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Let $n$ be an integer such that $n = 5$ or $n \geq 7$. In this article, weintroduce a recipe for a certain infinite family of non-singular plane curvesof degree $n$ which violate the local-global principle. Moreover, each familycontains infinitely many members which are not geometrically isomorphic to eachother. Our construction is based on two arithmetic objects; that is, primenumbers of the form $X^{3}+NY^{3}$ due to Heath-Brown and Moroz and the Fermattype equation of the form $x^{3}+Ny^{3} = Lz^{n}$, where $N$ and $L$ aresuitably chosen integers. In this sense, our construction is an extension ofthe family of odd degree $n$ which was previously found by Shimizu and theauthor. The previous construction works only if the given degree $n$ has aprime divisor which satisfies a certain indivisibility conjecture ofAnkeny-Artin-Chowla-Mordell type. In this time, we focus on the complementarycases, namely the cases of even degrees and exceptional odd degrees.Consequently, our recipe works well as a whole. This means that we canunconditionally obtain infinitely many non-singular plane curves of everydegree $n = 5$ or $n \geq 7$ which violate the local-global principle. Thisgives a conclusion of the classical story of searching explicit ternary formsviolating the local-global principle, which was originated by Selmer (1951) andextended by Fujiwara (1972) and others.

Primes of the form $X^{3}+NY^{3}$ and a family of non-singular plane curves which violate the local-global principle