On Algebraic Hyperbolicity of Log Varieties
Fix a very general hypersurface D in P^n of degree at least 2n + 1 and we show that the complement P^n - D does not contain any algebraic torus C^*.
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arXiv subjects
Xi Chen
Publications and source records attributed to Xi Chen.
A Simple Proof that Rational Curves on K3 are Nodal
We give a simple proof of the statement that every rational curve in the primitive class of a general K3 surface is nodal.
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On Algebraic Hyperbolicity of Log Surfaces
We call a log variety (X, D) algebraically hyperbolic if there exists a positive number e such that 2g(C) - 2 + i(C, D) >= e deg(C) for all curves C on X, where i(C, D) is the number of the intersections between D and the normalization of C. Among other things, we proved that (P^2, D) is algebraically hyperbolic for a very general curve D of degree at least 5. More specifically, we showed that 2g(C) - 2 + i(C, D) >= (deg(D) - 4) deg C for all curves C on P^2. For example, fix a very general quintic curve D and then for any map f: C = P^1 --> P^2, there are at least deg(f) + 2 distinct points on C that map to points on D by f.
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On the Intersection of Two Plane Curves
We study the following question: fix a sufficient general curve D of degree d in P^2, what is the least number of intersections between D and an irreducible curve of degree m? G. Xu proved this number i(d, m) is at least d - 2 for all m. This problem can be regarded as the algebraic part of Kobayashi conjecture on the hyperbolicity of P^2 D. We first improved Xu's bound with m fixed and then generalized his result to rational ruled surfaces.
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Singularities of Rational Curves on K3 surfaces
We proved that every rational curves in the primitive class of a general K3 surface of any genus is nodal.
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Rational Curves on K3 Surfaces
We proved the existence of rational curves in every linear system on a general K3 surface and that all rational curves in the hyperplane class are nodal on a general K3 surface of small genus.
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