Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

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15
votes
2answers
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 ...
17
votes
2answers
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 ...
19
votes
2answers
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 $v$ ...
4
votes
3answers
796 views

Toric Fano manifolds with Picard number 1

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 ...
2
votes
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 $(-K_X)^n$ ...
4
votes
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 ...
4
votes
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 $(-...
2
votes
0answers
97 views

Descent of flatness from algebras to monoids II

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 ...
2
votes
1answer
129 views

Locally toric resolutions of compactifications

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 ...