## 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 $$(X,L)$$.

### MSC:

 14C20 Divisors, linear systems, invertible sheaves 14J25 Special surfaces 14J10 Families, moduli, classification: algebraic theory
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