Polynomials with integral coefficients, equivalent to a given polynomial (after Ja. Kollar)

Consider the following problem: given a homogeneous polynomial $f(x_0,\ldots,x_n)$ with rational coefficients, construct a polynomial $F(x_0,\ldots,x_n)$ with integral coefficients which is “equivalent” to f and as “simple” as possible. The exact definitions and the solution to this problems are given in terms of stability conditions coming from Geometric invariant theory. As an application we discuss the good models of the cubic surface fibrations.