Fibrations by nonsmooth genus three curves in characteristic three
The classification of complete, geometrically integral, regular and nonsmooth curves \(C\) over a field \(K\) is equivalent to the classification of nonconservative separably generated function fields, whose field of coefficients is algebraically closed in it. A function field \(K(C)/K\) is nonconservative if the genus decreases when we pass to \(\overline{K}(C)/\overline{K}\). By a theorem of \textit{J. Tate} [Proc. Am. Math. Soc. 3, 400--406 (1952; Zbl 0047.03901)], there is an upper bound for the characteristic \(p\) of the base field \(K\), in terms of its geometric genus \(g\), namely \(p\leq 2g+1\). Nonconservative function fields of genus one and two were classified by \textit{C. S. Queen} [Arch. Math. 22, 612--623 (1971; Zbl 0251.14010)] and \textit{H. Borges Neto} [Change of genus and classification of fields of genus two. Doctoral thesis, IMPA (1979)]. The case \(g=3\) and \(p=5,7\) was solved by \textit{H. Stichtenoth} [J. Reine Angew. Math. 301, 30--45 (1978; Zbl 0375.12014)] and \textit{K. O. Stöhr} and \textit{M. L. Torres Villela} [Manuscr. Math. 66, No. 1, 61--71 (1989; Zbl 0699.14031)]. The paper under review starts the classification of nonconservative function fields of genus \(g=3\) in characteristic \(p=3\). There is a close relation between nonsmoothness of an algebraic curve and nonregularity of its image under the relative Frobenius morphism. Assuming a certain good behaviour with respect to the Frobenius pullback normalization, the main theorem states that any complete, geometrically integral, regular and nonsmooth curve \(C\) of genus \(3\) over a field \(K\) of characteristic \(3\) is isomorphic to a plane quartic defined by the homogeneous polynomial \(ZY^3-aZ^4-bZX^3-X^4\), where \(a\in K\setminus K^3\) and \(b\in K\). The paper focuses on nonsmooth fibrations. In analogy to the Kodaira-Néron classification of special fibers of minimal fibrations by elliptic curves, the author constructs the minimal proper regular model of some fibrations by nonsmooth projective plane quartic curves, determines the structure of the bad fibers, and studies the global geometry of the total spaces.
scientific article; zbMATH DE number 5886353
scientific article