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Transactions of the American Mathematical Society

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Sets of points which project to complete intersections, and unexpected cones


Authors: Luca Chiantini and Juan Migliore
Journal: Trans. Amer. Math. Soc. 374 (2021), 2581-2607
MSC (2020): Primary 14M10; Secondary 14N20, 14N05, 14M07
DOI: https://doi.org/10.1090/tran/8290
Published electronically: January 20, 2021
MathSciNet review: 4223027
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Abstract: The paper is devoted to the description of those non-degenerate sets of points $Z$ in $\mathbb P^3$ whose general projection to a general plane is a complete intersection of curves in that plane. One large class of such $Z$ is what we call $(a,b)$-grids. We relate this problem to the unexpected cone property $\mathcal {C}(d)$, a special case of the unexpected hypersurfaces which have been the focus of much recent research. After an analysis of $\mathcal {C}(d)$ for small $d$, we show that a non-degenerate set of $9$ points has a general projection that is the complete intersection of two cubics if and only if the points form a $(3,3)$-grid. However, in an appendix we describe a set of $24$ points that are not a grid but nevertheless have the projection property. These points arise from the $F_4$ root system. Furthermore, from this example we find subsets of $20$, $16$ and $12$ points with the same feature.


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Additional Information

Luca Chiantini
Affiliation: Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, 53100 Siena, Italy
MR Author ID: 194958
ORCID: 0000-0001-5776-1335
Email: luca.chiantini@unisi.it

Juan Migliore
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556; Dipartimento di Matematica, Università di Trento, 38123 Povo (TN), Italy; Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, 53100 Siena, Italy; Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada; DISMA-Department of Mathematical Sciences, Politecnico di Torino, 10129 Torino, Italy; Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0130; Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556; Department of Mathematics, Pedagogical University of Cracow, PL-30-084 Krakow, Poland; Institute of Mathematics, Polish Academy of Sciences, PL-00-656 Warszawa, Poland
MR Author ID: 124490
ORCID: 0000-0001-5528-4520
Email: migliore.1@nd.edu

Keywords: Cones, complete intersections, projections, special linear systems, unexpected varieties, base loci.
Received by editor(s): November 7, 2019
Received by editor(s) in revised form: June 3, 2020
Published electronically: January 20, 2021
Additional Notes: The first author was partially supported by the Italian INdAM-GNSAGA. The second author was partially supported by Simons Foundation grant #309556.
Dedicated: With an appendix by A. Bernardi, L. Chiantini, G. Dedham, G. Favacchio, B. Harbourne, J. Migliore, T. Szemberg, and J. Szpond
Article copyright: © Copyright 2021 American Mathematical Society