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Higher Noether-Lefschetz loci of elliptic surfaces. (English) Zbl 1141.14019
Let \({\mathcal M} _{n}\) be the coarse moduli space of elliptic surfaces \(\pi :X \to \mathbb{P}^1\) over \(\mathbb{C}\) with a section such that the geometric genus of \(X\) is \(n-1\) and \(\pi\) has at least one singular fiber. It is known that the Picard number \(\rho(X)\) satisfies \(2 \leq \rho(X) \leq h^{1,1} = 10n\). In this paper the author studies the loci \(\text{NL}_{r}=\{[\pi :X \to \mathbb{P}^1]\in {\mathcal M} _{n} \mid \rho(X)\geq r\}\), which he calls higher Noether-Lefschetz loci in analogy with the article by D. A. Cox [Am. J. Math. 112, No. 2, 289–329 (1990; Zbl 0721.14017)]. The author proves that when \(n \geq 2\) and \(2 \leq r \leq 10n\), the inequality \(\dim \text{NL}_{r}\geq \dim {\mathcal M} _{n} - (r - 2)=10n-r\) holds. Moreover, suppose \(U\) is the locus of elliptic surfaces with non-constant \(j\)-invariant, we have \(\dim \text{NL}_{r}\cap\;U=10n-r\). For the lower bound, the author constructs elliptic surfaces with high Picard number; he begins with a rational elliptic surface with four multiplicative singular fibers, and takes a cyclic base change to obtain a surface with reducible singular fibers with many components. To obtain the upper bound, he regards an elliptic surface over \(\mathbb{P}^1\) as a surface \(Y\) in the weighted projective space \(\mathbb{P}(1,1,2n,3n)\), and uses Griffiths-Steenbring identification of the Hodge filtration on \(H^{2}(Y,\mathbb{C})\).

14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14J15 Moduli, classification: analytic theory; relations with modular forms
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