2021-04-28_Mock_AMS_2020_Ideas

Talks Subjects/Algebraic Geometry

Ideas for Spaces

  • Curves
    • elliptic curve
    • Higher genus
    • hyperelliptic curves
    • ../modular curve
  • Surfaces
    • Compact Riemann surface
      • Bolza Surface (Genus 2) Bolza surface
      • Klein Quartic (Genus 3) Klein quartic
      • Hurwizt Surfaces Hurwitz surface
    • Kummer surfaces Kummer surface
    • Unsorted/K3 surfaces
  • Compact Complex Surfaces
    • Rational ruled ruled surfaces
    • Enriques Surfaces Enriques surface
    • \(K3\)
      • ../Kähler
    • Kodaira manifold
    • Toric variety
    • Hyperelliptic
    • Properly quasi-elliptic quasi-elliptic surface
    • General type
    • Type VII
  • Fake projective planes
  • Conic
  • Hurwitz schemes
  • Topological Galois groups, e.g. \(G(\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu /F )\) for \(F = {\mathbf{Q}}, { \mathbf{F} }_p\).
  • \(\operatorname{Spec}(R)\) for \(R\) a ../DVR (a Sierpinski space)
  • Quiver Grassmannians
  • Rigid analytic spaces
  • Affine line with two origins
  • ../moduli stack of elliptic curves
  • ../moduli stack of abelian varieties
  • Fano variety
  • Fano Varieties
  • Curves: isomorphic to \({\mathbf{P}}^1\)
  • Surfaces: del Pezzo surface
  • weighted projective space
  • ../Grassmannian
  • flag variety
  • moduli

Due to Kunihiko Kodaira’s classification of complex surfaces, we know that any compact hyperkähler 4-manifold is either a K3 surface or a compact torus T^{4}. (Every Calabi-Yau manifold in 4 (real) dimensions is a hyperkähler manifold, because SU(2) is isomorphic to Sp(1).)

As was discovered by Beauville, the Hilbert scheme of k points on a compact hyperkähler 4-manifold is a hyperkähler manifold of dimension 4k. This gives rise to two series of compact examples: Hilbert schemes of points on a K3 surface and generalized Kummer varieties.