Positive Ulrich Sheaves

TL;DR

This paper introduces a criterion for identifying Ulrich sheaves using bilinear forms, and extends the concept to positive Ulrich sheaves over real numbers, linking algebraic geometry with real algebraic certificates.

Contribution

It provides a unified theoretical framework for positive Ulrich sheaves, connecting algebraic geometry with real algebraic geometry and classical theorems.

Findings

Criterion for Ulrich sheaves via bilinear forms

Extension to positive Ulrich sheaves over real numbers

Applications to Hilbert's theorem and Lax conjecture

Abstract

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