Abstract
Let X be a fixed projective scheme which is flat over a base scheme S. The association taking a quasi-projective S-scheme Y to the scheme parametrizing S-morphisms from X to Y is functorial. We prove that this functor preserves limits, and both open and closed immersions. As an application, we determine a partition of schemes parametrizing rational curves on the blow-ups of projective spaces at finitely many points. We compute the dimensions of its components containing rational curves outside the exceptional divisor and the ones strictly contained in it. Furthermore, we provide an upper bound for the dimension of the irreducible components intersecting the exceptional divisors properly.
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Acknowledgements
I would like to thank Vladimir Guletskiĭ for suggesting the problem and for many useful discussions. I am grateful to Thomas Eckl, Roy Skjelnes and the anonymous referee for helpful suggestions. This work was supported by CNPq, National Council for Scientific and Technological Development under the Grant [159845/2019-0].
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Communicated by Eduardo Esteves.
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das Dores, L. Properties of schemes of morphisms and applications to blow-ups. São Paulo J. Math. Sci. 15, 790–811 (2021). https://doi.org/10.1007/s40863-021-00258-9
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DOI: https://doi.org/10.1007/s40863-021-00258-9