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- Title
Worst Singularities of Plane Curves of Given Degree.
- Authors
Cheltsov, Ivan
- Abstract
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 $$\frac{5}{2d}$$ .
- Publication
Journal of Geometric Analysis, 2017, Vol 27, Issue 3, p2302
- ISSN
1050-6926
- Publication type
Article
- DOI
10.1007/s12220-017-9762-y