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On the secant varieties to the tangential varieties of a Veronesean. (English) Zbl 0990.14021

From the introduction: The outstanding work by F. Zak: “Tangents and secants of algebraic varieties”, Trans. Math. Monogr. 127 (1993; Zbl 0795.14018)] has rekindled interest in the classical study of higher secant varieties to projective varieties. In this paper we study the secant varieties of tangent varieties to Veronesean varieties.
The Veronesean varieties \(V_{n,j}\), \(n\geq 1\), \(j\geq 2\), are the embeddings of \(\mathbb{P}^n\) into \(\mathbb{P}^N\), \(N={j+n\choose n}-1\), via the complete linear system \(S_j\), where \(S=k[x_0, \dots,x_n]\). Some of them (the first is actually the Veronese surface) have secant varieties which “do not have the expected dimension”. E.g. the closure of the union of the secant lines to the Veronese surface \(V_{2,2}\subset\mathbb{P}^5\) should, by a dimension count, fill up \(\mathbb{P}^5\), while the dimension of the secant variety is actually 4. For a variety \(V \subseteq \mathbb{P}^N\), let \({\mathcal S}^{s-1}(V)\) be the variety which is the closure in \(\mathbb{P}^N\) of \(\bigcup_{P_1, \dots,P_s \in V}\langle P_1, \dots,P_s \rangle\). If \(\dim V=d\), the “expected dimension” of \({\mathcal S}^{s-1} (V)\) is \(sd+s-1\) whenever this value is \(\leq N\), otherwise we expect \({\mathcal S}^{s-1} (V)= \mathbb{P}^N\). The “exceptional behaviour” of the dimension of secant varieties can be found among other varieties \({\mathcal S}^{s-1} (V_{n,j})\). The problem of determining which of them has a “defective” dimension is related (via the theory of inverse systems or, equivalently, apolarity) to determining the Hilbert functions of a 0-dimensional scheme \(X\subset \mathbb{P}^n\) made by the first infinitesimal neighbourhoods of \(s\) generic points [this is equivalent to what is classically known as Terracini’s lemma; cf. A. Terracini, Rend. Palermo 31, 392-396 (1911; JFM 42.0673.02)]. In this paper we look for other varieties that have this kind of peculiarity with respect to the dimension of their secant variables. We investigate the tangential varieties \(T_{n,j}\) of the Veronesean varieties \(V_{n,j}\) and relate the dimension of their \(s\)-secant varieties with the Hilbert function \(H(Z,j)\) of certain 0-dimensional scheme \(Z\subset\mathbb{P}^n\), supported at \(s\) generic points and whose structure is given by the intersection of their second infinitesimal neighbourhood with a double line (in a sense this is a generalization of Terracini’s lemma). More precisely, we will prove that \[ \dim{\mathcal S}^{s-1}(T_{n,j}) =\dim_k(L_1^{j-1}, \dots,L_s^{j-1}, L_1^{j-2}M_1, \dots,L_s^{j-2} M_s)_j-1= H(Z,j)-1 \] where \(L_1,\dots, L_s,M_1, \dots,M_s\) are \(2s\) generic linear forms in \(k[x_0,\dots, x_n]\).

MSC:

14N15 Classical problems, Schubert calculus
14M12 Determinantal varieties
14B10 Infinitesimal methods in algebraic geometry

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[1] Bjørn Ådlandsvik, Joins and higher secant varieties, Math. Scand. 61 (1987), no. 2, 213 – 222. · Zbl 0657.14034 · doi:10.7146/math.scand.a-12200
[2] J. Alexander and A. Hirschowitz, Polynomial interpolation in several variables, J. Algebraic Geom. 4 (1995), no. 2, 201 – 222. · Zbl 0829.14002
[3] Michael L. Catalano-Johnson, The possible dimensions of the higher secant varieties, Amer. J. Math. 118 (1996), no. 2, 355 – 361. · Zbl 0871.14043
[4] A. Capani, G. Niesi, L. Robbiano, CoCoA, a system for doing computations in Commutative Algebra (Available via anonymous ftp from: cocoa.dima.unige.it). · Zbl 0920.68060
[5] 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. · Zbl 0864.14031
[6] 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. · Zbl 0743.14005
[7] Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. · Zbl 0932.14001
[8] André Hirschowitz, La méthode d’Horace pour l’interpolation à plusieurs variables, Manuscripta Math. 50 (1985), 337 – 388 (French, with English summary). · Zbl 0571.14002 · doi:10.1007/BF01168836
[9] A. Iarrobino, Inverse system of a symbolic power. III. Thin algebras and fat points, Compositio Math. 108 (1997), no. 3, 319 – 356. · Zbl 0899.13016 · doi:10.1023/A:1000155612073
[10] 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. · Zbl 0942.14026
[11] Vassil Kanev, Chordal varieties of Veronese varieties and catalecticant matrices, J. Math. Sci. (New York) 94 (1999), no. 1, 1114 – 1125. Algebraic geometry, 9. · Zbl 0936.14035 · doi:10.1007/BF02367252
[12] 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. · JFM 40.0713.01
[13] K. Ranestad, F.O. Schreyer. Varieties of sums of powers, preprint. · Zbl 1078.14506
[14] 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. · JFM 42.0673.02
[15] 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. · Zbl 0795.14018
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