Zariski decompositions of numerical cycle classes
Authors:
Mihai Fulger and Brian Lehmann
Journal:
J. Algebraic Geom. 26 (2017), 43-106
DOI:
https://doi.org/10.1090/jag/677
Published electronically:
August 3, 2016
MathSciNet review:
3570583
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We construct a Zariski decomposition for cycle classes of arbitrary codimension. This decomposition is an analogue of well-known constructions for divisors. Examples illustrate how Zariski decompositions of cycle classes reflect the geometry of the underlying space. We also analyze the birational behavior of Zariski decompositions, leading to a Fujita approximation-type result for curve classes.
References
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- Sébastien Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 1, 45–76 (English, with English and French summaries). MR 2050205, DOI https://doi.org/10.1016/j.ansens.2003.04.002
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- A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020
- Mihai Fulger and Brian Lehmann, Kernels of numerical pushforwards, 2014, arXiv:1407.6455 [math.AG].
- ---, Positive cones of dual cycle classes, 2014, arXiv:1408.5154 [math.AG].
- Takao Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 106–110. MR 531454
- Takao Fujita, Zariski decomposition and canonical rings of elliptic threefolds, J. Math. Soc. Japan 38 (1986), no. 1, 19–37. MR 816221, DOI https://doi.org/10.2969/jmsj/03810019
- Takao Fujita, Approximating Zariski decomposition of big line bundles, Kodai Math. J. 17 (1994), no. 1, 1–3. MR 1262949, DOI https://doi.org/10.2996/kmj/1138039894
- William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620
- Mihai Fulger, The cones of effective cycles on projective bundles over curves, Math. Z. 269 (2011), no. 1-2, 449–459. MR 2836078, DOI https://doi.org/10.1007/s00209-010-0744-z
- Noah Giansiracusa, Conformal blocks and rational normal curves, J. Algebraic Geom. 22 (2013), no. 4, 773–793. MR 3084722, DOI https://doi.org/10.1090/S1056-3911-2013-00601-3
- Tom Graber, Enumerative geometry of hyperelliptic plane curves, J. Algebraic Geom. 10 (2001), no. 4, 725–755. MR 1838977
- 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
- Brendan Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), no. 2, 316–352. MR 1957831, DOI https://doi.org/10.1016/S0001-8708%2802%2900058-0
- Yujiro Kawamata, The Zariski decomposition of log-canonical divisors, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 425–433. MR 927965
- Yujiro Kawamata, Crepant blowing-up of $3$-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2) 127 (1988), no. 1, 93–163. MR 924674, DOI https://doi.org/10.2307/1971417
- Sean Keel, Intersection theory of moduli space of stable $n$-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545–574. MR 1034665, DOI https://doi.org/10.1090/S0002-9947-1992-1034665-0
- A. G. Khovanskiĭ, The Newton polytope, the Hilbert polynomial and sums of finite sets, Funktsional. Anal. i Prilozhen. 26 (1992), no. 4, 57–63, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 26 (1992), no. 4, 276–281 (1993). MR 1209944, DOI https://doi.org/10.1007/BF01075048
- Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 360616
- Seán Keel and James McKernan, Contractible extremal rays on $\overline M_{0,n}$, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 115–130. MR 3184175
- Alex Küronya and Catriona Maclean, Zariski decomposition of b-divisors, Math. Z. 273 (2013), no. 1-2, 427–436. MR 3010168, DOI https://doi.org/10.1007/s00209-012-1012-1
- Robert Lazarsfeld, Positivity in algebraic geometry. I, 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, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471
- Brian Lehmann, Comparing numerical dimensions, Algebra Number Theory 7 (2013), no. 5, 1065–1100. MR 3101072, DOI https://doi.org/10.2140/ant.2013.7.1065
- Brian Lehmann, Geometric characterizations of big cycles, 2013, arXiv:1309.0880 [math.AG].
