---
date: 2021-04-28 17:14:31
created: 2021-10-27T19:35
updated: 2024-01-10T12:17
---

[Talks](../000_Talks%20Index.md)
[Subjects/Algebraic Geometry](Obsidian/MOCs/Algebraic%20Geometry.md)

# Ideas for Spaces

- Curves
  - [elliptic curve](elliptic%20curve.md)
  - Higher [genus](genus)
  - [hyperelliptic](hyperelliptic) curves
  - [../modular curve](../modular%20curve.md)
- Surfaces
  - Compact [Riemann surface](../Riemann%20surface.md)
    - Bolza Surface (Genus 2) [Bolza surface](Bolza%20surface)
    - Klein Quartic (Genus 3) [Klein quartic](Klein%20quartic)
    - Hurwizt Surfaces [Hurwitz surface](Hurwitz%20surface)
  - Kummer surfaces [Kummer surface](Kummer%20surface)
  - [K3 surfaces](K3%20surfaces.md)
- Compact Complex Surfaces
  - Rational ruled [ruled surfaces](ruled%20surfaces)
  - Enriques Surfaces [Enriques surface](Enriques%20surface.md)
  - $K3$
    - [../Kähler](../Kähler.md)
  - [Kodaira manifold](Kodaira%20manifold)
  - [Toric variety](Toric%20variety)
  - Hyperelliptic
  - Properly quasi-elliptic [quasi-elliptic surface](quasi-elliptic%20surface)
  - General type
  - Type VII
- Fake projective planes 
- Conic
- Hurwitz schemes
- Topological Galois groups, e.g. $G(\bar F /F )$ for $F = \QQ, \FF_p$.
- $\spec (R)$ for $R$ a [../DVR](../DVR.md) (a Sierpinski space)
- Quiver Grassmannians
- Rigid analytic spaces
- Affine line with two origins
- [../moduli stack of elliptic curves](../moduli%20stack%20of%20elliptic%20curves.md)
- [../moduli stack of abelian varieties](../moduli%20stack%20of%20abelian%20varieties.md)
- [Fano variety](Fano%20variety)
- Fano Varieties
 - Curves: isomorphic to $\PP^1$
 - Surfaces: [del Pezzo surface](del%20Pezzo%20surface)
- [weighted projective space](weighted%20projective%20space)
- [../Grassmannian](../Grassmannian.md)
- [flag variety](flag%20variety)
- [moduli](moduli%20space.md)


> 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.