Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.


I would like to understand when a toric variety is connected. Given $\Delta$ a fan (possibly with infinitely many cones) in $\mathbb{R}^n$, $n\geq 2$ denote with $X_{\Delta}$ the associated toric variety. I would like to know if there exists some property $P$ for which a theorem as follows holds:

if $\Delta$ satisfies $P$ then $X_{\Delta}$ is connected.

share|improve this question
margarida? :) sei tu? –  diverietti Jun 16 '13 at 16:02
I don't even understand what it would mean for a disconnected variety to be toric. –  Dan Petersen Jun 16 '13 at 16:42
With the definition I know, every toric variety is connected. –  Angelo Jun 16 '13 at 17:00
The usual definition of "fan" implies that every two cone intersect in a common (non-empty) sub-cone: which implies that the open sets corresponding to those cones intersect in a non-empty open set. So the toric variety is connected by default. –  auniket Jun 16 '13 at 18:18
Wait... we will again have to discuss whether or not the empty scheme is connected. –  Fred Rohrer Jun 16 '13 at 20:35
show 4 more comments

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.