Geodesic
Rational Surfaces of Kodaira Type IV
Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 741-750.

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We study the geometry of a rational surface of Kodaira type IV by giving the nature of its integral curves of self-intersection less than zero, in particular we show that they are smooth and rational. Hence, under a reasonable assumption, we prove the finite generation of its monoid of effective divisor classes and in almost all cases its anticanonical complete linear system is of projective dimension zero and of self- intersection strictly negative. Furthermore, we show that if this condition is not fulfilled, the monoid may fail to be finitely generated.
Si studia la geometria di una superficie di tipo Kodaira IV descrivendo la natura delle sue curve integre di autointersezione minore di zero, in particolare si prova che esse sono regolari e razionali. Sotto un'opportuna ipotesi si dimostra che il monoide delle classi di divisori effettivi ad essa associato è finitamente generato e che in quasi tutti i casi il suo sistema lineare completo anticanonico è di dimensione proiettiva zero e auto-intersezione strettamente negativa. Infine si evidenzia che se tale condizione non è soddisfatta, il monoide può non essere finitamente generato. 
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Failla, Gioia; Lahyane, Mustapha; Molica Bisci, Giovanni. Rational Surfaces of Kodaira Type IV. Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 741-750. http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_3_a18/

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