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arXiv · 2011.00332

Smooth rational curves on rational surfaces

Lucas das Dores
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Abstract

Consider the scheme parametrizing non-constant morphisms from a fixed projective curve to a projective surface. There is a rational map between this scheme and the Chow variety of $1$-cycles on the surface. We prove that, if the curve is non-singular, then this rational map is a morphism. As a consequence, we obtain that, if the surface is rational and we fix a divisor class containing a non-singular rational curve, then the scheme parametrizing rational curves on this class is irreducible. Further, if the class has non-negative self-intersection, then the scheme of rational curves has expected dimension.

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Lucas das Dores. 2020-10-31. Smooth rational curves on rational surfaces. https://arxiv.org/abs/2011.00332

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