Morphisms between toric varieties (Q2763513)

The present paper is the author's dissertation, wherein he details his investigation of morphisms between toric varieties and provides a criterion of when they can be lifted to morphisms between quasiaffine varieties. Most of the material is also published elsewhere [\textit{F. Berchtold}, Manuscr. Math. 110, 33-44 (2003; Zbl 1014.14026)]. NEWLINENEWLINENEWLINEIf \(X\) is a toric variety, then there are several possibilities to represent \(X\) as a quotient of a quasiaffine variety \(\widehat{X}\), similarly to Cox's construction. By a paper by \textit{A. A'Campo-Neuen, J. Hausen} and \textit{S. Schröer} [Math. Nachr. 246-247, 5-19 (2000; Zbl 1064.14059)], those representations are parametrized by certain subgroups \(\widehat{M}\) of the torus-invariant Weil divisors of \(X\). Taking for \(\widehat{M}\) the group of \textit{all} those Weil divisors yields exactly Cox's construction. NEWLINENEWLINENEWLINEThe main result of the paper under review is that the lifts of a (non-necessarily toric) morphism \(f:X'\to X\) between toric varieties to the quasiaffine level associated to given divisor groups \(\widehat{M'}\) and \(\widehat{M}\) are in a 1-1-correspondence to the possibilities of pulling back the \(\widehat{M}\)-elements to Weil divisors from \(\widehat{M'}\) in a natural way. In particular, the author makes the latter notion precise. NEWLINENEWLINENEWLINEFinally, as an application, the special cases of the Cox construction and that arising from taking for \(\widehat{M}\) the invariant Cartier divisors are studied. This recovers some classical results of this subject which become much more natural by using this systematic treatment. Moreover, the author presents some new applications. For instance, he obtains that the fans of arbitrary toric varieties \(X\) are completely determined by \(X\) itself (forgetting the torus action).