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Algebraic Geometry

By P.E. Newstead

Backbone name: creation to moduli difficulties and orbit areas.

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2 Toric downgrades. The toric case, however, can also provide us with more interesting examples. ı; Nz / and fix a subtorus action T ,! Tz . ı; Nz / as a T -variety? Assuming that the embedding T ,! Tz is induced from a surjection of the corresponding character z ! groups p W M ! M , we denote the kernel by MY and obtain two mutually dual exact sequences (1) and (2). _ . On the dual Setting WD NQ \ ı, the map p gives us a surjection ı _ ! z side, denote the surjection N ! NY by q. ı/, we denote by † the coarsest fan refining the images of all faces of ı under the map q.

Süß, and R. Vollmert Proof. 8]. Remark 6. A special case of the above is when X and X 0 are toric varieties. Here, the 0 proposition simplifies to TV. / Š TV. / TV. 0 /. 1 Maps between toric varieties. We will now see that all constructions from the previous section are functorial. 1). A Z-linear map F W N 0 ! N satisfying FQ . F / W TV. 0 ; N 0 / ! TV. ; N / of affine toric varieties via F _ . _ \ M / Â . 0 /_ \ M 0 . For example, if E Â _ \ M is a Hilbert basis, then the embedding TV. / ,! 1) is induced by the map E W N !

N k/-dimensional variety Y which is a sort of quotient Y D X=T . Now, X can be described by presenting a “polyhedral” divisor D on Y with coefficients being not numbers but instead convex polyhedra in the vector space NQ WD N ˝Z Q where N is the lattice of one-parameter subgroups of T . 3 What this paper is about. The idea of the present paper is to give an introduction to this subject and to serve as a survey for the many recent papers on T -varieties. Moreover, since the notion of polyhedral divisors and the theory of T -varieties closely follows the concept of toric varieties, we will treat both cases in parallel.

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