On the secant varieties to the tangential varieties of a Veronesean
HTML articles powered by AMS MathViewer
- by M. V. Catalisano, A. V. Geramita and A. Gimigliano
- Proc. Amer. Math. Soc. 130 (2002), 975-985
- DOI: https://doi.org/10.1090/S0002-9939-01-06251-7
- Published electronically: October 12, 2001
- PDF | Request permission
Abstract:
We study the dimensions of the higher secant varieties to the tangent varieties of Veronese varieties. Our approach, generalizing that of Terracini, concerns 0-dimensional schemes which are the union of second infinitesimal neighbourhoods of generic points, each intersected with a generic double line.
We find the deficient secant line varieties for all the Veroneseans and all the deficient higher secant varieties for the quadratic Veroneseans. We conjecture that these are the only deficient secant varieties in this family and prove this up to secant projective 4-spaces.
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
- J. Alexander and A. Hirschowitz, Polynomial interpolation in several variables, J. Algebraic Geom. 4 (1995), no. 2, 201–222. MR 1311347
- 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
- A. Capani, G. Niesi, L. Robbiano, CoCoA, a system for doing computations in Commutative Algebra (Available via anonymous ftp from: cocoa.dima.unige.it).
- Anthony V. Geramita, Inverse systems of fat points: Waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals, 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. 2–114. MR 1381732
- Alessandro Gimigliano, Our thin knowledge of fat points, The Curves Seminar at Queen’s, Vol. VI (Kingston, ON, 1989) Queen’s Papers in Pure and Appl. Math., vol. 83, Queen’s Univ., Kingston, ON, 1989, pp. Exp. No. B, 50. MR 1036032
- 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
- André Hirschowitz, La méthode d’Horace pour l’interpolation à plusieurs variables, Manuscripta Math. 50 (1985), 337–388 (French, with English summary). MR 784148, DOI 10.1007/BF01168836
- A. Iarrobino, Inverse system of a symbolic power. III. Thin algebras and fat points, Compositio Math. 108 (1997), no. 3, 319–356. MR 1473851, DOI 10.1023/A:1000155612073
- Anthony Iarrobino and Vassil Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, vol. 1721, Springer-Verlag, Berlin, 1999. Appendix C by Iarrobino and Steven L. Kleiman. MR 1735271, DOI 10.1007/BFb0093426
- Vassil Kanev, Chordal varieties of Veronese varieties and catalecticant matrices, J. Math. Sci. (New York) 94 (1999), no. 1, 1114–1125. Algebraic geometry, 9. MR 1703911, DOI 10.1007/BF02367252
- F. Palatini, Sulle varietà algebriche per le quali sono di dimensione minore dell’ordinario, senza riempire lo spazio ambiente, una o alcuna delle varietà formate da spazi seganti. Atti Accad. Torino Cl. Scienze Mat. Fis. Nat. 44 (1909), 362-375.
- K. Ranestad, F.O. Schreyer. Varieties of sums of powers, preprint.
- A. Terracini. 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.
- 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
Bibliographic Information
- M. V. Catalisano
- Affiliation: D.I.M.E.T., Università di Genova, P.le Kennedy, 16129 Genova, Italy
- Email: catalisa@dima.unige.it
- A. V. Geramita
- Affiliation: Department of Mathematics, Queen’s University, Kingston, Ontario, Canada K7L 3N6 and Dipartimento di Matematica, Università di Genova, 16146 Genova, Italy
- MR Author ID: 72575
- Email: tony@mast.queensu.ca, geramita@dima.unige.it
- A. Gimigliano
- Affiliation: Dipartimento di Matematica, Università di Bologna, 40126 Bologna, Italy
- Email: gimiglia@dm.unibo.it
- Received by editor(s): February 25, 2000
- Received by editor(s) in revised form: October 26, 2000
- Published electronically: October 12, 2001
- Additional Notes: The first author was supported in part by MURST funds.
The second author was supported in part by MURST funds, and by the Natural Sciences and Engineering Research Council of Canada.
The third author was supported in part by the University of Bologna, funds for selected research topics, and by the P.R.R.N.I. “Geometria Algebrica e Algebra Commutativa". - Communicated by: Michael Stillman
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 975-985
- MSC (2000): Primary 14N15; Secondary 14M12
- DOI: https://doi.org/10.1090/S0002-9939-01-06251-7
- MathSciNet review: 1873770