Positive Ulrich sheaves

Abstract We provide a criterion for coherent sheaf to be an Ulrich in terms of certain bilinear form on its global sections. When working over the real numbers, we call it positive if this is symmetric or Hermitian and positive-definite. In that case, our result provides common theoretical framework several results algebraic geometry concerning existence certificates geometric properties. For instance, implies Hilbert’s theorem nonnegative ternary quartics, via del Pezzo surfaces, solution Lax conjecture plane hyperbolic curves due Helton Vinnikov.