Multisecant subspaces to smooth projective varieties in arbitrary characteristic

Noma, Atsushi

doi:10.1090/S0002-9939-09-09977-8

Multisecant subspaces to smooth projective varieties in arbitrary characteristic
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by Atsushi Noma

Proc. Amer. Math. Soc. 137 (2009), 3985-3990

DOI: https://doi.org/10.1090/S0002-9939-09-09977-8

Published electronically: July 1, 2009

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Abstract:

Let $X \subseteq \mathbb {P}^{N}$ be a projective variety of dimension $n\geq 1$, degree $d$, and codimension $e$, not contained in any hyperplane, defined over an algebraically closed field $\Bbbk$ of arbitrary characteristic. We show that if a $k$-dimensional linear subspace $M$ meets $X$ at the smooth locus such that $X\cap M$ is finite and locally lies on a smooth curve, then the length $l(X\cap M)$ does not exceed $d-e+k-\min \{g,e-k\}$ for the sectional genus $g$ of $X$.

References

Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461, DOI 10.1007/BFb0060932
Marie-Amélie Bertin, On the regularity of varieties having an extremal secant line, J. Reine Angew. Math. 545 (2002), 167–181. MR 1896101, DOI 10.1515/crll.2002.032
David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89–133. MR 741934, DOI 10.1016/0021-8693(84)90092-9
H. Flenner, L. O’Carroll, and W. Vogel, Joins and intersections, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. MR 1724388, DOI 10.1007/978-3-662-03817-8
T. Fujita, Defining equations for certain types of polarized varieties, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 165–173. MR 0437533
Takao Fujita, Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, Cambridge, 1990. MR 1162108, DOI 10.1017/CBO9780511662638
L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), no. 3, 491–506. MR 704401, DOI 10.1007/BF01398398
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
Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR 0282977, DOI 10.1007/BFb0067839
Sijong Kwak, Smooth projective varieties with extremal or next to extremal curvilinear secant subspaces, Trans. Amer. Math. Soc. 357 (2005), no. 9, 3553–3566. MR 2146638, DOI 10.1090/S0002-9947-04-03594-9
Atsushi Noma, A bound on the Castelnuovo-Mumford regularity for curves, Math. Ann. 322 (2002), no. 1, 69–74. MR 1883389, DOI 10.1007/s002080100265
Atsushi Noma, Castelnuovo-Mumford regularity for nonhyperelliptic curves, Arch. Math. (Basel) 83 (2004), no. 1, 23–26. MR 2079822, DOI 10.1007/s00013-003-4888-5
Atsushi Noma, Multisecant lines to projective varieties, Projective varieties with unexpected properties, Walter de Gruyter, Berlin, 2005, pp. 349–359. MR 2202263
A. Noma, Multisecant lines to smooth Del Pezzo varieties, preprint, 2007.
Oscar Zariski, Introduction to the problem of minimal models in the theory of algebraic surfaces, Publications of the Mathematical Society of Japan, vol. 4, Mathematical Society of Japan, Tokyo, 1958. MR 0097403