relative geometric invariant theory · geometric invariant theory alexander schmitt surfaces in...
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Relative geometric invariant theory
Alexander SchmittFreie Universitat Berlin
MS Seminar (Mathematics - String Theory), July 16, 2019
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Relativegeometricinvarianttheory
AlexanderSchmitt
Surfaces inthreedimen-sionalprojectivespace
Kummer’s quartic
![Page 3: Relative geometric invariant theory · geometric invariant theory Alexander Schmitt Surfaces in threedimen-sional projective space Semistable cubic surfaces Theorem (Hilbert 1893)](https://reader034.vdocuments.us/reader034/viewer/2022042912/5f485ab89debb312842ce6e1/html5/thumbnails/3.jpg)
Relativegeometricinvarianttheory
AlexanderSchmitt
Surfaces inthreedimen-sionalprojectivespace
Semistable cubic surfaces
Theorem (Hilbert 1893)
A cubic form f ∈ V3,3 \ {0} is semistable if and only ifV (f ) ⊂ P3 is smooth or has only isolated singuarities of typeA1 (conical nodes) and A2 (binodes).
Double points of surfaces
Conical node binode uninode
A1 Ak , k ≥ 2 Dk , k ≥ 4, E6, E7, E8
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Relativegeometricinvarianttheory
AlexanderSchmitt
Surfaces inthreedimen-sionalprojectivespace
The most singular semistable surface