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Zbl 0664.14018
Kleinschmidt, Peter
A classification of toric varieties with few generators. (English)
[J] Aequationes Math. 35, No.2-3, 254-266 (1988). ISSN 0001-9054; ISSN 1420-8903/e

A d-dimensional toric variety is a T-invariant subvariety of a T-invariant completion of the torus $T=(k\sp*)\sp d$ (k can be ${\bbfC}$ or any algebraically closed field). Such a variety can be described by a fan, a finite system of cones in ${\bbfR}\sp d$ spanned by integer lattice points. In the paper a classification of complete smooth toric varieties with arbitrary dimension but relatively few generators is obtained. All 3-dimensional smooth toric varieties with up to 8 ``generators'' have been previously classified by T. Oda. \par Let $X\sb d(a\sb 1,...,a\sb r)$ be the toric d-variety corresponding to a certain fan in ${\bbfR}\sp d$, defined in {\S}2. The main result is the following theorem: \par Every complete nonsingular toric d-variety with $d+2$ generators is isomorphic to precisely one of the varieties $X\sb d(a\sb 1,...,a\sb t)$. Among those varieties, precisely those with $a\sb 1=a\sb 2=...=a\sb{r- 1}=0$ and $a\sb r=1$ can be further blown down to ${\bbfP}\sp d.$ \par Furthermore the following properties of the varieties above are shown. \par 1.i) All varieties $X\sb d(a\sb i,...,a\sb r)$ are projective; \par ii) A variety $X\sb d(a\sb 1,...,a\sb r)$ is a Fano variety if and only if $\sum\sb{i=1,...,r}a\sb i<d-r+1.$ \par 2. The variety $X\sb d(a\sb 1,...,a\sb r)$ is isomorphic to the variety given by the following system of equations with homogeneous coordinates in ${\bbfP}\sb{sr}\times {\bbfP}\sb{s-1}$ $(={\bbfP}[x\sb 0,x\sb{11},...,x\sb{s1},s\sb{12},...,x\sb{s2},...,x\sb{1r},...,x\sb{sr}]\times {\bbfP}[y\sb 1,...,y\sb s]):$ $x\sb{mi}y\sp{ai/n}=x\sb{ni}y\sp{ai/m}$ for all triples (m,n,i) such that $1\le i\le r$; $1\le m,n\le s$; $m\ne n.$ \par 3. There exists a compact smooth toric d-variety X which can be obtained from both $X\sb d(a\sb 1,...,a\sb r)$ and ${\bbfP}\sp d$ by a sequence of equivariant blow-ups. \par It should be noted that the property 3 settles affirmatively a stronger version of a conjecture of Oda and Miyake, previously proved by Danilov and Ewald.
[M.Beltrametti]

MSC 2000:: *14J10 Families, algebraic moduli, classification (surfaces)
14J40 Special n-folds

Keywords: fan; classification of complete smooth toric varieties; Fano variety

Cited in: Zbl 0838.14041

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