Families of K3 surfaces polarised by a rank 18 lattice

  • 7 June 2023
  • 1500-1600
  • Sch 1.05

Joseph Prebble (Loughborough)

K3 surfaces, defined with trivial canonical bundle and zero first cohomology group, admit a polarisation by sublattices of the K3 lattice. Choosing a lattice M with a hyperbolic lattice H and two copies of a rank 8 lattice E8, the intrinsic nature of the M-polarisation yields a birational morphism of a surface whose image is a quartic hypersurface in three-dimensional projective space. By allowing these coefficients to vary we draw out, after a six-fold cover over the base, a family of these surfaces. Such a family is defined uniquely by two invariants: a functional invariant defining a map into the moduli space of the surfaces, and a homological invariant describing the action of monodromy around singular fibres. Families with singular or otherwise unusual central fibre can then classified by the exceptional curves in their blowups.

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