Creating weak Fano toric varieties of Picard rank 3

Authors: - Zhengning Hu - Rohan Joshi

Bibtex citation: @misc{weakFano-repo, author = {Hu, Zhengning and Joshi, Rohan}, title = {{Weak-Fanos}}, howpublished = {\url{https://github.com/rohansjoshi/Weak-Fanos}}, year = {2025}, }

Publication: * Classifying weak Fano toric varieties of Picard rank 3, Zhengning Hu and Rohan Joshi. 2025.

This repository contains Macaulay 2 functions for constructive smooth projective toric varieties of Picard rank 3 (either as projective bundles or Batyrev varieties), following Batyrev. We also construct all such varieties that are weak Fano. Everything we have written is in a Macaulay2 package called RankThreeTorics.

There are functions that automate the process of producing the classification in any dimension, and removing redundant isomorphic copies. For example, you can produce all smooth weak Fano toric varieties of Picard rank 3 with three primitive collections (which are projective bundles) of dimension 3, say, and print out the list as output by running

``` i1 : loadPackage "RankThreeTorics"

o1 = RankThreeTorics

o1 : Package

i2 : printListName(weakFanoProjectiveBundleVarieties(3)) 1. (2, {0}, {{0, -2}}): 48 24
2. (2, {0}, {{0, -1}}): 48 24
3. (2, {0}, {{0, 0}}): 48 24
4. (2, {0}, {{1, -2}}): 40 24
5. (2, {0}, {{1, -1}}): 44 24
6. (2, {0}, {{1, 1}}): 52 24
7. (2, {0}, {{1, 2}}): 56 24
8. (2, {0}, {{2, -2}}): 32 24
9. (2, {0}, {{2, 2}}): 64 24
10. (2, {1}, {{0, -1}}): 48 24
11. (2, {1}, {{1, -1}}): 46 24
12. (2, {1}, {{1, 0}}): 50 24
13. (2, {1}, {{1, 1}}): 54 24
14. (2, {1}, {{2, -1}}): 48 24
15. (2, {1}, {{2, 0}}): 56 24
16. (2, {1}, {{2, 1}}): 64 24
17. (2, {2}, {{1, 0}}): 52 24
18. (2, {2}, {{2, 0}}): 64 24
```

Here the two columns are the computations of the Chern numbers $c1^4$ and $c1^2 c_2$. You can produce all smooth weak Fano toric varieties of Picard rank 3 with five primitive collections (Batyrev's construction) of dimension 3 by running

i3 : printListName(weakFanoBatyrevVarieties(3)) 1. ({1, 1, 1, 1, 2}, {0}, {}): 46 24 2. ({1, 1, 1, 1, 2}, {1}, {}): 48 24 3. ({1, 1, 2, 1, 1}, {0}, {0}): 46 24 4. ({1, 1, 2, 1, 1}, {0}, {1}): 46 24 5. ({1, 1, 2, 1, 1}, {1}, {0}): 46 24 6. ({1, 2, 1, 1, 1}, {0}, {}): 48 24 7. ({1, 2, 1, 1, 1}, {1}, {}): 54 24 8. ({1, 2, 1, 1, 1}, {2}, {}): 64 24 9. ({2, 1, 1, 1, 1}, {1}, {}): 50 24 10. ({2, 1, 1, 1, 1}, {2}, {}): 58 24

There are many routines in the RankThreeTorics package which are generally useful for working with toric varieties. For example, the function areIsomorphic, which checks if two toric varieties are isomorphic. To see the functions in this package, you can run viewHelp.

``` i1 : installPackage "RankThreeTorics"

o1 = RankThreeTorics

o1 : Package

i2 : viewHelp RankThreeTorics ```

Included in this repository is a package called ToricExtras which was written at the 2024 Macaulay2 workshop (and will eventually will be incorporated into NormalToricVarieties). Note that the function batyrevConstructor is in ToricExtras, not RankThreeTorics.