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Four Lectures on Secant Varieties

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Connections Between Algebra, Combinatorics, and Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 76))

Abstract

This paper is based on the first author’s lectures at the 2012 University of Regina Workshop “Connections Between Algebra and Geometry.” Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.

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Notes

  1. 1.

    Some authors use the notation S(X) for the (first) secant variety of X, which corresponds to σ 2(X), and S k (X) to denote the kth secant variety to X, which corresponds to σ k+1(X). We prefer to reference the number of points used rather than the dimension of their span because this is often more relevant for applications because of its connection to rank.

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Acknowledgments

The second and third authors would like to thank the first author for his lectures at the workshop “Connections Between Algebra and Geometry” held at the University of Regina in 2012. The second author would like to thank the third author for tutoring these lectures and for helping to write solutions to the exercises. All three authors would like to thank the organizers S. Cooper, S. Sather-Wagstaff and D. Stanley for their efforts in organizing the workshop and for securing funding to cover the costs of the participants. It is also our pleasure to acknowledge the lecture notes of Tony Geramita [99] which have influenced our exposition here. The three authors received partial support by different sources: Enrico Carlini by GNSAGA of INDAM, Nathan Grieve by an Ontario Graduate Fellowship, and Luke Oeding by NSF RTG Award # DMS-0943745. Finally, all the authors thank the anonymous referee for the improvement to the paper produced by the referee’s suggestions and remarks.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Politecnico di Torino, 10129, Turin, Italy

    Enrico Carlini

  2. School of Mathematical Sciences, Monash University, Melbourne, VIC, 3800, Australia

    Enrico Carlini

  3. Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada

    Nathan Grieve

  4. Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada

    Nathan Grieve

  5. Department of Mathematics, University of California, Berkeley, Berkeley, CA, USA

    Luke Oeding

  6. Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA

    Luke Oeding

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  1. Enrico Carlini
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Correspondence to Enrico Carlini .

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  1. Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan, USA

    Susan M. Cooper

  2. Department of Mathematics, North Dakota State University, Fargo, North Dakota, USA

    Sean Sather-Wagstaff

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Carlini, E., Grieve, N., Oeding, L. (2014). Four Lectures on Secant Varieties. In: Cooper, S., Sather-Wagstaff, S. (eds) Connections Between Algebra, Combinatorics, and Geometry. Springer Proceedings in Mathematics & Statistics, vol 76. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0626-0_2

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