  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.
