Non-classical Gorenstein curves of arithmetic genus three and four
A complete irreducible non-hyperelliptic Gorenstein curve \(C\) of arithmetic genus \(g\), canonically embedded in the projective space of dimension \(g-1\), is called non-classical, when for each non-singular point \(P\in C\) the osculating hyperplane of \(C\) at \(P\) has contact order larger than \(g-1\). This rather unexpected geometric behaviour can only happen when the characteristic of the constant field is a prime smaller than \(2g-2\). The authors classify the non-classical non-hyperelliptic Gorenstein curves of arithmetic genus three and four; the non-singular case was treated previously by \textit{K. Komiya} [Hiroshima Math. J. 8, 371-400 (1978; Zbl 0406.14007)]. If the arithmetic genus is equal to three, then the classification list consists of one non-singular curve, discovered by F. K. Schmidt, and of five non-classical singular curves. If \(g=4\), then in addition to Komiya's one-parameter family of non-singular curves and an example of F. K. Schmidt the authors obtain various new families of non-classical Gorenstein curves.
non-classical non-hyperelliptic Gorenstein curves
scientific article; zbMATH DE number 784084