Singularities of zero sets of semi-invariants for quivers. (English) Zbl 1495.16016
Let \(k\) be an algebraically closed field, let \(Q\) be a finite quiver, and let \(\alpha\) be a dimension vector. Denote by \(\mathrm{Rep}(Q,\alpha)\) the affine variety of representations of \(Q\) over \(k\) with dimension vector \(\alpha\). Suppose \(\alpha\) is prehomogeneous under the action of the product of general linear groups \(\mathrm{GL}(\alpha)\) acting on \(\mathrm{Rep}(Q,\alpha)\). By [C. Riedtmann and G. Zwara, Ann. Sci. Éc. Norm. Supér. (4) 36, No. 6, 969–976 (2003; Zbl 1067.16022)], the nullcone in \(\mathrm{Rep}(Q,N\cdot \alpha)\) becomes a complete intersection for large numbers \(N\). The author shows that it also becomes reduced and gives sharp bounds for \(N\) if \(Q\) is a Dynkin quiver. Using Bernstein-Sato polynomials, the author studies whether zero sets of semi-invariants have rational singularities. In particular, he shows that if \(Q\) is a Dynkin quiver then all codimension one orbit closures have rational singularities.
Reviewer: Frauke Bleher (Iowa City)
MSC:
| 16G20 | Representations of quivers and partially ordered sets |
| 14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |
Citations:
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