- Robert Lazarsfeld and Mircea Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5, 783–835 (English, with English and French summaries). MR 2571958, DOI https://doi.org/10.24033/asens.2109
- Atsushi Moriwaki, Semiampleness of the numerically effective part of Zariski decomposition, J. Math. Kyoto Univ. 26 (1986), no. 3, 465–481. MR 857230, DOI https://doi.org/10.1215/kjm/1250520879
- Mircea Mustaţă, The non-nef locus in positive characteristic, A celebration of algebraic geometry, Clay Math. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 2013, pp. 535–551. MR 3114955
- Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR 2104208
- Sam Payne, Stable base loci, movable curves, and small modifications, for toric varieties, Math. Z. 253 (2006), no. 2, 421–431. MR 2218709, DOI https://doi.org/10.1007/s00209-005-0923-5
- Yu. G. Prokhorov, On the Zariski decomposition problem, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry, 43–72 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 1(240) (2003), 37–65. MR 1993748
- V. V. Shokurov, Prelimiting flips, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry, 82–219; English transl., Proc. Steklov Inst. Math. 1(240) (2003), 75–213. MR 1993750
- Satoshi Takagi, Fujita’s approximation theorem in positive characteristics, J. Math. Kyoto Univ. 47 (2007), no. 1, 179–202. MR 2359108, DOI https://doi.org/10.1215/kjm/1250281075
- Hajime Tsuji, Analytic Zariski decomposition, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), no. 7, 161–163. MR 1193172
- Oscar Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560–615. MR 141668, DOI https://doi.org/10.2307/1970376
References
- Daniele Arcara, Aaron Bertram, Izzet Coskun, and Jack Huizenga, The minimal model program for the Hilbert scheme of points on $\mathbb {P}^2$ and Bridgeland stability, Adv. Math. 235 (2013), 580–626. MR 3010070, DOI https://doi.org/10.1016/j.aim.2012.11.018
- Thomas Bauer, Mirel Caibăr, and Gary Kennedy, Zariski decomposition: a new (old) chapter of linear algebra, Amer. Math. Monthly 119 (2012), no. 1, 25–41. MR 2877664 (2012m:15001), DOI https://doi.org/10.4169/amer.math.monthly.119.01.025
- Sébastien Boucksom, Jean-Pierre Demailly, Mihai Păun, and Thomas Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom. 22 (2013), no. 2, 201–248. MR 3019449, DOI https://doi.org/10.1090/S1056-3911-2012-00574-8
- Sébastien Boucksom, Charles Favre, and Mattias Jonsson, Differentiability of volumes of divisors and a problem of Teissier, J. Algebraic Geom. 18 (2009), no. 2, 279–308. MR 2475816 (2009m:14005), DOI https://doi.org/10.1090/S1056-3911-08-00490-6
- Th. Bauer, A. Küronya, and T. Szemberg, Zariski chambers, volumes, and stable base loci, J. Reine Angew. Math. 576 (2004), 209–233. MR 2099205 (2005h:14012), DOI https://doi.org/10.1515/crll.2004.090
- Sébastien Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 1, 45–76 (English, with English and French summaries). MR 2050205 (2005i:32018), DOI https://doi.org/10.1016/j.ansens.2003.04.002
- Steven D. Cutkosky, Zariski decomposition of divisors on algebraic varieties, Duke Math. J. 53 (1986), no. 1, 149–156. MR 835801 (87f:14004), DOI https://doi.org/10.1215/S0012-7094-86-05309-3
- Steven Dale Cutkosky, Teissier’s problem on inequalities of nef divisors, J. Algebra Appl. 14 (2015), no. 9, 1540002, 37. MR 3368254, DOI https://doi.org/10.1142/S0219498815400022
- Olivier Debarre, Lawrence Ein, Robert Lazarsfeld, and Claire Voisin, Pseudoeffective and nef classes on abelian varieties, Compos. Math. 147 (2011), no. 6, 1793–1818. MR 2862063, DOI https://doi.org/10.1112/S0010437X11005227
- A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020
- Mihai Fulger and Brian Lehmann, Kernels of numerical pushforwards, 2014, arXiv:1407.6455 [math.AG].
- ---, Positive cones of dual cycle classes, 2014, arXiv:1408.5154 [math.AG].
