We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration \({\pi : \mathbb{CP}^3 \rightarrow S^4}\). We prove three results about the topology of the twistor discriminant locus of an algebraic surface in \({\mathbb{CP}^3}\). First of all we prove that, with the exception of two special cases, the real dimension of the twistor discriminant locus of an algebraic surface is always equal to 2. Secondly we describe the possible intersections of a general surface with the family of twistor lines: we find that only 4 configurations are possible and for each of them we compute the dimension. Lastly we give a decomposition of the twistor discriminant locus of a given cone in terms of its singular locus and its dual variety.

中文翻译：

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4球面上的扭转判别轨迹的三个拓扑结果

我们利用经典（实数和复数）代数几何的技术来研究标准扭曲纤维\（{\ pi：\ mathbb {CP} ^ 3 \ rightarrow S ^ 4} \）。我们证明了\（{\ mathbb {CP} ^ 3} \）中代数曲面的扭曲判别轨迹的拓扑的三个结果。首先，我们证明，除两种特殊情况外，代数曲面的扭曲判别轨迹的实际尺寸始终等于2。其次，我们描述了广义曲面与扭曲线族的可能交点：我们发现只有4种配置是可行的，并且我们针对每种配置计算尺寸。最后，我们根据给定圆锥的奇异轨迹及其对偶变化，给出了其扭曲判别轨迹的分解。