On supercuspidal families of curves on a surface in positive characteristic
Let X be a nonsingular irreducible surface over an algebraically closed field k of characteristic \(p>0\), and \(\{C_ u\}_{u\in T}\) a nontrivial family of curves on X parametrized by a nonsingular irreducible curve T. For a closed point \(u\in T\), let \(\eta_ u: \tilde C_ u\to C_ u\) be the normalization, and \(N_ u\) the cokernel of \(\Theta_{\tilde C_ u}\to \eta^*_ u\Theta_ X\). We call a family \(\{C_ u\}\) supercuspidal if \(\deg (N_ u/torsion)<0\) for a general \(u\in T\). There are no supercuspidal families in characteristic 0, while in characteristic \(p>0\) we have following examples: (1) If \(\kappa\) (X)\(\geq 0\) and \(C_ u\) is rational, then \(\{C_ u\}\) is supercuspidal. (2) If a general fibre of a surjection from X to a curve D is singular, then the fibres form a supercuspidal family. In this paper, we prove a kind of normal form theorem for a singularity of general members of a supercuspidal family \(\{C_ u\}\). Let \({\mathcal C}\to T\) be the universal curve of \(\{C_ u\}\). There is a purely inseparable dominant morphism \(B\to T\) such that \(S:=\) the normalization of \(B\times_ T{\mathcal C}\) is nonsingular and \(S\to B\) is smooth. Let \(\phi: S\to X\) be a natural morphism. Then we have a commutative diagram such that F is a succession of Frobenius maps over k, \(\psi\) is purely inseparable, \(\tilde S\) is nonsingular in codimension 1, the generic fibre of \({\tilde \pi}\) is singular, and at a singular point of a general fibre of \({\tilde \pi}\), \(\pi\) is written in the form \(t=x^ p+y^{m+1}\) \((m\not\equiv -1\mod p)\) or \(t=x^ p+y^ q+x^{pm}y\) \((q=p^{\nu}(\nu \geq 1)\), \(m\geq 1)\) by suitable power series coordinates (x,y) of \(\tilde S\) and t of \(\tilde B.\) Thus a general member of \(\{C_ u\}\) must have a singularity which is an image of one of the above singularities.
scientific article; zbMATH DE number 4214195