On the notion of essential dimension for algebraic groups
Let \(G\) be a linear algebraic group defined over an algebraically closed field of characteristic zero. In this long paper consisting of 40 pages, the author defines the essential dimension of \(G\) to be the smallest integer \(d\) such that every principal \(G\)-bundle \(X\to B\) can be obtained (up to a birational isomorphism) by pull-back from a diagram \[ \begin{tikzcd} X\simeq B\times_CY\ar[r]\ar[d] & Y\ar[d]\\ B\ar[r] & C \rlap{\, ,}\end{tikzcd} \] where \(Y\to C\) is a principal \(G\)-bundle, \(B\to C\) is a dominant map, and \(\dim(C)\leq d\). The essential dimension is a numerical invariant of the group, which is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type. This notion naturally comes up in many interesting situations. The author develops a general theory, and his methods to compute or estimate the essential dimension of a group can be divided into three categories. The geometric approach is based on the (birational) invariant theory; the algebraic approach is based on the descent of ``structured spaces''; and the cohomological approach is mostly concerned with the vanishing of \(H^1\). Also, the author gives an exposition of some unpublished results of J.-P. Serre relating the essential dimension to Galois cohomology.
scientific article; zbMATH DE number 1530010
scientific article