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**On blowing-up of polarized surfaces.**
*(English)*
Zbl 0968.14003

Let \(X\) be a smooth projective variety defined over the complex number field, and let \(L\) be an ample line bundle on \(X\). Let \(p_{1},\dots , p_{r}\) be \(r\) points on \(X\) and let \(\pi : X''\to X\) be the blowing up at these \(r\) points. We put \(L''=\pi^{*}(L)-\sum_{i}a_{i}E_{i}\), where \(a_{i}\) is an integer and \(E_{i}\) is an exceptional divisor with \(\pi(E_{i})=p_{i}\) for each \(i\).

In this situation, there is the following classical question: Is the line bundle \(L''\) ample if \(p_{1}, \dots , p_{r}\) are in general positions?

By considering this problem, it is possible to make a new example of polarized varieties and so this problem is very important to study a classification of polarized varieties.

Recently a study of this problem has been developed. First we assume that \(\dim X=2\). In Math. Ann. 304, 151-155 (1996; Zbl 0834.14024), O. Küchle proved that \(L\) is ample if \((L'')^{2}>0\), \(L=mH\) for some ample line bundle \(H\) on \(X\), \(m\) an integer with \(m\geq 3\) and \(a_{i}=1\) for any \(i\). In Manuscr. Math. 86, 195-197 (1995; Zbl 0836.14004), G. Xu has obtained the same result if \(X\) is a 2-dimensional projective space. In J. Reine Angew. Math. 469, 199-209 (1995; Zbl 0833.14028), G. Xu also studied the case where \(X\) is a 2-dimensional projective space and \(a_{i}\geq 1\) for \(1\leq i\leq r\). In Rev. Roum. Math. Pures Appl. 42, 371-373 (1997; Zbl 0929.14001), E. Ballico considered the case in which \(L\) is a \(\mathbb{Q}\)-divisor, and obtained a criterion of ampleness of \(L''\) under some conditions.

In Proc. Am. Math. Soc. 127, 2527-2528 (1999; Zbl 0918.14002), E. Ballico treated the case where \(X\) is an \(n\)-dimensional projective space \(L=\mathcal{O}(d)\) with \(d\geq 2\), and \(a_{i}=1\) for \(1\leq i\leq r\).

In the paper under review, the author proves the following theorem: Let \(X\), \(L\), \(X''\), \(L''\), \(p_{i}\), \(E_{i}\), and \(a_{i}\) be as above. Assume that \(\dim X=2\). Then \(L''\) is ample if \((L'')^{2}>0\), the complete linear system \(|L|\) has an irreducible and reduced member \(C\), \(g(C)>h^{1}(\mathcal{O}_{X})\), and \(a_{i}=1\) for any \(i\).

[Here we remark that \(g(C)\geq h^{1}(\mathcal{O}_{X})\) holds in general. For example see Y. Fukuma, Geom. Dedicata 64, 229-251 (1997; Zbl 0897.14008)].

Furthermore by using this criterion, the author gives some examples of polarized surfaces \((X,L)\) with \(g(L)=2\), where \(g(L)\) is the sectional genus of \((X,L)\).

In this situation, there is the following classical question: Is the line bundle \(L''\) ample if \(p_{1}, \dots , p_{r}\) are in general positions?

By considering this problem, it is possible to make a new example of polarized varieties and so this problem is very important to study a classification of polarized varieties.

Recently a study of this problem has been developed. First we assume that \(\dim X=2\). In Math. Ann. 304, 151-155 (1996; Zbl 0834.14024), O. Küchle proved that \(L\) is ample if \((L'')^{2}>0\), \(L=mH\) for some ample line bundle \(H\) on \(X\), \(m\) an integer with \(m\geq 3\) and \(a_{i}=1\) for any \(i\). In Manuscr. Math. 86, 195-197 (1995; Zbl 0836.14004), G. Xu has obtained the same result if \(X\) is a 2-dimensional projective space. In J. Reine Angew. Math. 469, 199-209 (1995; Zbl 0833.14028), G. Xu also studied the case where \(X\) is a 2-dimensional projective space and \(a_{i}\geq 1\) for \(1\leq i\leq r\). In Rev. Roum. Math. Pures Appl. 42, 371-373 (1997; Zbl 0929.14001), E. Ballico considered the case in which \(L\) is a \(\mathbb{Q}\)-divisor, and obtained a criterion of ampleness of \(L''\) under some conditions.

In Proc. Am. Math. Soc. 127, 2527-2528 (1999; Zbl 0918.14002), E. Ballico treated the case where \(X\) is an \(n\)-dimensional projective space \(L=\mathcal{O}(d)\) with \(d\geq 2\), and \(a_{i}=1\) for \(1\leq i\leq r\).

In the paper under review, the author proves the following theorem: Let \(X\), \(L\), \(X''\), \(L''\), \(p_{i}\), \(E_{i}\), and \(a_{i}\) be as above. Assume that \(\dim X=2\). Then \(L''\) is ample if \((L'')^{2}>0\), the complete linear system \(|L|\) has an irreducible and reduced member \(C\), \(g(C)>h^{1}(\mathcal{O}_{X})\), and \(a_{i}=1\) for any \(i\).

[Here we remark that \(g(C)\geq h^{1}(\mathcal{O}_{X})\) holds in general. For example see Y. Fukuma, Geom. Dedicata 64, 229-251 (1997; Zbl 0897.14008)].

Furthermore by using this criterion, the author gives some examples of polarized surfaces \((X,L)\) with \(g(L)=2\), where \(g(L)\) is the sectional genus of \((X,L)\).

Reviewer: Yoshiaki Fukuma (Kochi)

### MSC:

14C20 | Divisors, linear systems, invertible sheaves |

14J25 | Special surfaces |

14J10 | Families, moduli, classification: algebraic theory |