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arXiv · 1706.00492

On the existence of birational surjective parametrizations of affine surfaces

Jorge Caravantes·J. Rafael Sendra·David Sevilla·Carlos Villarino
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Abstract

In this paper we show that not all affine rational complex surfaces can be parametrized birationally and surjectively. For this purpose, we prove that, if S is an affine complex surface whose projective closure is smooth, a necessary condition for S to admit a birational surjective parametrization from an open subset of the affine complex plane is that the infinity curve of S must contain at least one rational component. As a consequence of this result we provide examples of affine rational surfaces that do not admit birational surjective parametrizations.

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Jorge Caravantes, J. Rafael Sendra, David Sevilla, Carlos Villarino. 2017-06-01. On the existence of birational surjective parametrizations of affine surfaces. https://arxiv.org/abs/1706.00492

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