Projective varieties with nonbirational linear projections and applications
HTML articles powered by AMS MathViewer
- by Atsushi Noma PDF
- 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
Additional Information
- Atsushi Noma
- Affiliation: Faculty of Engineering Sciences, Department of Mathematics, Yokohama National University, Yokohama 240-8501 Japan
- MR Author ID: 315999
- Email: noma@\@ynu.ac.jp
- Received by editor(s): July 1, 2014
- Received by editor(s) in revised form: April 28, 2016
- Published electronically: December 18, 2017
- Additional Notes: This paper was partially supported by Grant-in-Aid for Scientific Research (C), 20540039, 23540043, and 26400041 Japan Society for the Promotion of Science.
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 2299-2320
- MSC (2010): Primary 14N05, 14N15
- DOI: https://doi.org/10.1090/tran/7086
- MathSciNet review: 3748569