F-threshold functions: Syzygy gap fractals and the two-variable homogeneous case

In this article we study F-pure thresholds (and, more generally, F-thresholds) of homogeneous polynomials in two variables over a field of characteristic p>0. Passing to a field extension, we factor such a polynomial into a product of powers of pairwise prime linear forms, and to this collection of linear forms we associate a special type of function called a syzygy gap fractal. We use this syzygy gap fractal to study, at once, the collection of all F-pure thresholds of all polynomials constructed with the same fixed linear forms. This allows us to describe the structure of the denominator of such an F-pure threshold, showing in particular that whenever the F-pure threshold differs from its expected value its denominator is a multiple of p. This answers a question of Schwede in the two-variable homogeneous case. In addition, our methods give an algorithm to compute F-pure thresholds of homogenous polynomials in two variables.