  We study computational aspects of the tight closure of a homogeneous primary
ideal in a two-dimensional normal standard-graded domain. We show how to use
slope criteria for the sheaf of syzygies for generators of the ideal to compute
the tight closure. Our method gives in particular an algorithm to compute the
tight closure of three elements under the condition that we are able to compute
the Harder-Narasimhan filtration. We apply this to the computation of
(x^a,y^a,z^a)^* in K[x,y,z]/(F), where F is a homogeneous polynomial.
