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Essential dimension of extensions of finite groups by tori. (English) Zbl 1509.14092

Let \(p\) be a prime. This paper calculates the essential \(p\)-dimension of algebraic groups \(G\) that may be placed in short exact sequences \(1 \to T \to G \to F \to 1\), where \(T\) is a (possibly nonsplit) torus and \(F\) is a finite diagonalizable algebraic \(p\)-group (possibly nonconstant). There is no harm in assuming that the ground field is \(p\)-closed (i.e., all finite extensions are of \(p\)-power degree).
Recall from [R. Lötscher et al., Algebra Number Theory 7, No. 8, 1817–1840 (2013; Zbl 1288.20061)] that a representation is \(p\)-faithful if the kernel is finite of order prime to \(p\), and that a \(p\)-faithful representation is \(p\)-generically free if the induced faithful representation is generically free.
Two integers are defined: \(\eta(G)\) and \(\rho(G)\). The first is the smallest dimension of a \(p\)-faithful representation and the second is the smallest dimension of a \(p\)-generically free representation. The main theorem of the paper is Theorem 1.2, which is in two parts. The first establishes \[ \mathrm{ed}(G; p) = \rho(G) - \dim G. \] This partly strengthens a result of [R. Lötscher et al., Algebra Number Theory 7, No. 8, 1817–1840 (2013; Zbl 1288.20061)], which says that right hand side is an upper bound for the left. (That paper does not assume the group \(F\) is diagonalizable.)
The second part of Theorem 1.2 gives a technique for deducing \(\rho(G)\) from \(\eta(G)\): Suppose \(V\) is a \(p\)-faithful representation of \(G\) of minimal dimension, and suppose \(S_V\) is the stabilizer group of a sufficiently general point of \(V_{\bar k}\) (a precise definition is given in Section 2), then \(\rho(G) - \eta(G)\) is the \(p\)-rank of the group \(S_V\). This reduces the problem of calculating the essential \(p\)-dimension of \(G\) to finding a \(p\)-faithful representation of minimal dimension.
The proof of Theorem 1.2 involves a sort of equivariant resolution theorem which seems to be of considerable general applicability. Let \(G\) be a smooth linear algebraic group over a field and \(f: X \dashrightarrow Y\) a dominant rational map of \(G\)-variaties. Assume \(X\), \(Y\) are normal and \(Y\) is complete. Let \(D\) denote a prime divisor of \(X\) whose image in \(Y\) is not dense. Then Theorem 7.2 asserts that \(f\) can be factored \(G\)-equivariantly as \(f' :X \to Y'\) (again dominant and rational with complete codomain) and \(\pi: Y' \to Y\) (birational and \(G\)-equivariant) so that \(\overline{f'(D)}\) is again a divisor.
In the last section of the paper, section 9, the essential \(p\)-dimensions are computed for normalizers of \(k\)-split maximal tori in the split simple groups \(SL_n\) and \(SO_{4n}\), thus completing the work of [A. Meyer and Z. Reichstein, Algebra Number Theory 3, No. 4, 467–487 (2009; Zbl 1222.11056)] and [M. L. MacDonald, Transform. Groups 16, No. 4, 1143–1171 (2011; Zbl 1263.20046)].

MSC:

14L30 Group actions on varieties or schemes (quotients)
20G15 Linear algebraic groups over arbitrary fields
14E05 Rational and birational maps
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