On the secant varieties to the tangential varieties of a Veronesean

Authors:
M. V. Catalisano, A. V. Geramita and A. Gimigliano

Journal:
Proc. Amer. Math. Soc. **130** (2002), 975-985

MSC (2000):
Primary 14N15; Secondary 14M12

Published electronically:
October 12, 2001

MathSciNet review:
1873770

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[A]**Bjørn Ådlandsvik,*Joins and higher secant varieties*, Math. Scand.**61**(1987), no. 2, 213–222. MR**947474****[AH]**J. Alexander and A. Hirschowitz,*Polynomial interpolation in several variables*, J. Algebraic Geom.**4**(1995), no. 2, 201–222. MR**1311347****[CJ]**Michael L. Catalano-Johnson,*The possible dimensions of the higher secant varieties*, Amer. J. Math.**118**(1996), no. 2, 355–361. MR**1385282****[CoCoA]**A. Capani, G. Niesi, L. Robbiano,*CoCoA, a system for doing computations in Commutative Algebra*(Available via anonymous ftp from: cocoa.dima.unige.it).**[Ge]**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****[Gi]**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****[Hr]**Joe Harris,*Algebraic geometry*, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR**1182558****[H]**André Hirschowitz,*La méthode d’Horace pour l’interpolation à plusieurs variables*, Manuscripta Math.**50**(1985), 337–388 (French, with English summary). MR**784148**, 10.1007/BF01168836**[I]**A. Iarrobino,*Inverse system of a symbolic power. III. Thin algebras and fat points*, Compositio Math.**108**(1997), no. 3, 319–356. MR**1473851**, 10.1023/A:1000155612073**[IK]**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****[K]**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**, 10.1007/BF02367252**[Pa]**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.**[RS]**K. Ranestad, F.O. Schreyer.*Varieties of sums of powers*, preprint.**[Te]**A. Terracini.*Sulle**per cui la varietà degli**-seganti ha dimensione minore dell'ordinario.*Rend. Circ. Mat. Palermo**31**(1911), 392-396.**[Z]**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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
14N15,
14M12

Retrieve articles in all journals with MSC (2000): 14N15, 14M12

Additional 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

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

DOI:
http://dx.doi.org/10.1090/S0002-9939-01-06251-7

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

Article copyright:
© Copyright 2001
American Mathematical Society