Geodesic
Construction of arrangements of a cubic and a quartic by patchworking
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 5, Tome 267 (2000), pp. 119-132
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B. Sturmfels modified Viro's patchworking method and applied it for construction of complete intersections. We use this modification for construction of decomposable curves. We realize 11 new arrangements of an $M$-cubic and an $M$-quartic with 12 common points lying on the odd branch of the cubic and on an oval of the quartic. 
@article{ZNSL_2000_267_a5,
     author = {M. A. Gushchin and A. N. Korobeinikov and G. M. Polotovsky},
     title = {Construction of arrangements of a cubic and a quartic by patchworking},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {119--132},
     year = {2000},
     volume = {267},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_267_a5/}
}
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M. A. Gushchin; A. N. Korobeinikov; G. M. Polotovsky. Construction of arrangements of a cubic and a quartic by patchworking. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 5, Tome 267 (2000), pp. 119-132. http://geodesic.mathdoc.fr/item/ZNSL_2000_267_a5/