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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Weakly defective varieties
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by L. Chiantini and C. Ciliberto PDF
Trans. Amer. Math. Soc. 354 (2002), 151-178 Request permission

Abstract:

A projective variety $X$ is ‘$k$-weakly defective’ when its intersection with a general $(k+1)$-tangent hyperplane has no isolated singularities at the $k+1$ points of tangency. If $X$ is $k$-defective, i.e. if the $k$-secant variety of $X$ has dimension smaller than expected, then $X$ is also $k$-weakly defective. The converse does not hold in general. A classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension. We start this classification here, describing all weakly defective irreducible surfaces. Our method also provides a new proof of the classical Terracini’s classification of $k$-defective surfaces.
References
  • Bjørn Ådlandsvik, Joins and higher secant varieties, Math. Scand. 61 (1987), no. 2, 213–222. MR 947474, DOI 10.7146/math.scand.a-12200
  • Enrico Arbarello and Maurizio Cornalba, Footnotes to a paper of Beniamino Segre: “On the modules of polygonal curves and on a complement to the Riemann existence theorem” (Italian) [Math. Ann. 100 (1928), 537–551; Jbuch 54, 685], Math. Ann. 256 (1981), no. 3, 341–362. MR 626954, DOI 10.1007/BF01679702
  • Bronowski J., Surfaces whose prime sections are hyperelliptic, J. London Math. Soc. 8 (1933), 308-312.
  • Michael L. Catalano-Johnson, The possible dimensions of the higher secant varieties, Amer. J. Math. 118 (1996), no. 2, 355–361. MR 1385282, DOI 10.1353/ajm.1996.0012
  • Michael Catalano-Johnson, When do $k$ general double points impose independent conditions on degree $d$ plane curves, The Curves Seminar at Queen’s, Vol. X (Kingston, ON, 1995) Queen’s Papers in Pure and Appl. Math., vol. 102, Queen’s Univ., Kingston, ON, 1996, pp. 166–181. MR 1381737
  • Luca Chiantini and Ciro Ciliberto, A few remarks on the lifting problem, Astérisque 218 (1993), 95–109. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). MR 1265310
  • Ciro Ciliberto and André Hirschowitz, Hypercubiques de $\textbf {P}^4$ avec sept points singuliers génériques, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 3, 135–137 (French, with English summary). MR 1121575
  • Ciro Ciliberto, Angelo Felice Lopez, and Rick Miranda, Some remarks on the obstructedness of cones over curves of low genus, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 167–182. MR 1463178
  • Ciro Ciliberto, Hilbert functions of finite sets of points and the genus of a curve in a projective space, Space curves (Rocca di Papa, 1985) Lecture Notes in Math., vol. 1266, Springer, Berlin, 1987, pp. 24–73. MR 908707, DOI 10.1007/BFb0078177
  • Ciro Ciliberto and Edoardo Sernesi, Singularities of the theta divisor and congruences of planes, J. Algebraic Geom. 1 (1992), no. 2, 231–250. MR 1144438
  • M. Dale, Terracini’s lemma and the secant variety of a curve, Proc. London Math. Soc. (3) 49 (1984), no. 2, 329–339. MR 748993, DOI 10.1112/plms/s3-49.2.329
  • Dale M., On the secant variety of an algebrac surface, University of Bergen, Dept. of Math. preprint no. 33 (1984).
  • Di Gennaro V., Self intersection of the canonical bundle of a projective variety, to appear in Comm. in Alg..
  • Joe Harris, Curves in projective space, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 85, Presses de l’Université de Montréal, Montreal, Que., 1982. With the collaboration of David Eisenbud. MR 685427
  • Federigo Enriques and Oscar Chisini, Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche. 1. Vol. I, II, Collana di Matematica [Mathematics Collection], vol. 5, Zanichelli Editore S.p.A., Bologna, 1985 (Italian). Reprint of the 1915 and 1918 editions. MR 966664
  • Barbara Fantechi, On the superadditivity of secant defects, Bull. Soc. Math. France 118 (1990), no. 1, 85–100 (English, with French summary). MR 1077089, DOI 10.24033/bsmf.2137
  • Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
  • Phillip Griffiths and Joseph Harris, Algebraic geometry and local differential geometry, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 3, 355–452. MR 559347, DOI 10.24033/asens.1370
  • Joe Harris, A bound on the geometric genus of projective varieties, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 35–68. MR 616900
  • T. Matsusaka, On a theorem of Torelli, Amer. J. Math. 80 (1958), 784–800. MR 97398, DOI 10.2307/2372783
  • Palatini F., Sulle varietà algebriche per le quali sono di dimensione minore dell’ordinario, senza riempire lo spazio ambiente, una o alcune delle varietà formate da spazi seganti, Atti. Accad. Torino 44 (1909), 362-374.
  • Palatini F., Sulle superficie algebriche i cui $S_{h}$ $(h+1)$-seganti non riempiono lo spazio ambiente, Atti. Accad. Torino 41 (1906), 634-640.
  • Scorza G., Determinazione delle varietá a tre dimensioni di $S-r$, $r\ge 7$, i cui $S_{3}$ tangenti si tagliano a due a due., Rend. Circ. Mat. Palermo 25 (1908), 193-204.
  • Scorza G., Un problema sui sistemi lineari di curve appartenenti a una superficie algebrica, Rend. R. Ist. Lombardo (2) 41 (1908), 913-920.
  • Segre C., Preliminari di una teoria delle varietá luoghi di spazi, Rend. Circ. Mat. Palermo 30 (1910), 87-121.
  • Terracini A., Sulle $V_{k}$ per cui la varietà degli $S_{h}$ $(h+1)$-seganti ha dimensione minore dell’ ordinario, Rend. Circ. Mat. Palermo 31 (1911), 392-396.
  • Terracini A., Su due problemi, concernenti la determinazione di alcune classi di superficie, considerati da G. Scorza e F. Palatini, Atti Soc. Natur. e Matem. Modena (V) 6 (1921-22), 3-16.
  • F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127, American Mathematical Society, Providence, RI, 1993. Translated from the Russian manuscript by the author. MR 1234494, DOI 10.1090/mmono/127
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Additional Information
  • L. Chiantini
  • Affiliation: Department of Mathematics, University of Siena, Via del Capitano 15, 53100 Siena, Italy
  • MR Author ID: 194958
  • ORCID: 0000-0001-5776-1335
  • Email: chiantini@unisi.it
  • C. Ciliberto
  • Affiliation: Department of Mathematics, University of Rome II, Viale della Ricerca Scientifica, 16132 Rome, Italy
  • MR Author ID: 49480
  • Email: cilibert@axp.mat.uniroma2.it
  • Received by editor(s): March 1, 2000
  • Published electronically: July 13, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 151-178
  • MSC (2000): Primary 14E25
  • DOI: https://doi.org/10.1090/S0002-9947-01-02810-0
  • MathSciNet review: 1859030