Effective divisor classes and blowings-up of \({\mathbb{P}}^ 2\)
Let \(X_ n\to^{\pi}{\mathbb{P}}^ 2\) be the monoidal transformation of the (complex) projective plane centered at distinct points \(P_ 1,...,P_ n\) of \({\mathbb{P}}^ 2\). We recall that the Néron-Severi group of \(X_ n\) is freely generated by the divisor class [L] of the proper transform L of a line in \({\mathbb{P}}^ 2\) and by the classes \([E_ i]\) of the ''exceptional'' fibers \(E_ i\) over \(P_ i\); the intersection pairing is given by \([L]^ 2=1\); \([L]\cdot [E_ i]=0\); \([E_ i]\cdot [E_ j]=- \delta_{i,j}\). Let \({\mathcal M}(X_ n)\) denote the monoid of elements F in the Néron-Severi group with the property that F contains an effective divisor. In this paper we (1) construct a finite generating set for \({\mathcal M}(X_ n)\) for \(n\leq 8\), and give a particularly simple geometric description of the generators when \(P_ 1,...,P_ n\) are in ''general position''; (2) show that, for \(n\geq 9\), \({\mathcal M}(X_ n)\) need not be finitely generated, despite the finite generation of the whole Néron-Severi group; (3) prove the related result that if a nonsingular surface X contains an infinite number of exceptional curves of the first kind, then X is necessarily rational.
monoidal transformation of the complex projective plane
scientific article; zbMATH DE number 3899047