Combinatorial symbolic powers

By Seth Sullivant. 2006

70 citations recorded by the source corpus.

Abstract Symbolic powers are studied in the combinatorial context of monomial ideals. When the ideals are generated by quadratic squarefree monomials, the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blowups. As a result, perfect graphs play an important role in the theory, dual to the role played by perfect graphs in the theory of secants of monomial ideals. We use Grobner degenerations as a tool to reduce questions about symbolic powers of arbitrary ideals to the monomial case. Among the applications are a new, unified approach to the Grobner bases of symbolic powers of determinantal and Pfaffian ideals.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/396cc7e44b4572a201463ef4c25391a09817e937

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In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Quantifying singularities with differential operators (2018, methodology): “ows from the characterization of diﬀerential powers in this case [DDSG+18]. The formula in Proposition 10.6 is similar in spirit to the characterization of symbolic powers in terms of joins of ideals [Sul08]. The key advantage of the formula above is that the computation involves only twice (not thr”
Resolutions of square-free monomial ideals via facet ideals: a survey (2006, background): “To learn more about the properties of edge ideals, one should see [1, 2, 6, 16, 17, 25, 26, 27, 36, 38, 39, 40, 41, 44, 46]; further properties of facet ideals can be found in the sequels [12, 13, 14] to Faridi’s paper cited above.”