Essential dimension and genericity for quiver representations (Q2187343)

In this article, the author continues the study of the essential dimension of an algebraic stack introduced in [\textit{P. Brosnan} et al., J. Eur. Math. Soc. (JEMS) 13, No. 4, 1079--1112 (2011; Zbl 1234.14003)], focusing on (the stack of) representations of a quiver with a fixed dimension vector. In particular, the author investigates the question of when the essential dimension and the generic essential dimension (the essential dimension of a generic object) agree. This is known to hold in the case of smooth algebraic stacks with reductive automorphism groups, but, as the author shows, also holds in many cases beyond this (and fails to hold in many cases). The article gives an upper bound on the generic essential dimension for the representations of a quiver when the dimension vector is taken to be a root of the quiver. Conjecturally this upper bound is an equality. The author shows that for quivers of finite type, or those with at least one loop at every vertex, the essential and generic essential dimensions agree for every dimension vector, and moreover shows that this fails for at least one dimension vector for all other quivers. A detailed study is also made of the essential dimensions of the (generalised) Kronecker quivers.