Abstract
We study the relation between Cremona transformations in space and quadratic line complexes. We show that it is possible to associate a space Cremona transformation to each smooth quadratic line complex once we choose two distinct lines contained in the complex. Such Cremona transformations are cubo-cubic and we classify them in terms of the relative position of the lines chosen. It turns out that the base locus of such a transformation contains a smooth genus two quintic curve. Conversely, we show that given a smooth quintic curve C of genus 2 in ℙ3 every Cremona transformation containing C in its base locus factorizes through a smooth quadratic line complex as before. We consider also some cases where the curve C is singular, and we give examples both when the quadratic line complex is smooth and singular.
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The first author is partially supported by Acordo de Cooperação Franco-brasileira, the second author is partially supported by Capes-Cofecub and the third author is partially supported by CNPq-Grant: 307833/2006-2, Capes-Cofecub
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Avritzer, D., Gonzalez-Sprinberg, G. & Pan, I. On Cremona transformations and quadratic complexes. Rend. Circ. Mat. Palermo 57, 353–375 (2008). https://doi.org/10.1007/s12215-008-0026-3
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DOI: https://doi.org/10.1007/s12215-008-0026-3