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Register a new userA stratification on the moduli of K3 surfaces in positive characteristic
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Authors
Gerard van der Geer
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We review the results on the cycle classes of the strata defined by the height and the Artin invariant on the moduli of K3 surfaces in positive characteristic obtained in joint work with Katsura and Ekedahl. In addition we prove a new irreducibility result for these strata.
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We study isogenies between K3 surfaces in positive characteristic. Our main result is a characterization of K3 surfaces isogenous to a given K3 surface $X$ in terms of certain integral sublattices of the second rational $ell$-adic and crystalline cohomology groups of $X$. This is a positive characteristic analog of a result of Huybrechts, and extends results of the second author. We give applications to the reduction types of K3 surfaces and to the surjectivity of the period morphism. To prove these results we describe a theory of B-fields and Mukai lattices in positive characteristic, which may be of independent interest. We also prove some results on lifting twisted Fourier--Mukai equivalences to characteristic 0, generalizing results of Lieblich and Olsson.
Let $F$ be a moduli space of lattice-polarized K3 surfaces. Suppose that one has chosen a canonical effective ample divisor $R$ on a general K3 in $F$. We call this divisor recognizable if its flat limit on Kulikov surfaces is well defined. We prove that the normalization of the stable pair compactification $overline{F}^R$ for a recognizable divisor is a Looijenga semitoroidal compactification. For polarized K3 surfaces $(X,L)$ of degree $2d$, we show that the sum of rational curves in the linear system $|L|$ is a recognizable divisor, giving a modular semitoroidal compactification of $F_{2d}$ for all $d$.
We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genus. 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 improved to 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 with certain cubic fourfolds, Gushel-Mukai fourfolds, and hyperkaehler fourfolds.
We give a notion of ordinary Enriques surfaces and their canonical lifts in any positive characteristic, and we prove Torelli-type results for this class of Enriques surfaces.
We show that the K-moduli spaces of log Fano pairs $(mathbb{P}^1timesmathbb{P}^1, cC)$ where $C$ is a $(4,4)$-curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ complete intersection curves in $mathbb{P}^3$. This, together with recent results by Laza-OGrady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$-curves on $mathbb{P}^1timesmathbb{P}^1$ and the Baily-Borel compactification of moduli of quartic hyperelliptic K3 surfaces.
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