Abstract: F-signature plays a crucial role when measuring singularities of varieties in positive characteristics. For example, if R is a local ring, s(R) = 1 implies R is regular, and 0 < s(R) < 1 implies R is strongly F-regular which is a char p analog of klt singularities. For a globally F-regular variety X, the F-signature function of an ample invertible sheaf is defined in an analogous way to the F-signature of the section ring along the invertible sheaf over X. In this talk, we will discuss how the F-signature of pairs compares along proper birational maps and the positivity of the F-signature function on big divisors. This is joint work with Suchitra Pande.