Abstract
A pseudoeffective divisor is said to have a weak Zariski decomposition if it can be, up to a birational transformation, numerically written as the sum of a nef and an effective divisor. In this paper we consider the problem of the existence of a weak Zariski decomposition for each pseudoeffective divisor on a variety \(X= \mathbb {P}(\fancyscript{E})\), where \(\fancyscript{E}\) is a vector bundle on a smooth complex projective variety \(Z\) of Picard number one. We prove the existence of such a decomposition in a number of meaningful situations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Banica, C., Manolache, N.: Rank 2 stable vector bundles on \({\mathbb{P}}^3({\mathbb{C}})\) with chern classes \(c_1=-1,\;c_2=4\). Math. Z. 190(3), 315–339 (1985). doi:10.1007/BF01215133
Birkar, C.: On existence of log minimal models and weak Zariski decompositions. Math. Ann. 354(2), 787–799 (2012). doi:10.1007/s00208-011-0756-y
Fujita, T.: On Zariski problem. Proc. Japan Acad. Ser. A Math. Sci. 55(3), 106–110 (1979). URL http://projecteuclid.org/getRecord?id=euclid.pja/1195517399
Fulger, M.: The cones of effective cycles on projective bundles over curves. Math. Z. 269(1–2), 449–459 (2011). doi:10.1007/s00209-010-0744-z
Fulton, W.: Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 2, 2 edn. Springer (1998)
González, J.L.: Projectivized rank two toric vector bundles are Mori dream spaces. Comm. Algebra 40(4), 1456–1465 (2012). doi:10.1080/00927872.2011.551900
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, No. 52. Springer, New York (1977)
Hartshorne, R.: Stable vector bundles of rank 2 on \(\mathbb{P}^3\). Math. Ann. 238, 229–280 (1978). doi:10.1007/BF01420250
Hartshorne, R.: Stable reflexive sheaves. Math. Ann. 254(2), 121–176 (1980). doi:10.1007/BF01467074
Hu, Y., Keel, S.: Mori dream spaces and GIT. Mich. Math. J. 48, 331–348 (2000). doi:10.1307/mmj/1030132722. Dedicated to William Fulton on the occasion of his 60th birthday
Kollár, J.: Rational curves on algebraic varieties, 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. 32. Springer, Berlin (1996). doi:10.1007/978-3-662-03276-3
Lazarsfeld, R.: Positivity in algebraic geometry, 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. 48, 49. Springer, Berlin (2004)
Lesieutre, J.: The diminished base locus is not always closed. Compos. Math. 150(10), 1729–1741 (2014). doi:10.1112/S0010437X14007544
Mehta, V.B., Ramanathan, A.: Semistable sheaves on projective varieties and their restriction to curves. Math. Ann. 258(3), 213–224 (1981/1982). doi:10.1007/BF01450677
Miyaoka, Y.: The Chern classes and Kodaira dimension of a minimal variety. In: Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, pp. 449–476. North-Holland, Amsterdam (1987)
Muñoz, R., Occhetta, G., Solá Conde, L.E.: On rank 2 vector bundles on Fano manifolds. Kyoto J. Math. 54(1), 167–197 (2014). doi:10.1215/21562261-2400310
Nakayama, N.: Zariski-decomposition and Abundance, MSJ Memoirs, vol. 14. Mathematical Society of Japan, Tokyo (2004)
Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3. Birkhäuser Boston, Mass (1980)
Prokhorov, Y.G.: On the Zariski decomposition problem. Tr. Mat. Inst. Steklova 240(Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry), 43–72 (2003)
Schwarzenberger, R.L.E.: Vector bundles on the projective plane. Proc. London Math. Soc. 3(11), 623–640 (1961)
Vallès, J.: Fibrés de Schwarzenberger et coniques de droites sauteuses. Bull. Soc. Math. France 128(3), 433–449 (2000)
Zariski, O.: The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface. Ann. Math. 2(76), 560–615 (1962)
Author information
Authors and Affiliations
Corresponding author
Additional information
First and third authors supported by the Spanish project MTM2012-32670. Third author supported by Polish National Science Center project 2013/08/A/ST1/00804.
Rights and permissions
About this article
Cite this article
Muñoz, R., Di Sciullo, F. & Solá Conde, L.E. On the existence of a weak Zariski decomposition on projectivized vector bundles. Geom Dedicata 179, 287–301 (2015). https://doi.org/10.1007/s10711-015-0082-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-015-0082-8