Projective varieties with nonbirational linear projections and applications

Noma, Atsushi

doi:10.1090/tran/7086

Projective varieties with nonbirational linear projections and applications
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Trans. Amer. Math. Soc. 370 (2018), 2299-2320 Request permission

Abstract:

We work over an algebraically closed field of characteristic zero. The purpose of this paper is to characterize a nondegenerate projective variety $X$ with a linear projection which induces a nonbirational map to its image. As an application, for smooth $X$ of degree $d$ and codimension $e$, we prove the “semiampleness” of the $(d-e+1)$th twist of the ideal sheaf. This improves a linear bound of the regularity of smooth projective varieties by Bayer–Mumford–Bertram–Ein–Lazarsfeld, and gives an asymptotic regularity bound.

References

E. Ballico, Special inner projections of projective varieties, Ann. Univ. Ferrara Sez. VII (N.S.) 50 (2004), 23–26 (English, with English and Italian summaries). MR 2159804
Dave Bayer and David Mumford, What can be computed in algebraic geometry?, Computational algebraic geometry and commutative algebra (Cortona, 1991) Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993, pp. 1–48. MR 1253986
Aaron Bertram, Lawrence Ein, and Robert Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. Amer. Math. Soc. 4 (1991), no. 3, 587–602. MR 1092845, DOI 10.1090/S0894-0347-1991-1092845-5
Alberto Calabri and Ciro Ciliberto, On special projections of varieties: epitome to a theorem of Beniamino Segre, Adv. Geom. 1 (2001), no. 1, 97–106. MR 1823955, DOI 10.1515/advg.2001.007
Marc Chardin and Bernd Ulrich, Liaison and Castelnuovo-Mumford regularity, Amer. J. Math. 124 (2002), no. 6, 1103–1124. MR 1939782, DOI 10.1353/ajm.2002.0035
Steven Dale Cutkosky, Lawrence Ein, and Robert Lazarsfeld, Positivity and complexity of ideal sheaves, Math. Ann. 321 (2001), no. 2, 213–234. MR 1866486, DOI 10.1007/s002080100220
S. Dale Cutkosky, Jürgen Herzog, and Ngô Viêt Trung, Asymptotic behaviour of the Castelnuovo-Mumford regularity, Compositio Math. 118 (1999), no. 3, 243–261. MR 1711319, DOI 10.1023/A:1001559912258
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
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
Takao Fujita, Semipositive line bundles, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), no. 2, 353–378. MR 722501
Katsuhisa Furukawa, Defining ideal of the Segre locus in arbitrary characteristic, J. Algebra 336 (2011), 84–98. MR 2802532, DOI 10.1016/j.jalgebra.2011.04.018
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 173675
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
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
Hajime Kaji, On the space curves with the same dual variety, J. Reine Angew. Math. 437 (1993), 1–11. MR 1212250, DOI 10.1515/crll.1993.437.1
Yujiro Kawamata, A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann. 261 (1982), no. 1, 43–46. MR 675204, DOI 10.1007/BF01456407
Steven L. Kleiman, The enumerative theory of singularities, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 297–396. MR 0568897
Vijay Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer. Math. Soc. 128 (2000), no. 2, 407–411. MR 1621961, DOI 10.1090/S0002-9939-99-05020-0
Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472, DOI 10.1007/978-3-642-18808-4
Atsushi Noma, Hypersurfaces cutting out a projective variety, Trans. Amer. Math. Soc. 362 (2010), no. 9, 4481–4495. MR 2645037, DOI 10.1090/S0002-9947-10-05054-3
B. Segre, On the locus of points from which an algebraic variety is projected multiply, Proceedings of the Phys.-Math. Soc. Japan Ser. III 18 (1936), 425–426.
Eckart Viehweg, Vanishing theorems, J. Reine Angew. Math. 335 (1982), 1–8. MR 667459, DOI 10.1515/crll.1982.335.1