Approximating curves on real rational surfaces

János Kollár (1) , Frédéric Mangolte (2)

1 Princeton University
2 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

It is known that every differentiable map from the circle to a rational variety S^1-> X can be approximated by an algebraic map RP^1 -> X. In particular, any simple closed curve on a rational surface S can be approximated by a rational curve on S. Note that the usual result is about maps of rational curves, so the image may have some extra isolated points. In this note, we get rid of these: Let L be a simple, connected, closed curve on a nonsingular rational surface. Then L can be approximated by a nonsingular rational curve. Furthermore, we give necessary and sufficient topological conditions for a simple closed curve on a rational surface to be approximated by a (-1)-curve. Note that (-1)-curves are quite rigid objects, hence approximating by (-1)-curves is a subtle problem.

Mots clés: rational curves; real algebraic surfaces; algebraic cycles

Domaines: Géométrie algébrique [math.AG]

János Kollár, Frédéric Mangolte. Approximating curves on real rational surfaces. 2012. ⟨hal-00743204v1⟩