Unobstructedness of filling secants and the Gruson-Peskine general projection theorem. (English) Zbl 1315.14068

The generic projection conjecture, which predicted that the projection of a smooth embedded projective variety from a generic point has only the expected singularities, has been recently proved by L. Gruson and C. Peskine (see [Duke Math. J. 162, No. 3, 553–578 (2013; Zbl 1262.14058)]). In the paper under review this result is generalized, showing smoothness and results of expected dimension for multiple point loci of generic projections, mainly from a point or a line, or for fibers of embedding dimension 2 or less. As stated in the introduction, three results extending Gruson and Peskine’s result are proved:
Theorem 4.1, where arbitrary ambient spaces are allowed and the result is presented as a general statement about deformation of rational curves on varieties constrained by contact conditions with a fixed subvariety;
Theorem 5.1, where arbitrary-dimension centers of projection are allowed, fibres with local embedding dimension 2;
Theorem 6.1, where arbitrary-centers and contact conditions are allowed, curvilinear fibers.
The approach is different from that of Gruson and Peskine: it encodes the secant or contact conditions in a sheaf \(M\) (on the secant plane or curve), which controls the corresponding deformations; the geometric hypotheses on these secant or contact conditions imply generic spannedness for \(M\); since the base is often linear, the generic spannedness implies spannedness and a good control on the deformations.


14N05 Projective techniques in algebraic geometry


Zbl 1262.14058


Full Text: DOI arXiv Euclid


[1] A. Alzati, A new Castelnuovo bound for codimension three subvarieties , Arch. Math. (Basel) 98 (2012), 219-227. · Zbl 1238.14034
[2] A. Alzati, personal communication, February 2013. · Zbl 1238.14034
[3] A. Alzati and G. Ottaviani, The theorem of Mather on generic projections in the setting of algebraic geometry , Manuscripta Math. 74 (1992), 391-412. · Zbl 0794.14018
[4] R. Beheshti and D. Eisenbud, Fibers of generic projections , Compos. Math. 146 (2010), 435-456. · Zbl 1189.14061
[5] J. Fogarty, Algebraic families on an algebraic surface , Amer. J. Math. 90 (1968), 511-521. · Zbl 0176.18401
[6] L. Gruson and C. Peskine, On the smooth locus of aligned Hilbert schemes, the \(k\)-secant lemma and the general projection theorem , Duke Math. J. 162 (2013), 553-578. · Zbl 1262.14058
[7] M. Lehn, “Lectures on Hilbert schemes” in Algebraic Structures and Moduli Spaces , CRM Proc. Lecture Notes 38 , Amer. Math. Soc., Providence, 2004, 1-30. · Zbl 1076.14010
[8] S. Lichtenbaum and M. Schlessinger, The cotangent complex of a morphism , Trans. Amer. Math. Soc. 128 , no. 1 (1967), 41-70. · Zbl 0156.27201
[9] J. N. Mather, Generic projections , Ann. of Math. (2) 98 (1973), 226-245. · Zbl 0242.58001
[10] Z. Ran, The (dimension \(+2\))-secant lemma , Invent. Math. 106 (1991), 65-71. · Zbl 0767.14024
[11] Z. Ran, Lie atoms and their deformations , Geom. Funct. Anal. 18 (2008), 184-221. · Zbl 1142.14007
[12] Z. Ran, Jacobi-Bernoulli cohomology and deformations of schemes and maps , Cent. Eur. J. Math. 10 (2012), 1541-1591. · Zbl 1279.14013
[13] Z. Ran, Structure of the cycle map for Hilbert schemes of families of nodal curves , preprint, [math.AG]. arXiv:0903.3693v2
[14] E. Sernesi, Deformations of Algebraic Schemes , Grundlehren Math. Wiss. 334 , Springer, Berlin, 2006. · Zbl 1102.14001
[15] F. L. Zak, Tangents and Secants of Algebraic Varieties , Transl. Math. Monogr. 127 , Amer. Math. Soc., Providence, 1993. · Zbl 0795.14018
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