Introduction to non-Abelian Patchworking

TL;DR

This paper introduces non-Abelian patchworking, a new geometric framework for constructing real algebraic surfaces in projective 3-space, expanding the understanding of tropical geometry for certain complex groups and reproducing all isotopy types up to degree three.

Contribution

It develops a novel non-Abelian patchworking method based on tropical geometry, providing explicit geometric input and analyzing the topology of primitive PGL_2 surfaces.

Findings

Primitive PGL_2 surfaces can have different Euler characteristics for the same degree.

The framework reproduces all isotopy types of surfaces up to degree three.

It offers a less combinatorial, more geometric approach compared to traditional methods.

Abstract

The note introduces a novel concept of non-Abelian patchworking arising as real locus of non-Abelian complex-phase tropical hypersurfaces, the theory of which is now developed enough to allow the proposed spin-off. Although, non-Abelian Tropical Geometry makes sense for an arbitrary reductive complex group, the state of the art is that of full understanding of tropicalizations of surfaces within three dimensional groups $PGL_2(\mathbb{C})$ and $SL_2(\mathbb{C}),$ which are closely related via the two-fold covering. We stress our point, that this is an announcement of a framework, taking care of explaining explicitly the input, which is more geometric and less combinatorial than in the original Viro's method, to construct possible types of real algebraic surfaces in the real projective 3-space, and verify that it reproduces all the existing isotopy types of surfaces up to degree three. We obtain two general theorems concerning the topology of primitive PGL2 surfaces, observing in particular that they may have different Euler charteristic for a fixed degree greater than one, not necessarily equal to the signature of the corresponding complex surface, which would be the case for primitive combinatorial patchworking due to a result of Itenberg.