Kanev and Todorov surfaces in toric 3-folds

TL;DR

This paper demonstrates explicit construction of Kanev and Todorov surfaces within toric 3-folds using toric methods, analyzing their properties and singularities through polytope-based techniques.

Contribution

It provides explicit toric constructions of Kanev and Todorov surfaces, linking their geometric properties to polytope data and computing singularities of their canonical models.

Findings

Constructed minimal and canonical models explicitly using toric methods.

Linked surface properties to polytope characteristics.

Computed singularities of the canonical models.

Abstract

In the first part of this article we show for some examples of surfaces of general type in toric 3-folds how to construct minimal and canonical models by toric methods explicitly. The examples we study turn out to be surfaces of general type, namely so called Kanev surfaces and Todorov surfaces. We show how properties of our examples of surfaces could be derived directly from properties of some polytopes and we compute the singularities of their canonical models.