On ruled surfaces with big anti-canonical divisor and numerically trivial divisors on weak log Fano surfaces
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Rikito Ohta · Shinnosuke Okawa
On ruled surfaces with big anti-canonical divisor and numerically trivial divisors on weak log Fano surfaces Received: 6 April 2019 / Accepted: 8 September 2020 Abstract. We investigate the structure of geometrically ruled surfaces whose anti-canonical class is big. As an application we show that the Picard group of a normal projective surface whose anti-canonical class is nef and big is a free abelian group of finite rank.
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability of vector bundles on curves . . . . . . . . . . . . . . . . . . . . . . . Geometrically ruled surface with big anti-canonical line bundle . . . . . . . . . Numerically trivial line bundles on normal surface with nef and big anti-canonical bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Proof via Corollary 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Proof via combination of known results . . . . . . . . . . . . . . . . . . . 4.3. What if ∈ {0, 1}? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Some examples of ruled surfaces whose anti-canonical sheaf is big . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction Let X be a smooth projective surface over a field k whose anti-canonical divisor −K X is big. Although the classification of such surfaces is completely understood if −K X is also nef, the question becomes much harder if we drop the nefness from the assumption. If X is rational, then it is known to be a Mori dream space as is shown in [16, Theorem 1]. They also gave a kind of structure theorem for such surfaces [16, Theorem 2]. Let us consider the case when X is not rational. Then X is obtained by repeatedly blowing up a geometrically ruled surface (over a curve of positive genus) with big anti-canonical bundle, since the Kodaira dimension of X is −∞. In the first part R. Ohta (B) · S. Okawa: Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka 560-0043, Japan. e-mail: [email protected] S. Okawa e-mail: [email protected] Mathematics Subject Classification: 14J26 · 14E30 · 14G17
https://doi.org/10.1007/s00229-020-01242-x
R. Ohta, S. Okawa
of this paper, we study the structure of geometrically ruled surfaces whose anticanonical divisor is big. We will show the following theorem. Theorem 1.1. (=Theorem 3.2) Let C be a smooth projective curve of genus g ≥ 1 and E be an unstable vector bundle of rank 2 on C. Then −K PC (E) is big if and only if E L ⊕ M for some line bundles L and M on C such that deg L −deg M > 2g −2. In Proposition 3.1 below, Theorem 1.1 is proven under the assumption that E is decomposable. In order to reduce the general case to this, we take a degeneration of E to the direct sum of its Harder-Narasimhan factors and then apply Proposition 3.1 to it.
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