for dimer models
Closed-string mirror symmetry for dimer models Open
For all punctured Riemann surfaces arising as mirror curves of toric Calabi--Yau threefolds, we show that their symplectic cohomology is isomorphic to the compactly supported Hochschild cohomology of the noncommutative Landau--Ginzburg mod…
View article: SU(n)-structures through quotient by torus actions
SU(n)-structures through quotient by torus actions Open
We show that if $(X,g,J,ω)$ is a Kähler manifold with an $SU(n+s)$-structure and a Hamiltonian holomorphic action of a compact torus $T^s$, then the usual symplectic quotient $Y$ inherits an $SU(n)$-structure provided the existence of spec…
View article: SU(n)-structures through quotient by torus actions
SU(n)-structures through quotient by torus actions Open
We show that if $(X,g,J,ω)$ is a Kähler manifold with an $SU(n+s)$-structure and a Hamiltonian holomorphic action of a compact torus $T^s$, then the usual symplectic quotient $Y$ inherits an $SU(n)$-structure provided the existence of spec…
View article: Complete Unconditional Resolution of the Hodge Conjecture via Classical Analytic and Arithmetic Methods
Complete Unconditional Resolution of the Hodge Conjecture via Classical Analytic and Arithmetic Methods Open
We establish an unconditional, purely classical resolution of the Hodge Conjecture within the Discrete Regulator (DR)–REFP framework. All analytic and arithmetic components are proved using classical Hodge theory, Demailly-type regularizat…
View article: Parking trees and the toric g-vector of nestohedra
Parking trees and the toric g-vector of nestohedra Open
We express the toric g-vector entries of any simple polytope as a nonnegative integer linear combination of its gamma-vector entries. Using this expression we obtain that the toric g-vector of the associahedron is the ascent statistic of 1…
View article: The Tropical Geometry of Graph Spanning Trees
The Tropical Geometry of Graph Spanning Trees Open
This paper explores the deep connections between the combinatorial structure of spanning trees in a graph and the geometric framework of tropical geometry. We demonstrate that the set of all spanning trees of a given graph can be represent…
View article: Parking trees and the toric g-vector of nestohedra
Parking trees and the toric g-vector of nestohedra Open
We express the toric g-vector entries of any simple polytope as a nonnegative integer linear combination of its gamma-vector entries. Using this expression we obtain that the toric g-vector of the associahedron is the ascent statistic of 1…
View article: Complete Unconditional Resolution of the Hodge Conjecture via Classical Analytic and Arithmetic Methods
Complete Unconditional Resolution of the Hodge Conjecture via Classical Analytic and Arithmetic Methods Open
We establish an unconditional, purely classical resolution of the Hodge Conjecture within the Discrete Regulator (DR)–REFP framework. All analytic and arithmetic components are proved using classical Hodge theory, Demailly-type regularizat…
View article: AdditiveToricVarieties: A Macaulay2 package for working with additive complete toric varieties
AdditiveToricVarieties: A Macaulay2 package for working with additive complete toric varieties Open
We introduce the AdditiveToricVarieties package for Macaulay2, a software system for algebraic geometry and commutative algebra, with methods for working with additive group actions on complete toric varieties. More precisely, we implement…
View article: Minimal Nilpotent Orbits and Toric Varieties
Minimal Nilpotent Orbits and Toric Varieties Open
Let $\overline{\mathcal{O}}_\textrm{min} \cap (\mathfrak n^+ \oplus \mathfrak n^-)$ be the collection of elements of $\mathfrak{sl}_{n+1}(\mathbb C)$ with rank less than or equal to $1$ and with all diagonal entries equal to zero. We show …
View article: Equivariant cohomology of juggling varieties in rank one
Equivariant cohomology of juggling varieties in rank one Open
We determine the ring structure of the torus-equivariant cohomology of rank-one juggling varieties with rational coefficients. By realizing these varieties as cyclic quiver Grassmannians, we construct a Knutson--Tao type basis for their eq…
View article: Equivariant cohomology of juggling varieties in rank one
Equivariant cohomology of juggling varieties in rank one Open
We determine the ring structure of the torus-equivariant cohomology of rank-one juggling varieties with rational coefficients. By realizing these varieties as cyclic quiver Grassmannians, we construct a Knutson--Tao type basis for their eq…