We prove that \(\frac{2}{d}, \frac{2d-3}{(d-1)^2}, \frac{2d-1}{d(d-1)}, \frac{2d-5}{d^2-3d+1}\) and \(\frac{2d-3}{d(d-2)}\) are the smallest log canonical thresholds of reduced plane curves of degree \(d\geqslant 3\), and we describe reduced plane curves of degree d whose log canonical thresholds are these numbers. As an application, we prove that \(\frac{2}{d}, \frac{2d-3}{(d-1)^2}, \frac{2d-1}{d(d-1)}, \frac{2d-5}{d^2-3d+1}\) and \(\frac{2d-3}{d(d-2)}\) are the smallest values of the \(\alpha \)-invariant of Tian of smooth surfaces in \({\mathbb {P}}^3\) of degree \(d\geqslant 3\). We also prove that every reduced plane curve of degree \(d\geqslant 4\) whose log canonical threshold is smaller than \(\frac{5}{2d}\) is GIT-unstable for the action of the group \(\mathrm {PGL}_3({\mathbb {C}})\), and we describe GIT-semistable reduced plane curves with log canonical thresholds \(\frac{5}{2d}\).

中文翻译：

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给定度的平面曲线的最差奇点。

我们证明\（\ frac {2} {d}，\ frac {2d-3} {（d-1）^ 2}，\ frac {2d-1} {d（d-1）}，\ frac { 2d-5} {d ^ 2-3d + 1} \）和\（\ frac {2d-3} {d（d-2）} \）是度数\（d的减小平面曲线的最小对数正则阈值\ geqslant 3 \），我们描述了度为d的简化平面曲线，其对数规范阈值为这些数字。作为应用程序，我们证明\（\ frac {2} {d}，\ frac {2d-3} {（d-1）^ 2}，\ frac {2d-1} {d（d-1）} ，\ frac {2d-5} {d ^ 2-3d + 1} \）和\（\ frac {2d-3} {d（d-2）} \）是\（\ alpha \ ） -度（\ d \ geqslant 3 \）的\（{\ mathbb {P}} ^ 3 \）中光滑表面的Tian不变性。我们还证明，对数\（d \ geqslant 4 \）的所有减小的平面曲线，其对数正则阈值均小于\（\ frac {5} {2d} \），对于该组\（\ mathrm {PGL} _3（{\ mathbb {C}}）\），我们描述了具有对数规范阈值\（\ frac {5} {2d} \）的GIT半稳定的简化平面曲线 。