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

HTML articles powered by AMS MathViewer

- by Luca Chiantini and Juan Migliore PDF
- Trans. Amer. Math. Soc.
**374**(2021), 2581-2607 Request permission

## 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.

## References

- J. Abbott, A. M. Bigatti, and L. Robbiano,
*CoCoA: A system for doing Computations in Commutative Algebra*, available at http://cocoa.dima.unige.it. - E. Ballico and Ph. Ellia,
*The maximal rank conjecture for nonspecial curves in $\textbf {P}^3$*, Invent. Math.**79**(1985), no. 3, 541–555. MR**782234**, DOI 10.1007/BF01388522 - Thomas Bauer, Sandra Di Rocco, David Schmitz, Tomasz Szemberg, and Justyna Szpond,
*On the postulation of lines and a fat line*, J. Symbolic Comput.**91**(2019), 3–16. MR**3860881**, DOI 10.1016/j.jsc.2018.06.010 - Thomas Bauer, Grzegorz Malara, Tomasz Szemberg, and Justyna Szpond,
*Quartic unexpected curves and surfaces*, Manuscripta Math.**161**(2020), no. 3-4, 283–292. MR**4060481**, DOI 10.1007/s00229-018-1091-3 - Anna Bigatti, Anthony V. Geramita, and Juan C. Migliore,
*Geometric consequences of extremal behavior in a theorem of Macaulay*, Trans. Amer. Math. Soc.**346**(1994), no. 1, 203–235. MR**1272673**, DOI 10.1090/S0002-9947-1994-1272673-7 - L. Chiantini and C. Ciliberto,
*Weakly defective varieties*, Trans. Amer. Math. Soc.**354**(2002), no. 1, 151–178. MR**1859030**, DOI 10.1090/S0002-9947-01-02810-0 - Ciro Ciliberto, Th. Dedieu, F. Flamini, R. Pardini, C. Galati, and S. Rollenske,
*Open problems*, Boll. Unione Mat. Ital.**11**(2018), no. 1, 5–11. MR**3782686**, DOI 10.1007/s40574-018-0158-0 - D. Cook II, B. Harbourne, J. Migliore, and U. Nagel,
*Line arrangements and configurations of points with an unexpected geometric property*, Compos. Math.**154**(2018), no. 10, 2150–2194. MR**3867298**, DOI 10.1112/s0010437x18007376 - Steven Diaz,
*Space curves that intersect often*, Pacific J. Math.**123**(1986), no. 2, 263–267. MR**840844** - Roberta Di Gennaro, Giovanna Ilardi, and Jean Vallès,
*Singular hypersurfaces characterizing the Lefschetz properties*, J. Lond. Math. Soc. (2)**89**(2014), no. 1, 194–212. MR**3174740**, DOI 10.1112/jlms/jdt053 - Michela Di Marca, Grzegorz Malara, and Alessandro Oneto,
*Unexpected curves arising from special line arrangements*, J. Algebraic Combin.**51**(2020), no. 2, 171–194. MR**4069339**, DOI 10.1007/s10801-019-00871-0 - David Eisenbud, Mark Green, and Joe Harris,
*Cayley-Bacharach theorems and conjectures*, Bull. Amer. Math. Soc. (N.S.)**33**(1996), no. 3, 295–324. MR**1376653**, DOI 10.1090/S0273-0979-96-00666-0 - Alessandro Gimigliano,
*ON LINEAR SYSTEMS OF PLANE CURVES*, ProQuest LLC, Ann Arbor, MI, 1987. Thesis (Ph.D.)–Queen’s University (Canada). MR**2635606** - Salvatore Giuffrida,
*On the intersection of two curves in $\textbf {P}^r$*, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.**122**(1988), no. 3-4, 139–143 (English, with Italian summary). MR**1013253** - Brian Harbourne,
*The geometry of rational surfaces and Hilbert functions of points in the plane*, Proceedings of the 1984 Vancouver conference in algebraic geometry, CMS Conf. Proc., vol. 6, Amer. Math. Soc., Providence, RI, 1986, pp. 95–111. MR**846019** - B. Harbourne, J. Migliore, U. Nagel, and Z. Teitler,
*Unexpected hypersurfaces and where to find them*, preprint arXiv:1805.10626 2018, to appear in Michigan J. Math. - B. Harbourne, J. Migliore, and H. Tutaj-Gasińska,
*New constructions of unexpected hypersurfaces in $\mathbb P^n$*, to appear in Revista Matemática Complutense. - Joe Harris,
*Algebraic geometry*, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR**1182558**, DOI 10.1007/978-1-4757-2189-8 - Robin Hartshorne and André Hirschowitz,
*Droites en position générale dans l’espace projectif*, Algebraic geometry (La Rábida, 1981) Lecture Notes in Math., vol. 961, Springer, Berlin, 1982, pp. 169–188 (French). MR**708333**, DOI 10.1007/BFb0071282 - André Hirschowitz,
*Une conjecture pour la cohomologie des diviseurs sur les surfaces rationnelles génériques*, J. Reine Angew. Math.**397**(1989), 208–213 (French). MR**993223**, DOI 10.1515/crll.1989.397.208 - Juan C. Migliore,
*Introduction to liaison theory and deficiency modules*, Progress in Mathematics, vol. 165, Birkhäuser Boston, Inc., Boston, MA, 1998. MR**1712469**, DOI 10.1007/978-1-4612-1794-7 - Beniamino Segre,
*Alcune questioni su insiemi finiti di punti in geometria algebrica.*, Atti Convegno Internaz. Geometria Algebrica (Torino, 1961) Rattero, Turin, 1962, pp. 15–33 (Italian). MR**0146714** - J. Szpond,
*Unexpected hypersurfaces with multiple fat points*, preprint 2019, to appear in J. Symb. Comp.

## 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
- 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.
- © Copyright 2021 American Mathematical Society
- 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
- MathSciNet review: 4223027

Dedicated: With an appendix by A. Bernardi, L. Chiantini, G. Dedham, G. Favacchio, B. Harbourne, J. Migliore, T. Szemberg, and J. Szpond