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On Włodarczyk’s embedding theorem. (English) Zbl 0991.14019

J. Włodarczyk introduced the notion of a toric prevariety as the non-separated analogue of a toric variety, that means an algebraic prevariety with a faithful action of an algebraic torus having a dense orbit. He proved that every smooth (respectively \(\mathbb Q\)-factorial or normal) algebraic variety over an algebraically closed field is embeddable into a smooth (respectively \(\mathbb Q\)-factorial or normal) toric prevariety [J. Włodarczyk, J. Algebr. Geom. 2, 705-726 (1993; Zbl 0809.14043)]. In the article under review the author refines Włodarczyk’s methods to prove the following equivariant version of his theorem:
Every normal complex algebraic variety with an algebraic \(\mathbb C^*\)-action admits an equivariant closed embedding into a toric prevariety on which \(\mathbb C^*\) acts as a one-parameter subgroup of the big torus. If in addition the initial variety is \(\mathbb Q\)-factorial then the toric prevariety can be chosen to be simplicial having an affine diagonal morphism.
Note that in the last statement one can even achieve the toric prevariety to be smooth with affine diagonal morphism. This follows from a recent article of J. Hausen, where using a different approach he characterizes embeddability of varieties into smooth toric varieties and prevarieties [Can. J. Math. 54, No. 3, 554-570 (2002)]. There he shows that every normal divisorial variety with an action of any connected algebraic group admits an equivariant embedding into a smooth toric prevariety with an affine diagonal morphism.

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14E25 Embeddings in algebraic geometry

Citations:

Zbl 0809.14043
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References:

[1] DOI: 10.2748/tmj/1178224848 · Zbl 0942.14028 · doi:10.2748/tmj/1178224848
[2] DOI: 10.1070/SM1997v188n05ABEH000222 · Zbl 0895.14015 · doi:10.1070/SM1997v188n05ABEH000222
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[6] Wlodarczyk J., J. Algeb. Geom. 2 pp 705– (1993)
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