CEU Electronic Theses and Dissertations, 2015
| Author | Joo, Daniel |
|---|---|
| Title | Toric Quiver Varieties |
| Summary | Toric quiver varieties arise as GIT moduli spaces of quiver representations when the dimension vector is fixed to have value $1$ on every vertex, and come with a canonical embedding into projective space associated to a quiver polyhedron. We outline a procedure for their classification and show that up to isomorphism there are only finitely many $d$-dimensional toric quiver varieties in each fixed dimension $d$. We study the homogeneous toric ideals of projective toric quiver varieties in the canonical embedding associated to the GIT construction. It is shown that these toric ideals are always generated by elements of degree at most $3$. We demonstrate a method of subdividing quiver polytopes into 0-1 polytopes to obtain an estimate on the minimal degree of the generators in their toric ideals. As an application of this method it is then shown that up to dimension $4$ the toric ideal of every quiver polytope can be generated in degree $2$, with the single exception of the Birkhoff polytope $B_3$. We then investigate 0-1 polytopes arising from general toric GIT constructions and prove that under certain assumptions on the arrangement of singular points their toric ideals are generated in degree $2$. Finally, departing from the toric case, we prove a characterization of triples consisting of a quiver, a dimension vector, and a weight vector that yield smooth GIT moduli spaces in terms of forbidden descendants, which is in the flavour of characterizing classes of graphs by forbidden minors. |
| Supervisor | Domokos, Mátyás |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2015/joo_daniel.pdf |
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