On minimal models of algebraic curves

Nguyen Khac Viet

Sbornik. Mathematics, Tome 67 (1990) no. 1, pp. 65-74

Résumé

Let be an odd prime number. Consider the algebraic curves (normalizations of their projective closures): $$ x^p+y^p=1, \qquad y^p=x^s(1-x), \quad s=1,\dots,p-2. $$ Let $\zeta$ be a primitive $p$th root of $1$. The Galois group $\operatorname{Gal}(\mathbf Q_p(\zeta)/\mathbf Q_p)$ acts on the minimal models of these curves over $\mathbf Z_p(\zeta)$. This idea is used here to study their minimal models over $\mathbf Z_p$. The action of $\operatorname{Gal}(\mathbf Q_p(\zeta)/\mathbf Q_p)$, passage to the quotient modulo this action, the resolution of singularities on the quotients, and the contraction of exceptional curves of genus $1$ are described. All of this leads to minimal models of the indicated curves over $\mathbf Z_p$. Bibliography: 6 titles.