Density conjecture for rational curves on K3 surfaces

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The density conjecture for rational curves on K3 surfaces

   Let $(X,H)$ be a polarized complex K3 surface. A rational curve on $X$ means a curve $C\subset X$ that is rational. The density conjecture. The union

   $$\bigcup C\subset X$$

   of all rational curves $C\subset X$ is dense in the Zariski topology, or, more strongly, in the classical topology. This is known for several general classes of polarized K3 surfaces, including general $(X,H)$ in the classical topology, but the stronger form remains open in general.

   source: Daniel Huybrechts and Claire Voisin, “Curves and cycles on K3 surfaces”, arXiv:1303.4564 (2013).

2. The density conjecture for rational curves on K3 surfaces

   Let $X$ be a K3 surface over the complex numbers. Consider the union of all rational curves contained in $X$, and equip $X$ with its classical topology.

   The density conjecture for rational curves on K3 surfaces. The union of all the rational curves is dense in $X$ with respect to the classical topology.

   This is a different assertion from the infinitude conjecture: it concerns the topological density of the rational curves rather than merely their number. The source presents it as another conjecture and does not state a resolution.

   source: Takeo Nishinou, “Degeneration and curves on K3 surfac”, arXiv:1510.03350 (2019).