Integer Angles of Polygons and Global Relations for Toric Singularities

Abstract

This chapter is dedicated to one important application of continued fractions to algebraic geometry, to complex projective toric surfaces. Toric surfaces are described in terms of convex polygons, and their singularities are in a straightforward correspondence with continued fractions. So it is really important to know relations between continued fractions of integer angles in a convex polygon. In Chap. 6 we proved a necessary and sufficient criterion for a triple of integer angles to be the angles of some integer triangle. In this chapter we prove the analogous statement for the integer angles of convex polygons. After a brief introduction of the main notions and definitions of complex projective toric surfaces, we discuss two problems related to singular points of toric varieties using integer geometry techniques. As an output one has global relations on toric singularities for toric surfaces.