[Paper Review] Singularity Properties of Graph Varieties

Guy Kapon|arXiv (Cornell University)|May 14, 2019

Advanced Graph Theory Research2 references4 citations

TL;DR

This paper investigates the singularity properties of graph varieties—algebraic varieties parameterizing assignments of vectors to graph vertices subject to symplectic orthogonality constraints on edges. Using degeneration techniques, dimension bounds, and canonical bundle calculations, it establishes that for graphs of bounded degeneracy and sufficiently large symplectic space dimension, the varieties are irreducible, their projective versions are smooth precisely for forests, and the original varieties have rational singularities under improved dimension bounds, with explicit results for trees and general graphs.

ABSTRACT

For a graph $G=(V,E)$, and a symplectic vector space $(W, \left)$, we define a variety $X(G,W)$ consisting of all functions $w:V o W$ satisfying $\left = 0$ for any edge $\{u,v\}$ in $G$. We study the singularities of this varieties.

Motivation & Objective

• To determine the dimension and irreducibility of graph varieties $X(G,W)$ for graphs $G$ and symplectic vector spaces $W$.
• To characterize when the projective version $\widetilde{X}(G,W)$ is smooth, particularly in relation to graph structure.
• To improve bounds on the dimension of $W$ ensuring $X(G,W)$ has rational singularities, especially for forests and general graphs.
• To apply these results to representation theory, refining bounds on the growth of irreducible representations of algebraic groups.

Proposed method

• Define the graph variety $X(G,W)$ as the zero locus of a map $\phi: W^V \to \mathbb{F}^E$ sending $f$ to $\langle f(u), f(v) \rangle$ for each edge $\{u,v\}$.
• Use projection maps and fiber dimension arguments to prove irreducibility and dimension formulas under degeneracy and dimension constraints.
• Introduce a projective version $\widetilde{X}(G,W)$ via blow-up at the origin, and characterize its smoothness via combinatorial analysis of singular points.
• Apply degeneration techniques by splitting the graph into subgraphs (e.g., edges and levels), using inductive coloring and weighting schemes on vertices and edges.
• Construct explicit resolutions of singularities for trees and use Kodaira vanishing and canonical bundle calculations to prove rational singularities.
• Derive combinatorial coloring bounds using $p_4(D) = \frac{D^2(D+1)^2}{2} - 1$ to ensure rational singularities for general graphs when $\dim(W) \geq 2p_4(D)$.

Experimental results

Research questions

• RQ1 Under what conditions on the graph $G$ and symplectic space $W$ is the graph variety $X(G,W)$ irreducible and of dimension $\dim(W)|V| - |E|$?
• RQ2 When is the projective graph variety $\widetilde{X}(G,W)$ smooth, and how does this relate to the graph structure?
• RQ3 What is the minimal dimension of $W$ ensuring $X(G,W)$ has rational singularities, and how can this bound be improved for specific graph classes like forests or trees?
• RQ4 How do the singularity properties of $X(G,W)$ relate to representation-theoretic bounds on algebraic groups?

Key findings

• For a $d$-degenerate graph $G$ with maximal degree $D$, if $\dim(W) \geq D + d$, then $X(G,W)$ is irreducible and has dimension $\dim(W)|V| - |E|$.
• The projective graph variety $\widetilde{X}(G,W)$ is smooth if and only if $G$ is a forest, when $\dim(W) \geq D + d$.
• If $G$ is a forest and $\dim(W) \geq D + 1$, then $X(G,W)$ has rational singularities.
• For general graphs, $X(G,W)$ has rational singularities when $\dim(W) \geq 2p_4(D)$, where $p_4(D) = \frac{D^2(D+1)^2}{2} - 1$, improving prior bounds.
• For trees, the bound can be further improved to $\dim(W) \geq D$, showing rational singularities hold under a tighter dimension condition.
• The results refine bounds in [AA15] and [AA18], leading to improved growth estimates for irreducible representations of algebraic groups.