Nested Hilbert schemes on Hirzebruch surfaces and quiver varieties

Abstract

Hilbert schemes were introduced by Grothendieck. They are a fundamental example of the notion of moduli spaces of geometric structures. The work of Nakajima on the properties of the Hilbert schemes of points of the complex plane has been the basis of many works that try to understand the properties of Hilbert schemes of other 2- dimensional varieties and also for higher dimensions. Furthermore, the nested Hilbert scheme of points on the complex plane was studied by von Flach, Jardim and Lanza. Moreover, Bartocci, Bruzzo, Lanza and Rava obtained a quiver description to the Hilbert scheme of points of the total space Ξn of appropriate line bundles over the projective line. In this work we show that the nested Hilbert scheme of points on the last varieties, parameterizing pairs of nested 0-cycles, is the quiver variety associated with a suitable quiver with relations, generalizing previous work about nested Hilbert schemes on the complex plane, in one direction, and about the Hilbert schemes of points of Ξn in another direction.