Affine curves on which every point is a set-theoretic complete intersection
Given a 1-dimensional function field K and field of constants k, let A be an entire k-algebra of finite type with field of fractions K, and X the affine curve Spec A. Well known facts: Pic A is a torsion group (i.e., A is prefactorial) \(\Leftrightarrow\) each point of X is set-theoretically the complete intersection of X and a hypersurface (in one, and so every, affine embedding of X); Pic A is torsion if k is algebraic over a finite field. Two theorems for the case of algebraically closed k (the first due to Cunnea, the second to Murthy and Pedrini): \(Pic A=0\) (i.e., A is factorial) \(\Leftrightarrow each\) point of X is principal (i.e., scheme- theoretically the complete intersection of X and a hypersurface) \(\Leftrightarrow X\) is rational (i.e., \(K=k(t))\) and smooth \(\Leftrightarrow X\) is the affine line minus a finite set of points; in case char \(k=0\), Pic A is torsion \(\Rightarrow Pic A=0.\) This paper treats related results which do not necessarily restrict k as much. Among the main technical tools used are: the units-Pic Mayer-Vietoris sequence arising from the ''conductor context'' [as in the Murthy-Pedrini treatment; cf. \textit{M. P. Murthy} and \textit{C. Pedrini}, Algebr. K-Theory II, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 342, 109-121 (1973; Zbl 0283.14003)]; in the case of normal X, the exact sequence (due essentially to F. K. Schmidt) involving Pic A, the group of units of A, the group of degree 0 divisors of K mod linear equivalence, and the set of points at infinity with respect to X; and certain classical geometric methods - albeit thoroughly disguised - involving the linear series on a projective completion of X. To convey something of the flavor of the results obtained, here are an example and, for the case of algebraically closed k, three theorems which supplement the two stated above: Example. Let \(X=C-S\), where C is the complete plane cuspidal cubic over k and S is a finite set of simple points of C. Let r be the rank of (the abstract abelian group) Pic A. Then: \(r=0\) if \(char k\neq 0;\) r is finite if k is an algebraic number field; r is infinite otherwise. In case k is an algebraic number field, \(r=0\) if S is ''correctly chosen'' - which at least requires that the number of points of S exceed the degree of k. - Theorem. Suppose that for some affine embedding of X there is an integer d and an infinite set of points of X, each the set-theoretic complete intersection of X and a hypersurface of degree at most d. Then X is rational and smooth. - Theorem. Suppose \(char k\neq 0\) and k is not algebraic over its prime field. Then: Pic A is torsion \(\Leftrightarrow X\) is rational and its singular points are unibranch \(\Leftrightarrow there\) exists a finite birational morphism \(X\to Y\) such that Y is rational and its singular points are ''cusps''. Moreover: in case Pic A is torsion, there is a unique such Y having the same conductor (viewed as a divisor of K) and set of points at infinity as has X. - Theorem. Suppose \(char k\neq 0\) and k is algebraic over its prime field. Then: Pic A\(=0 \Leftrightarrow X\) has infinitely many principal points.
scientific article; zbMATH DE number 3861318
scientific article