Talks Subjects/Algebraic Geometry
Ideas for Spaces
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Curves
- elliptic curve
- Higher genus
- hyperelliptic curves
- ../modular curve
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Surfaces
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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
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Compact Riemann surface
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Compact Complex Surfaces
- Rational ruled ruled surfaces
- Enriques Surfaces Enriques surface
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\(K3\)
- ../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.