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Relative geometric invariant theory Alexander Schmitt Freie Universit at Berlin MS Seminar (Mathematics - String Theory), July 16, 2019

Relative geometric invariant theory · geometric invariant theory Alexander Schmitt Surfaces in threedimen-sional projective space Semistable cubic surfaces Theorem (Hilbert 1893)

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Page 1: Relative geometric invariant theory · geometric invariant theory Alexander Schmitt Surfaces in threedimen-sional projective space Semistable cubic surfaces Theorem (Hilbert 1893)

Relative geometric invariant theory

Alexander SchmittFreie Universitat Berlin

MS Seminar (Mathematics - String Theory), July 16, 2019

Page 2: Relative geometric invariant theory · geometric invariant theory Alexander Schmitt Surfaces in threedimen-sional projective space Semistable cubic surfaces Theorem (Hilbert 1893)

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)

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

Page 4: Relative geometric invariant theory · geometric invariant theory Alexander Schmitt Surfaces in threedimen-sional projective space Semistable cubic surfaces Theorem (Hilbert 1893)

Relativegeometricinvarianttheory

AlexanderSchmitt

Surfaces inthreedimen-sionalprojectivespace

The most singular semistable surface

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