On the finite generation of the effective monoid of rational surfaces |
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Authors: | Mustapha Lahyane |
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Affiliation: | Instituto de Física y Matemáticas (IFM), Universidad Michoacana de San Nicolás de Hidalgo (U.M.S.N.H.), Edificio C-3, Ciudad Universitaria. C. P. 58040 Morelia, Michoacán, Mexico |
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Abstract: | We give a numerical criterion for ensuring the finite generation of the effective monoid of the surfaces obtained by a blowing-up of the projective plane at the supports of zero dimensional subschemes assuming that these are contained in a degenerate cubic. Furthermore, this criterion also ensures the regularity of any numerically effective divisor on these surfaces. Thus the dimension of any complete linear system is computed. On the other hand, in particular and among these surfaces, we obtain ringed rational surfaces with very large Picard numbers and with only finitely many integral curves of strictly negative self-intersection. These negative integral curves except two (−1)-curves are all contained in the support of an anticanonical divisor. Thus almost all the geometry of such surfaces is concentrated in the anticanonical class. |
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Keywords: | 14C20 14C40 14C17 14F17 14E05 14E15 14E30 14N05 |
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