This short introductory survey covers basic topics of toric geometry paying special attention to blowing-up and to resolution of singularities. Starting with the standard definition of a toric variety by gluing together affine toric varieties corresponding to the cones of a fan, the author discusses the relationship between orbital structure of toric varieties and combinatorics of their fans. The criteria of completeness and smoothness of a toric variety in terms of its fan are given. Further, the theory of Weil and Cartier divisors on a toric variety is considered. Support functions of invariant Cartier divisors are defined and it is shown that a divisor on a projective toric variety is globally generated (ample) iff its support function is (strictly) convex. On a smooth toric variety, every ample divisor is very ample (Demazure), and on any \(n\)-dimensional toric variety (\(n>1\)), the \((n-1)\)-th multiple of any ample divisor is very ample [see G. Ewald and W. Wessels, Result. Math. 19, No. 3/4, 275-278 (1991; 739.14031)]. Global sections of Cartier divisors are described, and it is shown that a projective toric variety \(X\) is recovered from the polytope of global sections of an ample divisor on \(X\).
Alternative constructions of toric varieties are given: as \(\text{Proj}\) of a graded ring (whose \(\text{Spec}\) is the affine cone over a projective toric variety), as a quotient of an open subset of an affine space by a quasitorus [see D. A. Cox, J. Algebr. Geom. 4, No. 1, 17-50 (1995; Zbl 0846.14032)], and by toric ideals (in the affine case). Non-normal toric varieties are discussed. In terms of subdivisions of fans, the blow-up of a smooth toric variety along an invariant subvariety is described, and the resolution of singularities of any toric variety \(X\), bijective over \(X^{\text{reg}}\), is constructed. Finally, rationality of singularities of any toric variety is proven.
The survey contains no proofs, except for the construction of a toric desingularization and the rationality of singularities, where detailed proofs are given. Instead, simple instructive examples illustrate definitions and theorems of toric geometry are given.
For the entire collection see [Zbl 0932.00042].