Polynomial curves on trinomial hypersurfaces

Abstract

We prove that every rational trinomial affine hypersurface admits a horizontal polynomial curve. This result provides an explicit non-trivial polynomial solution to a trinomial equation. Also we show that a trinomial affine hypersurface admits a Schwartz-Halphen curve if and only if the trinomial comes from a platonic triple. It is a generalization of Schwartz-Halphen's Theorem for Pham-Brieskorn surfaces.

Cylinders in Fano varieties

“…Affine varieties that do not admit effective G a -actions are often called rigid [7,8,24,65,82,118]. Applying [120,Corollary 2.1.4] and [10,Proposition 4.1] to affine cones over projective varieties, we obtain the following result: Theorem 1.20.…”