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By Francesco Scattone

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Extra resources for On the Compactification of Moduli Spaces for Algebraic K3 Surfaces

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L. )/r . L } D,/r are We are r = r. = 0 (L. )/0 (L. ) /0_ (L, ) • Since this analysis is more involved than the corresponding analysis for sublattices of rank one, we shall avoid any attempt at generalizations, and restrict our attention to the polarized K3 lattices L^ ; thus, throughout this section we use the notation L = L k = <-2k>@H2©Eg often omitting any explicit reference to the integer k . A major role in the description of the equivalence class of a lattice E /E . E e I~(L) is played by the isomorphism class of the lattice Namely, let's assume for simplicity that 50 E is also primitive MODULI OF K3 SURFACES as a sublattice of L E /E = F /F F e I 9 (L) <{) e 0_(L) E /E , for mapping .

3]). Nikulin has determined the exact conditions for the existence of an even lattice with prescribed signature and discriminant form. 2]). signature (n + ,n_) An even lattice with and discriminant form q: G -• Q/2Z exists if the following conditions are simultaneously satisfied: (1) n (2) n + > 0 , n_ > 0 , n + +n_ > h(G) where h(G) group G . - n_ = sign q (mod 8) is the minimum number of generators of the finite abelian The theorem above will be used in conjunction with the following result on primitive embeddings into even uniraodular lattices (cfr.

3. 3. Suppose L is a nondegenerate even lattice containing at least two hyperbolic planes. automorphisms of L containing <^(L) Let = {±1}«{ image of x: r -• 0(G)} < °( G L ) * Zv -• v induces a bisection • Then the association - KG L )/r I^U/r Classification. 2 r is a polarized K3 lattice, then divides GL L 2 q (m) = -m /2k + 2Z N > 0 , \L squarefree. elements of GT = Z/2kZ . Let's write k = kN In order to determine the isotropic we note that, for each positive integer 2k , there exists a unique subgroup H, < G d d 2 divides k <=> d that of order generated by [2k/d] .

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