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Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers

Steven Kleiman·Bernd Ulrich
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Abstract

Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade 2 as those with a Hilbert--Burch matrix that has a maximal symmetric subblock. We also prove that every Gorenstein perfect algebra of grade 1 can be presented, as a module, by a symmetric matrix. Both results are derived from the same elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor, a symmetric syzygy matrix. In addition, we establish a correspondence, roughly speaking, between Gorenstein perfect algebras of grade 1 that are birational onto their image, on the one hand, and self-linked perfect ideals of grade 2 that have one of the self-linking elements contained in the second symbolic power, on the other hand. Finally, we provide another characterization of these ideals in terms of their symbolic Rees algebras, and we prove a criterion for these algebras to be normal.

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Steven Kleiman, Bernd Ulrich. 1995-09-10. Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers. https://arxiv.org/abs/alg-geom/9509005

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