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.