# 2-faces of reflexive Delzant polytopes

Question 1. Can a reflexive Delzant polytope of some dimension contain a $$2$$-face with more than $$11$$ edges?

Motivation. I would like more generally to get an answer to the following question:

Question 2. Suppose $$X$$ is a smooth Fano variety with a $$\mathbb C^*$$-action. Let $$Y\subset X$$ be the connected component of $$X^{\mathbb C^*}$$ on which all the weights of the action are positive. Is it true that variety $$Y$$ can be deformed to a Fano variety?

A positive answer to the first question gives a negative to the second one.

Explanation. Indeed, a reflexive Delzant polytope by definition corresponds to a toric Fano. For any face of a Delzant polytope we can find a rational linear support function. This will give us a $$\mathbb C^*$$ action as in Question 2. A surface that deforms to a Fano surface has $$b_2\le 9$$. Finally, a toric surface corresponding to a polygon with $$n$$ sides has $$b_2=n-2$$.

• May I suggest that you remove the term "Obviously" ? There is nothing obvious here (and few obvious things in mathematics in general). Commented Oct 16, 2022 at 17:51
• Thanks for your comment Nicolas. Indeed, I wrote "Obviously" without thinking twice. Question 2 was the reason for Question 1, I was thinking of how to get a counter-example. Anyway, I replaced "Obviously" by an explanation Commented Oct 23, 2022 at 11:28