Combinatorial secant varieties

By Bernd Sturmfels, Seth Sullivant. 2005

91 citations recorded by the source corpus.

The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal of the secant ideal coincides with the secant ideal of the initial ideal. For toric varieties, this leads to the notion of delightful triangulations of convex polytopes.

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

View Paper: https://www.semanticscholar.org/paper/911a73425e1306378511835298294ef910b98091

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Citations in the corpus

An Invariant Regarding Waring’s Problem for Cubic Polynomials (2007, background): “We are indebted to S. Sullivant, for his beautiful lectures at Nordfjordeid in 2006 about [ SS ], where a representation of the Aronhold invariant is found with combinatorial techniques.”
A Beginner’s Guide to Edge and Cover Ideals (2013, background): “It relies on generalized Alexander duality, polarization and depolarization of monomial ideals, and a result of Sturmfels and Sullivant [60].”
Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals (2009, methodology): “By Lemma 3.1, we know that”
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.”
A tropical approach to secant dimensions (2006, unknown): “A G ] 5 S ep 2”