Rational quartic spectrahedra
DOI:
https://doi.org/10.7146/math.scand.a-121456Abstract
Rational quartic spectrahedra in $3$-space are semialgebraic convex subsets in $\mathbb{R} ^3$ of semidefinite, real symmetric $(4 \times 4)$-matrices, whose boundary admits a rational parameterization. The Zariski closure in $\mathbb{C}\mathbb{P} ^3$ of the boundary of a rational spectrahedron is a rational complex symmetroid. We give necessary conditions on the configurations of singularities of the corresponding real symmetroids in $\mathbb{R} \mathbb{P} ^3$ of rational quartic spectrahedra. We provide an almost exhaustive list of examples realizing the configurations, and conjecture that the missing example does not occur.
References
Cayley, A., On the cyclide, Quart. J. of Pure and Appl. Math 12 (1873), 148–165.
Chandru, V., Dutta, D., and Hoffmann, C. M., On the geometry of Dupin cyclides, The Visual Computer 5 (1989), no. 5, 277–290. https://doi.org/10.1007/BF01914786
Degtyarev, A. and Itenberg, I., On real determinantal quartics, in “Proceedings of the Gökova Geometry-Topology Conference 2010”, Int. Press, Somerville, MA, 2011, pp. 110–128.
Helsø, M., Maximality of quartic symmetroids with a double quadric of codimension $1$, eprint arXiv:1905.01091 [math.AG], 2019.
Helsø, M., Rational quartic symmetroids, Adv. Geom. 20 (2020), no. 1, 71–89. https://doi.org/10.1515/advgeom-2018-0037
Iliev, A., Kapustka, G., Kapustka, M., and Ranestad, K., Hyper-Kähler fourfolds and Kummer surfaces, Proc. Lond. Math. Soc. (3) 115 (2017), no. 6, 1276–1316. https://doi.org/10.1112/plms.12063
Jessop, C. M., Quartic surfaces with singular points, Cambridge University Press, 1916.
Maxwell, J. C., On the cyclide, Quart. J. of Pure and Appl. Math 9 (1868), 111–126.
Noether, M., Ueber die rationalen Flächen vierter Ordnung, Math. Ann. 33 (1889), no. 4, 546–571. https://doi.org/10.1007/BF01444033
Ottem, J. C., Ranestad, K., Sturmfels, B., and Vinzant, C., Quartic spectrahedra, Math. Program. 151 (2015), no. 2, Ser. B, 585–612. https://doi.org/10.1007/s10107-014-0844-3
Parrilo, P. A., Polynomial optimization, sums of squares, and applications, in “Semidefinite optimization and convex algebraic geometry”, MOS-SIAM Ser. Optim., vol. 13, SIAM, Philadelphia, PA, 2013, pp. 47–157.
Stevens, J., private communication, 2020.
Wall, C. T. C., Singularities of nets of quadrics, Compositio Math. 42 (1980/81), no. 2, 187–212.
Chandru, V., Dutta, D., and Hoffmann, C. M., On the geometry of Dupin cyclides, The Visual Computer 5 (1989), no. 5, 277–290. https://doi.org/10.1007/BF01914786
Degtyarev, A. and Itenberg, I., On real determinantal quartics, in “Proceedings of the Gökova Geometry-Topology Conference 2010”, Int. Press, Somerville, MA, 2011, pp. 110–128.
Helsø, M., Maximality of quartic symmetroids with a double quadric of codimension $1$, eprint arXiv:1905.01091 [math.AG], 2019.
Helsø, M., Rational quartic symmetroids, Adv. Geom. 20 (2020), no. 1, 71–89. https://doi.org/10.1515/advgeom-2018-0037
Iliev, A., Kapustka, G., Kapustka, M., and Ranestad, K., Hyper-Kähler fourfolds and Kummer surfaces, Proc. Lond. Math. Soc. (3) 115 (2017), no. 6, 1276–1316. https://doi.org/10.1112/plms.12063
Jessop, C. M., Quartic surfaces with singular points, Cambridge University Press, 1916.
Maxwell, J. C., On the cyclide, Quart. J. of Pure and Appl. Math 9 (1868), 111–126.
Noether, M., Ueber die rationalen Flächen vierter Ordnung, Math. Ann. 33 (1889), no. 4, 546–571. https://doi.org/10.1007/BF01444033
Ottem, J. C., Ranestad, K., Sturmfels, B., and Vinzant, C., Quartic spectrahedra, Math. Program. 151 (2015), no. 2, Ser. B, 585–612. https://doi.org/10.1007/s10107-014-0844-3
Parrilo, P. A., Polynomial optimization, sums of squares, and applications, in “Semidefinite optimization and convex algebraic geometry”, MOS-SIAM Ser. Optim., vol. 13, SIAM, Philadelphia, PA, 2013, pp. 47–157.
Stevens, J., private communication, 2020.
Wall, C. T. C., Singularities of nets of quadrics, Compositio Math. 42 (1980/81), no. 2, 187–212.
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Published
2021-02-17
How to Cite
Helsø, M., & Ranestad, K. (2021). Rational quartic spectrahedra. MATHEMATICA SCANDINAVICA, 127(1), 79–99. https://doi.org/10.7146/math.scand.a-121456
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