Geodesic
Legendrian curves in $\mathbb C P^3$: cubics and curves on a quadric surface
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 13, Tome 476 (2018), pp. 92-110

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We prove that the number of Legendrian rational cubics in ${\mathbb C P^3}$ through three generic points and a line is three. Also, we classify all Legendrian curves on a quadric surface. Additionally, several computations are verified using Macaulay2 computer algebra system. 
@article{ZNSL_2018_476_a5,
     author = {N. Kalinin},
     title = {Legendrian curves in $\mathbb C P^3$: cubics and curves on  a quadric surface},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {92--110},
     publisher = {mathdoc},
     volume = {476},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_476_a5/}
}
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N. Kalinin. Legendrian curves in $\mathbb C P^3$: cubics and curves on  a quadric surface. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 13, Tome 476 (2018), pp. 92-110. http://geodesic.mathdoc.fr/item/ZNSL_2018_476_a5/