An almost strongly minimal non-Desarguesian projective plane
An almost strongly minimal non-Desarguesian projective plane
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几乎强极小非笛沙格射影平面
DOI：
10.1090/s0002-9947-1994-1165085-8
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发表时间：
1994
影响因子：
1.3
通讯作者：
J. Baldwin
中科院分区：
文献类型：
--
作者：
J. Baldwin
存在一个几乎强极小射影平面，它不是Despermesian的. Zil'ber指出，每一个强极小集都是“平凡的”、“类域的”或“类模的”。这个猜想被Hrushovski反驳了[4]。改变他的结构，我们在这里反驳两个更精确的版本的猜想。Zil'ber [8]称一个强极小集M是域似的，如果在M中存在一个可定义的伪平面。（伪平面是一种关联结构，使得每对线仅相交于100个点，并且对偶地，只有100个线穿过一对点。如果下面的猜想是正确的，这种命名法就有道理了。[8]的猜想B。每一个不可数范畴伪平面在代数闭域中是可定义的，并且域在伪平面中是可定义的。这个猜想有几个方面。在模型论的脉络中，它限制了不可数范畴伪平面的多样性。特别是，它断言只有可数多个这样的（因为每个都必须在代数闭域中定义）。这方面的猜想已经被[4]驳斥。（Hrushovski表明存在21 O个不是局部模的强极小集。根据Zil'ber的可分性定理，它们是类场的。）一个更几何方面的问题是措辞在另一个Zil'ber猜想。[8]的猜想C。每一个不可数范畴仿射平面都是Despermesian的，因此是代数闭域上的仿射平面。我们证明了这里构造的射影平面不能解释一个群，因此不能是Despermesian的。与这个射影平面相关联的仿射平面也不是Deserkesian的，所以猜想C被反驳。最后，在猜想B的部分后面还有一个更代数的几何猜想，它断言每个类场结构都可以在域中定义。编辑于1989年10月23日收到，修订版于1992年5月14日收到。1991年数学学科分类。小学03 C60;中学51 C I 0。部分由NSF资助8602558。在时间，因为这篇论文提交（1989年10月）的各种作者，包括Herwig，Poizat，和瓦格纳，提出了替代方法来组织证明，通用模型是共饱和的，没有无限群是解释的结构构造的Hrushovski的方法。01994美国数学学会0002-9947/94 $1.00 + $.25每页
There is an almost strongly minimal projective plane which is not Desarguesian. Zil'ber conjectured that every strongly minimal set is 'trivial', 'field-like', or 'module-like'. This conjecture was refuted by Hrushovski [4]. Varying his construction, we refute here two more precise versions of the conjecture. Zil'ber [8] calls a strongly minimal set M field-like if there is a pseudoplane definable in M. (A pseudoplane is an incidence structure such that each pair of lines intersect in only finitely many points and dually there are only finitely many lines passing through a pair of points.) This nomenclature would have been justified if the following conjecture were correct. Conjecture B of [8]. Every uncountably categorical pseudoplane is definable in an algebraically closed field and the field is definable in the pseudoplane. This conjecture has several aspects. In a model theoretic vein it limits the variety of uncountably categorical pseudoplanes. In particular, it asserts that there are only countably many such (as each must be defined in an algebraically closed field). This aspect of the conjecture is already refuted by [4]. (Hrushovski shows there are 21O strongly minimal sets which are not locally modular. By Zil'ber's trichotomy theorem they are thus field-like.) A more geometric aspect of the problems is phrased in another Zil'ber conjecture. Conjecture C of [8]. Every uncountably categorical affine plane is Desarguesian and hence is an affine plane over an algebrically closed field. We show that the projective plane constructed here does not interpret a group and thus cannot be Desarguesian. The affine plane associated with this projective plane also fails to be Desarguesian so Conjecture C is refuted. Finally there is a more algebraic geometric conjecture behind the part of Conjecture B asserting every field-like structure is definable in a field. Received by the editors October 23, 1989 and, in revised form, May 14, 1992. 1991 Mathematics Subject Classification. Primary 03C60; Secondary 51 C I0. Partially supported by NSF grant 8602558. In the time since this paper was submitted (October 1989) various authors, including Herwig, Poizat, and Wagner, have suggested alternative methods to organize the proof that the generic model is co-saturated and that no infinite group is interpretable in a structure constructed by Hrushovski's method. 01994 American Mathematical Society 0002-9947/94 $1.00 + $.25 per page