Projective toric varieties as fine moduli spaces of quiver representations
This paper proves that every projective toric variety is the fine moduli space for stable
representations of an appropriate bound quiver. To accomplish this, we study the quiver $ Q
$ with relations $ R $ corresponding to the finite-dimensional algebra $\mathop {\rm
End}\nolimits (\textstyle\bigoplus\nolimits_ {i= 0}^{r} L_i) $ where ${\cal L}:=({\scr O} _X,
L_1,\ldots, L_r) $ is a list of line bundles on a projective toric variety $ X $. The quiver $ Q $
defines a smooth projective toric variety, called the multilinear series $|{\cal L}| $, and a map …
representations of an appropriate bound quiver. To accomplish this, we study the quiver $ Q
$ with relations $ R $ corresponding to the finite-dimensional algebra $\mathop {\rm
End}\nolimits (\textstyle\bigoplus\nolimits_ {i= 0}^{r} L_i) $ where ${\cal L}:=({\scr O} _X,
L_1,\ldots, L_r) $ is a list of line bundles on a projective toric variety $ X $. The quiver $ Q $
defines a smooth projective toric variety, called the multilinear series $|{\cal L}| $, and a map …
This paper proves that every projective toric variety is the fine moduli space for stable representations of an appropriate bound quiver. To accomplish this, we study the quiver with relations corresponding to the finite-dimensional algebra where is a list of line bundles on a projective toric variety . The quiver defines a smooth projective toric variety, called the multilinear series , and a map . We provide necessary and sufficient conditions for the induced map to be a closed embedding. As a consequence, we obtain a new geometric quotient construction of projective toric varieties. Under slightly stronger hypotheses on , the closed embedding identifies with the fine moduli space of stable representations for the bound quiver .
Project MUSE