arXiv subjects

Irem Portakal

Publications and source records attributed to Irem Portakal.

22 records · Page 2Linked to original sources

arXiv2022

Geometry of Dependency Equilibria

An $n$-person game is specified by $n$ tensors of the same format. We view its equilibria as points in that tensor space. Dependency equilibria are defined by linear constraints on conditional probabilities, and thus by determinantal quadrics in the tensor entries. These equations cut out the Spohn variety, named after the philosopher who introduced dependency equilibria. The Nash equilibria among these are the tensors of rank one. We study the real algebraic geometry of the Spohn variety. This variety is rational, except for $2 \times 2$ games, when it is an elliptic curve. For $3 \times 2$ games, it is a del Pezzo surface of degree two. We characterize the payoff regions and their boundaries using oriented matroids, and we develop the connection to Bayesian networks in statistics.

math.AG↗

arXiv2021

Rigid Gorenstein toric Fano varieties arising from directed graphs

A directed edge polytope $\mathcal{A}_G$ is a lattice polytope arising from root system $A_n$ and a finite directed graph $G$. If every directed edge of $G$ belongs to a directed cycle in $G$, then $\mathcal{A}_G$ is terminal and reflexive, that is, one can associate this polytope to a Gorenstein toric Fano variety $X_G$ with terminal singularities. It is shown by Totaro that a toric Fano variety which is smooth in codimension $2$ and $\mathbb{Q}$-factorial in codimension $3$ is rigid. In the present paper, we classify all directed graphs $G$ such that $X_G$ is a toric Fano variety which is smooth in codimension $2$ and $\mathbb{Q}$-factorial in codimension $3$.

math.CO↗

arXiv2020

On rigidity of toric varieties arising from bipartite graphs

One can associate to a bipartite graph a so-called edge ring whose spectrum is an affine normal toric variety. We characterize the faces of the (edge) cone associated to this toric variety in terms of some independent sets of the bipartite graph. By applying to this characterization the combinatorial study of deformations of toric varieties by Altmann, we present certain criteria for their rigidity purely in terms of graphs.

math.AG↗

arXiv2018

A note on deformations and mutations of fake weighted projective planes

It has been shown by Hacking and Prokhorov that if the projective surface X with quotient singularities and self-intersection number 9 has a smoothing to the projective plane, then X is the general fiber of a Q-Gorenstein deformation of the weighted projective plane with weights giving solutions to the Markov equation. This result has been understood and generalized by combinatorial mutations of Fano triangles by Akhtar, Coates, Galkin, and Kasprzyk. In this note, we study this result by utilizing polarized T-varieties and describe the associated deformation explicitly in terms of certain Minkowski summands of so-called divisorial polytopes.

math.AG↗