On the secant varieties to the tangential varieties of a Veronesean

Catalisano, M.; Geramita, A.; Gimigliano, A.

doi:10.1090/S0002-9939-01-06251-7

On the secant varieties to the tangential varieties of a Veronesean
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by M. V. Catalisano, A. V. Geramita and A. Gimigliano PDF

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