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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 $$H^{2}(Y,\mathbb{C})$$.

##### MSC:
 14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations 14J15 Moduli, classification: analytic theory; relations with modular forms
##### Keywords:
Noether-Lefschetz loci; Hurwitz spaces
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