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Equivariant compactification of a torus (after Brylinski and Künnemann). (English) Zbl 1078.14076

The main goal of the paper under review is to give a combinatorial proof for the existence of a smooth equivariant compactification of an algebraic torus defined over an arbitrary field. It is well known that this problem reduces to the following one: given a lattice \(L\), a finite group \(G\) of automorphisms of \(L\) and a projective fan \(A\) in \(L_{\mathbb R}\), to construct a smooth, projective, \(G\)-invariant fan in \(L_{\mathbb R}\) which is a subdivision of \(A\). A solution of this problem given by J.-L. Brylinski [C. R. Acad. Sci. Paris, Sér. A 288, 137–139 (1979; Zbl 0406.14022)] contains a gap.
The authors present a detailed proof where this gap is filled. A key argument is the following one. Given \(L\), \(G\), and \(A\) as above, suppose that \(A\) is \(G\)-invariant. Then there is a \(G\)-invariant subdivision \(B\) of \(A\) such that the following property holds: if \(g\in G\) and \(\sigma\in B\) are such that \(\sigma\) and \(g\cdot\sigma\) are faces of a common cone of \(B\), then \(\sigma = g\cdot\sigma\).

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)

Citations:

Zbl 0406.14022
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References:

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