Overview
The research focuses on the computation of the stability degree for specific algebraic geometric objects: curves of genus $g$ and semi-abelian varieties. For curves of genus $g$, the stability degree is determined and found to align with the classical Minkowski bound. Additionally, the stability degree for semi-abelian varieties is calculated, taking into account their prescribed toric and abelian ranks.
Research Context
The study operates within the mathematical framework of degenerating families and finite monodromy groups. It addresses the stability degree, a specific invariant in algebraic geometry. The explicit reference to the 'classical Minkowski bound' suggests a connection to established results within number theory or geometry.
Approach
The methodology employed for these computations relies on a two-pronged strategy. A central component is a specialization result concerning finite monodromy groups. This result is applied within the context of degenerating families. The derivation of this specialization result is attributed to a variant of Krasner's lemma. Furthermore, the proof incorporates an explicit construction specifically at the prime 2.
Findings
- The stability degree of curves of genus $g$ has been computed.
- This computed stability degree for curves of genus $g$ is equal to the classical Minkowski bound.
- The stability degree of semi-abelian varieties, characterized by their prescribed toric and abelian ranks, has also been computed.