# Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

9 questions
Filter by
Sorted by
Tagged with
6k views

### The canonical line bundle of a normal variety

I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical ...
2k views

### What are some open problems in toric varieties?

In light of the nice responses to this question, I wonder what are some open problems in the area of toric geometry? In particular, What are some open problems relating to the algebraic ...
1k views

### About a Delzant polytope. (In particular dodecahedron)

Hi. I have a question. Definition. Delzant polytope $P$ is a rational convex simple polytope with the smooth condition. Here, "smooth" means that for each vertex $v$, the $n$ edges containing $v$ ...
796 views

As far as I know, toric Fano manifolds are classified only up to dimension 4. In dimension one the projective line is the only example. In dimension two we have five examples: $\mathbb P ^2$, $\mathbb ... 2answers 394 views ### Bound on the (anticanonical) degree of toric Fano varieties Does there exists a universal constant$C \geq 1$such that if$X$is any a smooth, toric, Fano$n$-dimensional manifold admitting a Kähler-Einstein metric, then its anticanonical degree$(-K_X)^n$... 2answers 528 views ### Is there an extremal metric on toric Fano manifolds which have nonzero Futaki invariant? According the work by Wang & Zhu, on toric Fano manifolds there exist Kaehler-Ricci solitons. If Futaki=0, there also exist CSCK metrics. But if the Futaki invariant does not vanish, what about ... 3answers 618 views ### Number of$(-1)$curves on toric surfaces Hello. My question is: Is it possible that a smooth complete toric surface has infinitely many$(-1)$-curves. I know that there is a blow-up of$\mathbb P^2$in 9 points containing infintely many$(-...
This is a follow-up question to this one. There, Benjamin Steinberg showed that a morphism of commutative monoids $u$ need not be flat if the induced morphism of $R$-algebras $R[u]$ is flat for some ...
Suppose $U$ is a smooth, open $n$-dimensional variety over $\mathbb{C}.$ Say $X, X'$ are two proper normal-crossings compactifications of $U$. Call a map $m: X'\to X$ a modification if it is an ...