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On minimal models of algebraic curves
Nguyen Khac Viet
Abstract:
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.
Received: 07.04.1987
Citation:
Nguyen Khac Viet, “On minimal models of algebraic curves”, Mat. Sb., 180:5 (1989), 625–634; Math. USSR-Sb., 67:1 (1990), 65–74
Linking options:
https://www.mathnet.ru/eng/sm1624https://doi.org/10.1070/SM1990v067n01ABEH002085 https://www.mathnet.ru/eng/sm/v180/i5/p625
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Abstract page: | 290 | Russian version PDF: | 110 | English version PDF: | 2 | References: | 32 | First page: | 2 |
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