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Bounds on Seshadri constants on surfaces with Picard number 1. (English) Zbl 1248.14010

Let \(X\) be a smooth projective \(n\)-dimensional variety of Picard number \(1\), and \(L\) be an ample line bundle on \(X\) that generates the Néron-Severi group. The Seshadri constant \(\varepsilon (L;x)\) of \(L\) at \(x\in X\) reaches the maximal value \(\varepsilon (L;1)\) at very general points. For \(r\) points in \(X\) and \(L\), the maximum \(\varepsilon (L;r)\) of multipoint Seshadri constant is similarly defined. Estimating and determining the values \(\varepsilon (L;1)\) and \(\varepsilon (L;r)\) are important problems that arise in the study of local positivity of line bundles. The upper bound \(\varepsilon (L;1)\leq \sqrt[n]{L^n}\) is so far known for arbitrary \(n\). Besides, the lower bound of \(\varepsilon (L;r)\) is also introduced when \(X\) is a surface of Picard number \(\geq 1\) and \(L\) is a very ample line bundle.
The paper under review only considers a surface \(X\) of Picard number \(1\). The first theorem gives the new lower bound \(\varepsilon (L;1)\geq \frac{p_0}{m_0}N\) with \(N:=L^2\) being not a square. The values \(p_0\) and \(m_0\) are explicitely determined in terms of \(N\) by the equations (8)(9) in the paper. The second theorem estimates \(\varepsilon (L;r)\) by giving the lower bound \(\varepsilon (L;r)\geq \left\lfloor\sqrt{\frac{N}{r}}\right\rfloor\).
The author states a conjecture about the lower bound of \(\varepsilon (L;1)\) in the relation with the Pell’s equation.

MSC:

14C20 Divisors, linear systems, invertible sheaves
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References:

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