scholarly journals K3 polytopes and their quartic surfaces

2021 ◽  
Vol 21 (1) ◽  
pp. 85-98
Author(s):  
Gabriele Balletti ◽  
Marta Panizzut ◽  
Bernd Sturmfels
Keyword(s):  
Invariant Theory ◽  
Geometric Invariant Theory ◽  
Geometric Invariant ◽  
Combinatorial Objects ◽  
Reflexive Polytopes ◽  
Singular Loci ◽  
Quartic Surfaces

Abstract K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we classify K3 polytopes with up to 30 vertices. Their number is 36 297 333. We study the singular loci of quartic surfaces that tropicalize to K3 polytopes. These surfaces are stable in the sense of Geometric Invariant Theory.

scholarly journals Tits Buildings and K-Stability

2019 ◽  
Vol 62 (3) ◽  
pp. 799-815 ◽  
Author(s):  
Giulio Codogni
Keyword(s):  
Continuous Function ◽  
Invariant Theory ◽  
Geometric Invariant Theory ◽  
Geometric Invariant ◽  
Test Configuration ◽  
Direct System ◽  
Cauchy Sequences

AbstractA polarized variety is K-stable if, for any test configuration, the Donaldson–Futaki invariant is positive. In this paper, inspired by classical geometric invariant theory, we describe the space of test configurations as a limit of a direct system of Tits buildings. We show that the Donaldson–Futaki invariant, conveniently normalized, is a continuous function on this space. We also introduce a pseudo-metric on the space of test configurations. Recall that K-stability can be enhanced by requiring that the Donaldson–Futaki invariant is positive on any admissible filtration of the co-ordinate ring. We show that admissible filtrations give rise to Cauchy sequences of test configurations with respect to the above mentioned pseudo-metric.

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