A census of cubic fourfolds over $\mathbb {F}_2$
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- by Asher Auel, Avinash Kulkarni, Jack Petok and Jonah Weinbaum;
- Math. Comp.
- DOI: https://doi.org/10.1090/mcom/4010
- Published electronically: September 24, 2024
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Abstract:
We compute a complete set of isomorphism classes of cubic fourfolds over $\mathbb {F}_2$. Using this, we are able to compile statistics about various invariants of cubic fourfolds, including their counts of points, lines, and planes; all zeta functions of the smooth cubic fourfolds over $\mathbb {F}_2$; and their Newton polygons. One particular outcome is the number of smooth cubic fourfolds over $\mathbb {F}_2$, which we fit into the asymptotic framework of discriminant complements. Another motivation is the realization problem for zeta functions of $K3$ surfaces. We present a refinement to the standard method of orbit enumeration that leverages filtrations and gives a significant speedup. In the case of cubic fourfolds, the relevant filtration is determined by Waring representation and the method brings the problem into the computationally tractable range.References
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Bibliographic Information
- Asher Auel
- Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire
- MR Author ID: 932786
- Email: asher.auel@dartmouth.edu
- Avinash Kulkarni
- MR Author ID: 1250845
- ORCID: 0000-0002-4567-0396
- Email: avi.kulkarni.2.71@gmail.com
- Jack Petok
- Affiliation: Department of Mathematics, Colby College, Waterville, Maine
- MR Author ID: 1554543
- Email: jpetok@colby.edu
- Jonah Weinbaum
- Affiliation: Department of Computer Science, Dartmouth College, Hanover, New Hampshire
- Email: jonah.r.weinbaum.gr@dartmouth.edu
- Received by editor(s): August 15, 2023
- Received by editor(s) in revised form: July 23, 2024
- Published electronically: September 24, 2024
- Additional Notes: The first author received partial support from Simons Foundation grant 712097, National Science Foundation grant DMS-2200845, and a Walter and Constance Burke Award and a Senior Faculty Grant from Dartmouth College; the second author received support from the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation grant 550029; the fourth author received support from the Neukom Institute for Computational Science at Dartmouth College.
- © Copyright 2024 American Mathematical Society
- Journal: Math. Comp.
- MSC (2020): Primary 11M38, 14Q10, 14J70; Secondary 14E08, 14J28
- DOI: https://doi.org/10.1090/mcom/4010