Nonnegativity certificates on real algebraic surfaces

We introduce tools for transferring nonnegativity certificates for global sections between line bundles on real algebraic surfaces. As applications, we improve Hilbert's degree bounds on sum-of-squares multipliers for nonnegative ternary forms, give a complete characterization of nonnegative real forms of del Pezzo surfaces, and establish quadratic upper bounds for the degrees of sum-of-squares multipliers for nonnegative forms on real ruled surfaces.