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Fibers of generic projections

Published online by Cambridge University Press:  02 February 2010

Roya Beheshti
Affiliation:
Department of Mathematics, Washington University, St Louis, MO 63130, USA (email: [email protected])
David Eisenbud
Affiliation:
Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA (email: [email protected])
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Abstract

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Let X be a smooth projective variety of dimension n in Pr, and let π:XPn+c be a general linear projection, with c>0. In this paper we bound the scheme-theoretic complexity of the fibers of π. In his famous work on stable mappings, Mather extended the classical results by showing that the number of distinct points in the fiber is bounded by B:=n/c+1, and that, when n is not too large, the degree of the fiber (taking the scheme structure into account) is also bounded by B. A result of Lazarsfeld shows that this fails dramatically for n≫0. We describe a new invariant of the scheme-theoretic fiber that agrees with the degree in many cases and is always bounded by B. We deduce, for example, that if we write a fiber as the disjoint union of schemes Y and Y′′ such that Y is the union of the locally complete intersection components of Y, then deg Y+deg Y′′redB. Our method also gives a sharp bound on the subvariety of Pr swept out by the l-secant lines of X for any positive integer l, and we discuss a corresponding bound for highly secant linear spaces of higher dimension. These results extend Ran’s ‘dimension +2 secant lemma’.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Bertram, A., Ein, L. and Lazarsfeld, R., Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. Amer. Math. Soc. 4 (1991), 587602.CrossRefGoogle Scholar
[2]Boardman, J. M., Singularities of differentiable maps, Publ. Math. Inst. Hautes Études Sci. 33 (1967), 2157.CrossRefGoogle Scholar
[3]Buchsbaum, D. A. and Eisenbud, D., Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension three, Amer. J. Math. 99 (1977), 447485.Google Scholar
[4]Buchweitz, R., These d’Etat.Google Scholar
[5]Cartwright, D. A., Erman, D., Velasco, M. and Viray, B., Hilbert schemes of 8 points, Algebra Number Theory, to appear. Preprint (2008), arXiv:0803.0341.Google Scholar
[6]Conte, A. and Verra, A., Reye constructions for nodal enriques surfaces, Trans. Amer. Math. Soc. 336 (1993), 79100.Google Scholar
[7]Cossec, F., Reye congruences, Trans. Amer. Math. Soc. 280 (1983), 731751.Google Scholar
[8]Decker, W., Ein, L. and Schreyer, F.-O., Construction of surfaces in P 4, J. Algebraic Geom. 2 (1993), 185237.Google Scholar
[9]Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150 (Springer, New York, 1995).Google Scholar
[10]Eisenbud, D., Commutative algebra with a view toward algebraic geometry (Springer, New York, 1995).Google Scholar
[11]Eisenbud, D. and Goto, S., Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89133.CrossRefGoogle Scholar
[12]Eisenbud, D. and Harris, J., Powers of ideals and fibers of morphisms, Math. Res. Lett., to appear.Google Scholar
[13]Emsalem, J. and Iarrobino, A., Some zero-dimensional generic singularities: finite algebras having small tangent space, Compositio Math. 36 (1978), 145188.Google Scholar
[14]Gaeta, F., Quelques progrès récents dans la classification des variétés algébriques d’un espace projectif. Deuxieme Colloque de Géométrie Algébrique Liège. C.B.R.M. 145181 (1952).Google Scholar
[15]Grayson, D. and Stillman, M., Macaulay 2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.Google Scholar
[16]Gruson, L., Lazarsfeld, R. and Peskine, C., On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), 491506.CrossRefGoogle Scholar
[17]Hartshorne, R., Bull. Amer. Math. Soc. 80 (1974), 10171032.Google Scholar
[18]Hartshorne, R., Algebraic geometry (Springer, New York, 1977).Google Scholar
[19]Iarrobino, A., Reducibility of the family of zero-dimensional schemes on a variety, Invent. Math. 15 (1972), 7277.CrossRefGoogle Scholar
[20]Kwak, S., Generic projections, the equations defining projective varieties and Castelnuovo regularity, Math. Z. 234 (2000), 413434.CrossRefGoogle Scholar
[21]Lazarsfeld, R., A sharp Castelnuovo bound for smooth surfaces, Duke Math. J. 55 (1987), 423429.Google Scholar
[22]Lazarsfeld, R., Positivity in algebraic geometry (Springer, Berlin, 2004).Google Scholar
[23]Mather, J. N., Stability of C mappings. VI: the nice dimensions, in Proceedings of liverpool singularities-symposium, I (1969/70), Lecture Notes in Mathematics, vol. 192 (Springer, Berlin, 1971), 207253.Google Scholar
[24]Mather, J. N., Generic projections, Ann. of Math. (2) 98 (1973), 226245.CrossRefGoogle Scholar
[25]Peskine, C. and Szpiro, L., Liaison des variétés algébriques. I, Invent. Math. 26 (1974), 271302.CrossRefGoogle Scholar
[26]Popescu, S., Examples of smooth non-general type surfaces in P 4, Proc. London Math. Soc. (3) 76 (1998), 257275.Google Scholar
[27]Ran, Z., The (dimension +2)-secant lemma, Invent. Math. 106 (1991), 6571.CrossRefGoogle Scholar
[28]Roberts, J., Generic projections of algebraic varieties, Amer. J. Math. 93 (1971), 191214.Google Scholar
[29]Ulrich, B., Liaison and deformations, J. Pure Appl. Algebra 39 (1986), 165175.CrossRefGoogle Scholar
[30]Vasconcelos, W., Ideals generated by R-sequences, J. Algebra 6 (1967), 309316.CrossRefGoogle Scholar
[31]Watanabe, J., A note on Gorenstein rings of embedding codimension three, Nagoya Math. J. 50 (1973), 227232.CrossRefGoogle Scholar
[32]Zak, F., Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127 (American Mathematical Society, Providence, RI, 1993).Google Scholar