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The Kodaira dimension of some moduli spaces of elliptic \(K3\) surfaces. (English) Zbl 1473.14065

One of the first natural questions one may ask if it is interested in the geometry of moduli spaces regards the positivity of the canonical bundle of such spaces. The primary invariant that one computes in order to give a coarse answer to this question is the Kodaira dimension. Since the moduli space of \(K3\) surfaces that contain a fixed lattice in their Néron-Severi group has a nice description via the period map as a quotient of a bounded symmetric domain, this question is widely studied for such varieties. Originally this problem has been formulated for \(K3\) surfaces with a primitive polarisation of degree \(2d\), and several interesting results have been proved for such spaces. After that, it is natural to ask whether one can compute the Kodaira dimension of the moduli space of \(K3\) surfaces containing line bundles with prescribed intersection numbers.
In this paper, the authors consider \(K3\) surfaces which admit an elliptic fibration, i.e. a genus-one fibration with a distinguished zero section. Inside this \(18\)-dimensional family of \(K3\) surfaces, for any \(k>1\) one can consider the \(K3\) surfaces which admit another section of the elliptic fibration that meets in \(k-2\) points the zero section. Denote by \(\mathcal{M}_{2k}\) the moduli space that parametrizes these \(K3\) surfaces. By a general result on quotients of bounded symmetric domains, in order to prove that the Kodaira dimension of \(\mathcal{M}_{2k}\) is maximal, it is sufficient to find a non-zero cusp form of certain weight vanishing along with the ramification of the quotient map. The authors construct such cusp form for \(k\geq 208\) and some smaller \(k\), proving that \(\mathcal{M}_{2k}\) is of general type for such \(k\). In this direction, they also show that \(\mathrm{kod}(\mathcal{M}_{2k})\geq 0\) for some smaller \(k\). In the second part of the paper, the authors prove the unirationality of \(\mathcal{M}_{2k}\) for small values of \(k\) exhibiting a geometric realization of the parametrized \(K3\) surfaces. This proof is based on a case-by-case construction which is very explicit.

MSC:

14J15 Moduli, classification: analytic theory; relations with modular forms
14J28 \(K3\) surfaces and Enriques surfaces
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14M20 Rational and unirational varieties
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
32N15 Automorphic functions in symmetric domains
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References:

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