On the irrationality of moduli spaces of K3 surfaces

Agostini, Daniele; Barros, Ignacio; Lai, Kuan-Wen

doi:10.1090/tran/8830

On the irrationality of moduli spaces of K3 surfaces
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by Daniele Agostini, Ignacio Barros and Kuan-Wen Lai;

Trans. Amer. Math. Soc. 376 (2023), 1407-1426

DOI: https://doi.org/10.1090/tran/8830

Published electronically: November 9, 2022

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Abstract:

We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genus $g$. We prove that the growth is bounded by a polynomial function of degree $14+\varepsilon$ for any $\varepsilon >0$ and, for three sets of infinitely many genera, the bounds can be refined to polynomials of degree $10$. The main ingredients in our proof are the modularity of the generating series of Heegner divisors due to Borcherds and its generalization to higher codimensions due to Kudla, Millson, Zhang, Bruinier, and Westerholt-Raum. For special genera, the proof is also built upon the existence of K3 surfaces associated Hodge theoretically with certain cubic fourfolds, Gushel–Mukai fourfolds, and hyperkähler fourfolds.

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On the irrationality of moduli spaces of K3 surfaces
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Abstract: