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

Chiantini, Luca; Migliore, Juan

doi:10.1090/tran/8290

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