Unexpected hypersurfaces and where to find them

B. Harbourne, J. Migliore, U. Nagel, Z. Teitler

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


In the paper [7] by Cook et al., which introduced the concept of unexpected plane curves, the focus was on understanding the geometry of the curves themselves. Here, we expand the definition to hypersurfaces of any dimension and, using constructions which appeal to algebra, geometry, representation theory, and computation, we obtain a coarse but complete classification of unexpected hypersurfaces. In particular, we determine each (n, d,m) for which there is some finite set of points Z Pn with an unexpected hypersurface of degree d in Pn having a general point P of multiplicity m. Our constructions also give new insight into the interesting question of where to look for such Z. Recent work of Di Marca, Malara, and Oneto [10] and of Bauer, Malara, Szemberg, and Szpond [5] gives new results and examples in P2 and P3. We obtain our main results using a new construction of unexpected hypersurfaces involving cones. This method applies in Pn for n ≥ 3 and gives a broad range of examples, which we link to certain failures of the Weak Lefschetz Property. We also give constructions using root systems, both in P2 and Pn for n ≥ 3. Finally, we explain an observation of [5], showing that the unexpected curves of [7] are in some sense dual to their tangent cones at their singular point.

Original languageEnglish
Pages (from-to)301-339
Number of pages39
JournalMichigan Mathematical Journal
Issue number2
StatePublished - May 2021

Bibliographical note

Funding Information:
ACKNOWLEDGMENTS. Harbourne was partially supported by Simons Foundation grant #524858. Migliore was partially supported by Simons Foundation grant #309556. Nagel was partially supported by Simons Foundation grant #317096. Teitler was partially supported by Simons Foundation grant #354574. We thank M. Dyer for suggesting that we look at root systems and for bringing to our attention the H3 and H4 root systems. We thank Tomasz Szemberg and Justyna Szpond for helpful comments. And we thank Boise State University, where some of the work on this paper was done.

Publisher Copyright:
© 2021 University of Michigan. All rights reserved.

ASJC Scopus subject areas

  • Mathematics (all)


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