The homogeneous coordinate ring of a toric variety.

*(English)*Zbl 0846.14032
J. Algebr. Geom. 4, No. 1, 17-50 (1995); erratum ibid. 23, No. 2, 393-398 (2014).

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 Daniel Bühler (Diplomarbeit Zürich), this is generalized also to non-simplicial fans.

(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 Daniel Bühler (Diplomarbeit Zürich), this is generalized also to non-simplicial fans.

Reviewer: K.Altmann (Berlin)

##### MSC:

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

14L30 | Group actions on varieties or schemes (quotients) |

55U10 | Simplicial sets and complexes in algebraic topology |

14E07 | Birational automorphisms, Cremona group and generalizations |

14M17 | Homogeneous spaces and generalizations |