论文标题
阳性乌尔里希绳束
Positive Ulrich Sheaves
论文作者
论文摘要
我们为连贯的捆绑提供标准，使其成为其全球部分的某种双线性形式的乌尔里希捆。在实际数字上工作时，如果这种双线性形式是对称的或遗传学的，并且确定为正面，我们将其称为正面的Ulrich捆。在那种情况下，我们的结果为几种实际代数几何形状的几个结果提供了一个共同的理论框架，该几何形状是关于某些几何特性的代数证书的。例如，它暗示了希尔伯特（Hilbert）关于非负三元四重奏的定理，通过del pezzo表面的几何形状以及由于Helton和Vinnikov引起的平面双曲线曲线上的LAX猜想的解决方案。
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.
