an:00729195
Zbl 0846.14032
Cox, David A.
The homogeneous coordinate ring of a toric variety
EN
J. Algebr. Geom. 4, No. 1, 17-50 (1995); erratum ibid. 23, No. 2, 393-398 (2014).
00024544
1995
j
14M25 14L30 55U10 14E07 14M17
geometric quotients of a fan; projective toric varieties; Stanley-Reisner ring; automorphism group
Complex, quasi-smooth, projective toric varieties may be considered a generalization of the projective space \(\mathbb{P}^n\); they are obtained by glueing together affine pieces almost isomorphic to \(\mathbb{C}^n\). Those toric varieties may be given by a fan (a certain collection of rational, polyhedral cones in \(\mathbb{R}^n)\) containing all combinatorial information necessary for this process. In the present paper, the author describes a different method of synthesizing projective varieties as geometric quotients from the given fan. The relations to projective spaces are even more striking: First, assigning to each one-dimensional generator of the fan \(\Delta\) a coordinate, we obtain the affine space \(\mathbb{C}^{\Delta (1)}\) (in case of \(\mathbb{P} ^n\), this will be \(\mathbb{C}^{n + 1})\). Then, we have to do the following two jobs simultaneously:
(i) Construct a certain subgroup of \((\mathbb{C}^*)^{\Delta(1)}\) by using the exact knowledge of the one-dimensional cones in \(\Delta\). This subgroup is isomorphic to some \((\mathbb{C}^*)^k\) and clearly acts on \(\mathbb{C}^{\Delta (1)}\). (In case of \(\mathbb{P}^n\), we have \(k = 1\).)
(ii) Similarly to the construction of the Stanley-Reisner ring from a simplicial complex, the information which of the \(\Delta (1)\)-rays belongs to a common higher-dimensional cone (and which not) define a certain closed algebraic subset \(Z \subseteq \mathbb{C}^{\Delta (1)}\) of codimension at least two. \((Z \) equals \{0\} in case of \(\mathbb{P}^n\).)
Now, the main result is that the toric variety assigned to a simplicial fan \(\Delta\) equals the geometric quotient \([\mathbb{C}^{ \Delta (1)} \backslash Z] / (\mathbb{C}^*)^k\). Since the group action is defined without using the information about incidences of \(\Delta\)-cones, this description is very useful for studying flips and flops, i.e. for changing \(\Delta\) without changing \(\Delta (1)\).
Finally, this result is used for studying the automorphism group of a toric variety via considering \(\mathbb{C}^{\Delta (1)} \backslash Z\). In a paper of \textit{Daniel B??hler} (Diplomarbeit Z??rich), this is generalized also to non-simplicial fans.
K.Altmann (Berlin)