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It was proven by Batyrev in 1981

http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1581&option_lang=eng

that there exist exactly 18 smooth toric Fano three-folds. I would like to know if there is some place where the pictures of the corresponding 18 simple reflexive polytopes are presented.

More precisely, I would like to see the moment polytopes, i.e. the polytopes that are dual to 18 reflexive polytopes depicted in the reference given below by David. These polytopes should be Delzant, i.e., each vertex can be send by an element of $SL(3,\mathbb Z)$ to the standard vertex given by equations $x_i\ge 0$.

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    $\begingroup$ inspirehep.net/record/1591349/plots $\endgroup$
    – David G. Stork
    Commented Apr 7, 2018 at 17:54
  • $\begingroup$ Thanks! I wanted to see the dual pictures - the simple polytopes, or moment polytopes. But this is nice as well $\endgroup$
    – aglearner
    Commented Apr 7, 2018 at 19:05

3 Answers 3

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The article that was linked by David G. Stork actually contains information about how one can get these pictures. Namely, the SageMath computer algebra system contain a database of all reflexive polytopes and the figure from the article includes their position in this database (off by one). You can do much more with these objects in Sage than just getting their polar dual. Thus I can give you easily more than just a picture, I can give you a 3D model that you can rotate and zoom in your browser. Just paste the following code into SageMathCell and hit Evaluate. 

# change the number from 1 to 18
n = 3
polytopes_list = [1, 5, 6, 7, 8, 25, 26, 27, 28, 29, 30, 31, 82, 83, 84, 85, 219, 220]

P = ReflexivePolytope(3, polytopes_list[n-1]-1)
dP = P.polar()
dP.plot3d()

enter image description here

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  • $\begingroup$ Thanks a lot! I wonder if some of 18 numbers are not quite correct. Namely, when I plug 18 for $n$, which should correspond to number 220, I get the polytope with vertex 0 of degree 4. It follows that this polytope is not smooth (because a smooth one has all vertices of degree 3). So this polytope can not be dual to one of a smooth toric threefold. Do you know what is the problem here? $\endgroup$
    – aglearner
    Commented Apr 14, 2018 at 5:50
  • $\begingroup$ @aglearner Off by one error in database index. Sorry about that! The link and code are fixed now. If you want to see the polytope from the article, just change the last line to P.plot3d() $\endgroup$
    – Vít Tuček
    Commented Apr 16, 2018 at 9:16
  • $\begingroup$ It seems that Sage does not have a similar list for dimension 4, but is there a list available somewhere that can be imported to Sage? $\endgroup$
    – locally trivial
    Commented Feb 21 at 2:59
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Not sure if this will help...

Süß, Hendrik. "Fano Threefolds with 2-Torus Action: A Picture Book." Documenta Mathematica 19 (2014): 905-940. (PDF download.)
Abstract. ...we give a combinatorial description for smooth Fano threefolds admitting a $2$-torus action.

          BlowUp

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    $\begingroup$ It's interesting, the list in this theorem contains 19 Fanos. So only one of them doesn't admit a 3-torus action... I wonder which $\endgroup$
    – aglearner
    Commented Apr 7, 2018 at 22:06
  • $\begingroup$ @aglearner: Good question! $\endgroup$
    – Joseph O'Rourke
    Commented Apr 7, 2018 at 22:31
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    $\begingroup$ A Dr. Seuss picture book! $\endgroup$
    – Ben McKay
    Commented Apr 8, 2018 at 8:21
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    $\begingroup$ @aglearner: the quadric 3-fold is not toric. $\endgroup$
    – Pop
    Commented Apr 9, 2018 at 20:07
  • $\begingroup$ Pop, thanks. Indeed, quadric is not toric and it admits a $T^2$ action. But the flag variety admits a $T^2$ action as well and it is not toric (is not it). So it sounds like one example is missing from the list if I counted 19 correctly... $\endgroup$
    – aglearner
    Commented Apr 17, 2018 at 14:50
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The author of this and this might be able to send you lists of vertices, from which you can generate the pictures yourself, if you ask nicely.

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  • $\begingroup$ Thanks for the advise! I was somehow hopeful that the list of pictures was already printed somewhere. Maybe in someones Bsc on Msci project... But if this is not the case, I'll try to contact the author. $\endgroup$
    – aglearner
    Commented Apr 12, 2018 at 16:01

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