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Algebraic Geometry, Commutative Algebra and Combinatorics: Interactions and Open Problems

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Abstract

We survey recent research bridging algebraic geometry, commutative algebra and combinatorics, highlighting open questions, problems and conjectures of current interest. For expositional simplicity and focus, the emphasis is on homogeneous ideals of fat point subschemes of projective space. The survey is structured into three parts, each part starting with a question, conjecture or result of David Eisenbud. The first part is on the computability of semi-effectivity, the second part is on the containment problem of symbolic powers of ideals of fat points in their ordinary powers, and the third part is on splitting types of rank 2 bundles on rational curves.

Dedicated to David Eisenbud, on the occasion of his 75th birthday.

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Acknowledgements

The author was partially supported by Simons Foundation grant #524858. The author thanks everyone who looked over an early draft of this survey, and in particular M.-G. Ascenzi, S. Cooper, A. Dimca, E. Guardo, T. Ha, J. Migliore, J. Roé, A. Seceleanu, T. Szemberg, A. Van Tuyl and Y. Xie for helpful comments.

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  1. Department of Mathematics, University of Nebraska, Lincoln, NE, USA

    B. Harbourne

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  1. Department of Mathematics, Cornell University, Ithaca, NY, USA

    Irena Peeva

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Harbourne, B. (2021). Algebraic Geometry, Commutative Algebra and Combinatorics: Interactions and Open Problems. In: Peeva, I. (eds) Commutative Algebra. Springer, Cham. https://doi.org/10.1007/978-3-030-89694-2_14

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