Craw, Alastair; Smith, Gregory G.
December 2008
American Journal of Mathematics;Dec2008, Vol. 130 Issue 6, p1509
Academic Journal
This paper proves that every projective tone 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 End (⊕i=0r Li) where £ := ((Multiple line equation(s) cannot be represented in ASCII text)χ, L1 ,…Lr) is a list of line bundles on a projective tone variety X. The quiver Q defines a smooth projective tonic variety, called the multilinear series ∣L∣, and a map X →.. 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 tone varieties. Under slightly stronger hypotheses on .C, the closed embedding identifies X with the fine moduli space of stable representations for the bound quiver (Q, R).


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