Tritangents and Their Space Sextics

Türkü Özlüm Çelik, Avinash Kulkarni, Yue Ren, and Mahsa Sayyary Namin

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Submission date: 31. May. 2018
Pages: 26
published in: Journal of algebra, 538 (2019), p. 290-311 
DOI number (of the published article): 10.1016/j.jalgebra.2019.07.037
Bibtex
MSC-Numbers: 14Q05, 14H50
Keywords and phrases: algebraic curves, del Pezzo surfaces, theta characteristics, space sextic, tritangents
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Abstract:
Two classical results in algebraic geometry are that the branch curve of a del Pezzo surface of degree 1 can be embedded as a space sextic curve in ℙ³ and that every space sextic curve has exactly 120 tritangents corresponding to its odd theta characteristics. In this paper we revisit both results from the computational perspective. Specifically, we give an algorithm to construct space sextic curves that arise from blowing up ℙ² at eight points and provide algorithms to compute the 120 tritangents and their Steiner system of any space sextic. Furthermore, we develop efficient inverses to the aforementioned methods. We present an algorithm to either reconstruct the original eight points in ℙ² from a space sextic or certify that this is not possible. Moreover, we extend a construction of Lehavi which recovers a space sextic from its tritangents and Steiner system. All algorithms in this paper have been implemented in magma.