Highly versal torsors

Let G be a linear algebraic group over a field k. Recall that a G-torsor E-->X, where X is a k-variety, is said to be weakly versal if every G-torsor over a k-field is a specialization of E-->X. It is called versal, resp. strongly versal, when such specializations are abundant in a well-defined sense. Versal torsors are important to the study of essential dimension and also to cohomological invariants.

I will present some recent results about the existence of G-torsors admitting even stronger versality properties. For example, for every d>=0 there exist G-torsors which specialize to any torsor over an affine d-dimensional k-scheme, and such specializations are "abundant". Moreover, some algebraic groups even admit torsors which specialize to all torsors over all affine k-schemes; we characterize these groups when k is of char.0 as the unipotent groups.

Some applications to finiteness of the symbol length of local rings will also be discussed.

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062