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Simultaneous conjugacy classes of commuting nilpotents

William Haboush (University of Illinois)
Algebraisk geometriseminar
Tirsdag, 10 december, 2019, at 14:15-15:00, in Aud. D3 (1531-215)

We consider the space of simultaneous conjugacy classes of q-commuting n-nilpotents.

We show that the q-conjugacy classes satisfying a certain regularity condition constitute a dense open subset of the full space and we designate these the regular q-tuples. We first show that this space is isomorphic to the space of representations of the algebra of polynomials truncated at order n in the q-dimensional space V such that V is cyclic, that is spanned by a single element and that this space is in turn isomorphic to the space of curvilinear ideals in the punctual Hilbert scheme of ideals of coheight n.

We show that this space is a fiber space over projective (q-1)-space but not a vector bundle. This makes use of a result of Iarrobino and generalizes it.

We also give an algebraic flat deformation of the space to a product of twisted symmetric powers of the tangent bundle which implies that it is isomorphic to this sum in the smooth category.