Over the past forty years many papers have studied logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. Terao conjectured thirty years ago that when a curve is a finite set of distinct lines (i.e. a line arrangement) its freeness depends solely on its combinatorics, but this has only been proved for sets of up to 12 lines. In looking for a counter-example to Terao’s conjecture, the nearly free curves introduced by Dimca and Sticlaru arise naturally. We prove here that the logarithmic bundle associated to a nearly free curve possesses a minimal non-zero section that vanishes on one single point, P say, called the jumping point, and that this characterises the bundle. We then give a precise description of the behaviour of P. Based on detailed examples we then show that the position of P relative to its corresponding nearly free arrangement of lines may or may not be a combinatorial invariant, depending on the chosen combinatorics.

中文翻译：

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近乎自由的曲线和排列：向量丛的观点

在过去的四十年中，许多论文研究了与约数除数相关的对数滑轮，特别是与平面曲线相关的对数束。这些曲线的一个有趣的家族是所谓的自由曲线，其相关的对数层是两个线束的直接和。Terao 在 30 年前推测，当一条曲线是一组有限的不同线（即线排列）时，它的自由度仅取决于它的组合，但这仅在最多 12 条线的集合中得到证明。在寻找 Terao 猜想的反例时，Dimca 和 Sticlaru 引入的近乎自由的曲线自然而然地出现了。我们在这里证明了与几乎自由曲线相关的对数丛具有一个最小的非零截面，该截面在一个点上消失，P 说，称为跳跃点，这就是捆绑的特征。然后我们对 P 的行为给出一个精确的描述。基于详细的例子，我们表明 P 相对于其相应的几乎自由的线排列的位置可能是也可能不是组合不变量，这取决于所选择的组合。