Abstract

Let \(f:\mathbb{P}^{3}\longrightarrow\mathbb{P}^{3}\) be a morphism given by the linear system \(\mathcal{L}\) of quadrics. Using geometry of the Jacobian surface \(\widetilde{S}\) associated with \(\mathcal{L}\), we show that if \(\widetilde{S}\) is smooth, then \(f\) is not gradient (that is \(f\neq\mathrm{grad}F\) for any cubic polynomial \(F\)).

中文翻译：

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二次映射的梯度性质

摘要

令\(f:\mathbb{P}^{3}\longrightarrow\mathbb{P}^{3}\)是二次方程的线性系统\(\mathcal{L}\)给出的态射。使用与\(\mathcal{L }\)相关的雅可比表面\(\widetilde{S}\) 的几何形状，我们证明如果\(\widetilde{S}\)是光滑的，那么\(f\)是不是梯度（即\(f\neq\mathrm{grad}F\)对于任何三次多项式\(F\)）。