- Takao Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 106–110. MR 531454 (80j:14029)
- Takao Fujita, Zariski decomposition and canonical rings of elliptic threefolds, J. Math. Soc. Japan 38 (1986), no. 1, 19–37. MR 816221 (87e:14036), DOI https://doi.org/10.2969/jmsj/03810019
- Takao Fujita, Approximating Zariski decomposition of big line bundles, Kodai Math. J. 17 (1994), no. 1, 1–3. MR 1262949 (95c:14053), DOI https://doi.org/10.2996/kmj/1138039894
- William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620 (85k:14004)
- Mihai Fulger, The cones of effective cycles on projective bundles over curves, Math. Z. 269 (2011), no. 1-2, 449–459. MR 2836078 (2012h:14015), DOI https://doi.org/10.1007/s00209-010-0744-z
- Noah Giansiracusa, Conformal blocks and rational normal curves, J. Algebraic Geom. 22 (2013), no. 4, 773–793. MR 3084722, DOI https://doi.org/10.1090/S1056-3911-2013-00601-3
- Tom Graber, Enumerative geometry of hyperelliptic plane curves, J. Algebraic Geom. 10 (2001), no. 4, 725–755. MR 1838977 (2002d:14091)
- Robin Hartshorne, Ample subvarieties of algebraic varieties, notes written in collaboration with C. Musili, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. MR 0282977 (44 \#211)
- Brendan Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), no. 2, 316–352. MR 1957831 (2004b:14040), DOI https://doi.org/10.1016/S0001-8708%2802%2900058-0
- Yujiro Kawamata, The Zariski decomposition of log-canonical divisors, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 425–433. MR 927965 (89b:14052)
- Yujiro Kawamata, Crepant blowing-up of $3$-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2) 127 (1988), no. 1, 93–163. MR 924674 (89d:14023), DOI https://doi.org/10.2307/1971417
- Sean Keel, Intersection theory of moduli space of stable $n$-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545–574. MR 1034665 (92f:14003), DOI https://doi.org/10.2307/2153922
- A. G. Khovanskiĭ, The Newton polytope, the Hilbert polynomial and sums of finite sets, Funktsional. Anal. i Prilozhen. 26 (1992), no. 4, 57–63, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 26 (1992), no. 4, 276–281 (1993). MR 1209944 (94e:14068), DOI https://doi.org/10.1007/BF01075048
- Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 0360616 (50 \#13063)
- Seán Keel and James McKernan, Contractible extremal rays on $\overline M_{0,n}$, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 115–130. MR 3184175
- Alex Küronya and Catriona Maclean, Zariski decomposition of b-divisors, Math. Z. 273 (2013), no. 1-2, 427–436. MR 3010168, DOI https://doi.org/10.1007/s00209-012-1012-1
- Robert Lazarsfeld, Positivity in algebraic geometry. I, Classical setting: line bundles and linear series. 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, Springer-Verlag, Berlin, 2004. MR 2095471 (2005k:14001a)
- Brian Lehmann, Comparing numerical dimensions, Algebra Number Theory 7 (2013), no. 5, 1065–1100. MR 3101072, DOI https://doi.org/10.2140/ant.2013.7.1065
- Brian Lehmann, Geometric characterizations of big cycles, 2013, arXiv:1309.0880 [math.AG].
- Robert Lazarsfeld and Mircea Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5, 783–835 (English, with English and French summaries). MR 2571958 (2011e:14012)
- Atsushi Moriwaki, Semiampleness of the numerically effective part of Zariski decomposition, J. Math. Kyoto Univ. 26 (1986), no. 3, 465–481. MR 857230 (87j:14056)
- Mircea Mustaţă, The non-nef locus in positive characteristic, A celebration of algebraic geometry, Clay Math. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 2013, pp. 535–551. MR 3114955
- Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR 2104208 (2005h:14015)
- Sam Payne, Stable base loci, movable curves, and small modifications, for toric varieties, Math. Z. 253 (2006), no. 2, 421–431. MR 2218709 (2007m:14074), DOI https://doi.org/10.1007/s00209-005-0923-5
- Yu. G. Prokhorov, On the Zariski decomposition problem, Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 43–72 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 1 (240) (2003), 37–65. MR 1993748 (2004m:14021)
- V. V. Shokurov, Prelimiting flips, Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 82–219; English transl., Proc. Steklov Inst. Math. 1 (240) (2003), 75–213. MR 1993750 (2004k:14024)
- Satoshi Takagi, Fujita’s approximation theorem in positive characteristics, J. Math. Kyoto Univ. 47 (2007), no. 1, 179–202. MR 2359108 (2008i:14014)
- Hajime Tsuji, Analytic Zariski decomposition, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), no. 7, 161–163. MR 1193172 (93k:32014)
- Oscar Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560–615. MR 0141668 (25 \#5065)
Additional Information
Mihai Fulger
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544 – and – Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700, Bucharest, Romania
Email:
afulger@princeton.edu
Brian Lehmann
Affiliation:
Department of Mathematics, Rice University, Houston, Texas 77005
Address at time of publication:
Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
MR Author ID:
977848
Email:
lehmannb@bc.edu
Received by editor(s):
November 26, 2013
Received by editor(s) in revised form:
December 7, 2014, and January 5, 2015
Published electronically:
August 3, 2016
Additional Notes:
The second author was supported by NSF Award 1004363.
Article copyright:
© Copyright 2016
University Press, Inc.