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Generic simple coverings of the affine plane. (English) Zbl 0839.14005

In previous papers, ideas of Deligne are used to prove the factoriality of the surface \(Z^p = f(X,Y)\) for a generic choice of the polynomial \(f(X,Y)\) of arbitrary degree \(\geq 4\) (with \(p \geq 3)\). In this paper we study the class group of the surface \(Z^n = f(X,Y)\) for arbitrary positive integer \(n\). The above mentioned calculation leads us naturally to conjecture that the class group of \(Z^n = f(X,Y)\) is factorial for a generic choice of \(f\). To be more precise, let \(f = \sum T_{ij} X^iY^j\) be a generic polynomial with indeterminate coefficients and let \(A_n = K[X,Y,Z]/(Z^n - f)\) where \(K\) is the algebraic closure of \(\mathbb{F}_p (T_{ij})\) with \(\mathbb{F}_p\) the prime field of \(p\) elements \((p \geq 3)\). Assume the degree of \(f\) is at least 4. Then we conjecture: \[ \text{for all }n \in \mathbb{F}^+,\;A_n \text{ is factorial}. \tag{*} \] In this paper we prove that (*) reduces to the case \(\text{g.c.d.} (p,n) = 1\). In section 1 and 2, descent techniques are used to study the class group of arbitrary surfaces \(Z^n = f\). In section 3 the reduction of (*) to the case \(\text{g.c.d.} (p,n) = 1\) is accomplished by analyzing the action of \({\mathcal G} = \text{Gal} (K, \mathbb{F}_p (T_{ij}))\) on the divisor class group of \(Z^{pm} = f\).

MSC:

14E20 Coverings in algebraic geometry
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13C20 Class groups
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References:

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