Positive Ulrich sheaves
Canadian journal of mathematics, Tome 76 (2024) no. 3, pp. 881-914
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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.
Mots-clés :
Ulrich sheaf, hyperbolic polynomial, sums of squares, determinantal representations
Hanselka, Christoph; Kummer, Mario. Positive Ulrich sheaves. Canadian journal of mathematics, Tome 76 (2024) no. 3, pp. 881-914. doi: 10.4153/S0008414X23000263
@article{10_4153_S0008414X23000263,
author = {Hanselka, Christoph and Kummer, Mario},
title = {Positive {Ulrich} sheaves},
journal = {Canadian journal of mathematics},
pages = {881--914},
year = {2024},
volume = {76},
number = {3},
doi = {10.4153/S0008414X23000263},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X23000263/}
}
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